diff --git a/articles/NEWS.html b/articles/NEWS.html
index 9ed2201f9..dcb930e1a 100644
--- a/articles/NEWS.html
+++ b/articles/NEWS.html
@@ -8,7 +8,7 @@
 
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
 
-<title>The Marginal Effects Zoo (0.15.0) - 31&nbsp; News</title>
+<title>The Marginal Effects Zoo (0.15.1) - 31&nbsp; News</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -80,7 +80,7 @@
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 31&nbsp; News">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 31&nbsp; News">
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@@ -94,7 +94,7 @@
       <div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -462,31 +462,31 @@
     <h2 id="toc-title">Table of contents</h2>
    
   <ul>
-  <li><a href="#dev" id="toc-dev" class="nav-link active" data-scroll-target="#dev">dev</a></li>
-  <li><a href="#section" id="toc-section" class="nav-link" data-scroll-target="#section">0.15.0</a></li>
-  <li><a href="#section-1" id="toc-section-1" class="nav-link" data-scroll-target="#section-1">0.14.0</a></li>
-  <li><a href="#section-2" id="toc-section-2" class="nav-link" data-scroll-target="#section-2">0.13.0</a></li>
-  <li><a href="#section-3" id="toc-section-3" class="nav-link" data-scroll-target="#section-3">0.12.0</a></li>
-  <li><a href="#section-4" id="toc-section-4" class="nav-link" data-scroll-target="#section-4">0.11.2</a></li>
-  <li><a href="#section-5" id="toc-section-5" class="nav-link" data-scroll-target="#section-5">0.11.1</a></li>
-  <li><a href="#section-6" id="toc-section-6" class="nav-link" data-scroll-target="#section-6">0.11.0</a></li>
-  <li><a href="#section-7" id="toc-section-7" class="nav-link" data-scroll-target="#section-7">0.10.0</a></li>
-  <li><a href="#section-8" id="toc-section-8" class="nav-link" data-scroll-target="#section-8">0.9.0</a></li>
-  <li><a href="#section-9" id="toc-section-9" class="nav-link" data-scroll-target="#section-9">0.8.1</a></li>
-  <li><a href="#section-10" id="toc-section-10" class="nav-link" data-scroll-target="#section-10">0.8.0</a></li>
-  <li><a href="#section-11" id="toc-section-11" class="nav-link" data-scroll-target="#section-11">0.7.1</a></li>
-  <li><a href="#section-12" id="toc-section-12" class="nav-link" data-scroll-target="#section-12">0.7.0</a></li>
-  <li><a href="#section-13" id="toc-section-13" class="nav-link" data-scroll-target="#section-13">0.6.0</a></li>
-  <li><a href="#section-14" id="toc-section-14" class="nav-link" data-scroll-target="#section-14">0.5.0</a></li>
-  <li><a href="#section-15" id="toc-section-15" class="nav-link" data-scroll-target="#section-15">0.4.1</a></li>
-  <li><a href="#section-16" id="toc-section-16" class="nav-link" data-scroll-target="#section-16">0.4.0</a></li>
-  <li><a href="#section-17" id="toc-section-17" class="nav-link" data-scroll-target="#section-17">0.3.4</a></li>
-  <li><a href="#section-18" id="toc-section-18" class="nav-link" data-scroll-target="#section-18">0.3.3</a></li>
-  <li><a href="#section-19" id="toc-section-19" class="nav-link" data-scroll-target="#section-19">0.3.2</a></li>
-  <li><a href="#section-20" id="toc-section-20" class="nav-link" data-scroll-target="#section-20">0.3.1</a></li>
-  <li><a href="#section-21" id="toc-section-21" class="nav-link" data-scroll-target="#section-21">0.3.0</a></li>
-  <li><a href="#section-22" id="toc-section-22" class="nav-link" data-scroll-target="#section-22">0.2.0</a></li>
-  <li><a href="#section-23" id="toc-section-23" class="nav-link" data-scroll-target="#section-23">0.1.0</a></li>
+  <li><a href="#section" id="toc-section" class="nav-link active" data-scroll-target="#section">0.15.1</a></li>
+  <li><a href="#section-1" id="toc-section-1" class="nav-link" data-scroll-target="#section-1">0.15.0</a></li>
+  <li><a href="#section-2" id="toc-section-2" class="nav-link" data-scroll-target="#section-2">0.14.0</a></li>
+  <li><a href="#section-3" id="toc-section-3" class="nav-link" data-scroll-target="#section-3">0.13.0</a></li>
+  <li><a href="#section-4" id="toc-section-4" class="nav-link" data-scroll-target="#section-4">0.12.0</a></li>
+  <li><a href="#section-5" id="toc-section-5" class="nav-link" data-scroll-target="#section-5">0.11.2</a></li>
+  <li><a href="#section-6" id="toc-section-6" class="nav-link" data-scroll-target="#section-6">0.11.1</a></li>
+  <li><a href="#section-7" id="toc-section-7" class="nav-link" data-scroll-target="#section-7">0.11.0</a></li>
+  <li><a href="#section-8" id="toc-section-8" class="nav-link" data-scroll-target="#section-8">0.10.0</a></li>
+  <li><a href="#section-9" id="toc-section-9" class="nav-link" data-scroll-target="#section-9">0.9.0</a></li>
+  <li><a href="#section-10" id="toc-section-10" class="nav-link" data-scroll-target="#section-10">0.8.1</a></li>
+  <li><a href="#section-11" id="toc-section-11" class="nav-link" data-scroll-target="#section-11">0.8.0</a></li>
+  <li><a href="#section-12" id="toc-section-12" class="nav-link" data-scroll-target="#section-12">0.7.1</a></li>
+  <li><a href="#section-13" id="toc-section-13" class="nav-link" data-scroll-target="#section-13">0.7.0</a></li>
+  <li><a href="#section-14" id="toc-section-14" class="nav-link" data-scroll-target="#section-14">0.6.0</a></li>
+  <li><a href="#section-15" id="toc-section-15" class="nav-link" data-scroll-target="#section-15">0.5.0</a></li>
+  <li><a href="#section-16" id="toc-section-16" class="nav-link" data-scroll-target="#section-16">0.4.1</a></li>
+  <li><a href="#section-17" id="toc-section-17" class="nav-link" data-scroll-target="#section-17">0.4.0</a></li>
+  <li><a href="#section-18" id="toc-section-18" class="nav-link" data-scroll-target="#section-18">0.3.4</a></li>
+  <li><a href="#section-19" id="toc-section-19" class="nav-link" data-scroll-target="#section-19">0.3.3</a></li>
+  <li><a href="#section-20" id="toc-section-20" class="nav-link" data-scroll-target="#section-20">0.3.2</a></li>
+  <li><a href="#section-21" id="toc-section-21" class="nav-link" data-scroll-target="#section-21">0.3.1</a></li>
+  <li><a href="#section-22" id="toc-section-22" class="nav-link" data-scroll-target="#section-22">0.3.0</a></li>
+  <li><a href="#section-23" id="toc-section-23" class="nav-link" data-scroll-target="#section-23">0.2.0</a></li>
+  <li><a href="#section-24" id="toc-section-24" class="nav-link" data-scroll-target="#section-24">0.1.0</a></li>
   </ul>
 </nav>
     </div>
@@ -510,8 +510,8 @@ <h1 class="title"><span class="chapter-number">31</span>&nbsp; <span class="chap
 
 </header>
 
-<section id="dev" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="dev">dev</h2>
+<section id="section" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section">0.15.1</h2>
 <ul>
 <li><code>hypotheses()</code>: The <code>FUN</code> argument handles <code>group</code> columns gracefully.</li>
 <li>Native support for <code>Amelia</code> for multiple imputation.</li>
@@ -522,11 +522,11 @@ <h2 class="unnumbered anchored" data-anchor-id="dev">dev</h2>
 </ul>
 <p>Bug fix:</p>
 <ul>
-<li>Results of the <code>predictions()</code> function can be undesirable in version 0.15.0 when (a) <code>type=NULL</code>, (b) model is <code>glm()</code>, and (c) the <code>hypothesis</code> argument is non-numeric. This is because the default <code>type</code> was then set to <code>invlink(link)</code>, but the backtransformation using link function was applied to the quantity <em>after</em> applying the (non-linear) <code>hypothesis</code> test (ex: pairwise comparison or contrast). The <code>hypothesis</code> funtion should be applied to predictions on the response scale instead. A warning is now issued and the value of <code>type</code> is automatically switched to “response” when appropriate. Thanks to <a href="https://github.com/strengejacke">@strengejacke</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/903">[#903](https://github.com/vincentarelbundock/marginaleffects/issues/903)</a></li>
+<li>Results of the <code>predictions()</code> function could be inaccurate when (a) running version 0.15.0, (b) <code>type</code> is <code>NULL</code> or <code>invlink(link)</code>, (c) model is <code>glm()</code>, and (d) the <code>hypothesis</code> argument is non-numeric. Thanks to <a href="https://github.com/strengejacke">@strengejacke</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/903">[#903](https://github.com/vincentarelbundock/marginaleffects/issues/903)</a></li>
 </ul>
 </section>
-<section id="section" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section">0.15.0</h2>
+<section id="section-1" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-1">0.15.0</h2>
 <p>New:</p>
 <ul>
 <li>Conformal prediction via <code>inferences()</code></li>
@@ -549,8 +549,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section">0.15.0</h2>
 <li><code>newdata="median"</code> returned mean of binary variables. Thanks to <a href="https://github.com/jkhanson1970">@jkhanson1970</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/896">#896</a>.</li>
 </ul>
 </section>
-<section id="section-1" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-1">0.14.0</h2>
+<section id="section-2" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-2">0.14.0</h2>
 <p>Breaking changes:</p>
 <ul>
 <li>Row order of the output changes for some objects. Rows are not sorted alphabetically by <code>term</code>, <code>by</code>, and variables explicitly supplied to <code>datagrid</code>. This can affect hypothesis tests computed using the b1, b2, b3, and other indices.</li>
@@ -580,8 +580,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-1">0.14.0</h2>
 <li>Major refactor to simplify the code base and make maintenance easier.</li>
 </ul>
 </section>
-<section id="section-2" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-2">0.13.0</h2>
+<section id="section-3" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-3">0.13.0</h2>
 <p>Breaking change:</p>
 <ul>
 <li><code>glmmTMB</code>: Standard errors are no longer supported because they may have been erroneous. Follow Issue <a href="https://github.com/vincentarelbundock/marginaleffects/issues/810">#810</a> on Github for developments: https://github.com/vincentarelbundock/marginaleffects/issues/810</li>
@@ -605,8 +605,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-2">0.13.0</h2>
 <li>Degrees of freedom mismatch for joint hypothesis tests. Thanks to <a href="https://github.com/snhansen">@snhansen</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/789">#789</a>.</li>
 </ul>
 </section>
-<section id="section-3" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-3">0.12.0</h2>
+<section id="section-4" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-4">0.12.0</h2>
 <p>Breaking change:</p>
 <ul>
 <li>Row order of output has changed for many calls, especially those using the <code>by</code> argument. This may break hypothesis tests conducted by indexing <code>b1</code>, <code>b2</code>, etc. This was necessary to fix Issue <a href="https://github.com/vincentarelbundock/marginaleffects/issues/776">#776</a>. Thanks to <a href="https://github.com/marcora">@marcora</a> for the report.</li>
@@ -622,8 +622,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-3">0.12.0</h2>
 <li>Wrong order of rows in bayesian models with <code>by</code> argument. Thanks to <a href="https://github.com/shirdekel">@shirdekel</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/782">#782</a>.</li>
 </ul>
 </section>
-<section id="section-4" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-4">0.11.2</h2>
+<section id="section-5" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-5">0.11.2</h2>
 <ul>
 <li><code>vcov()</code> and <code>coef()</code> methods for <code>marginaleffects</code> objects.</li>
 <li>Strings in <code>wts</code> are accepted with the <code>by</code> argument.</li>
@@ -643,8 +643,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-4">0.11.2</h2>
 <li>Bug when <code>datagrid()</code> when called inside a user-written function. Thanks to <a href="https://github.com/NickCH-K">@NickCH-K</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/769">#769</a> and to <a href="https://github.com/capnrefsmmat">@capnrefsmmat</a> for the diagnostics.</li>
 </ul>
 </section>
-<section id="section-5" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-5">0.11.1</h2>
+<section id="section-6" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-6">0.11.1</h2>
 <p>Breaking change:</p>
 <ul>
 <li>Row orders are now more consistent, but may have changed from previous version. This could affect results from <code>hypothesis</code> with <code>b1</code>, <code>b2</code>, … indexing.</li>
@@ -673,8 +673,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-5">0.11.1</h2>
 <li><code>glmmTMB</code> models with zero-inflated components are supported. Thanks to <a href="https://github.com/Helsinki-Ronan">@Helsinki-Ronan</a> and <a href="https://github.com/strengejacke">@strengejacke</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/734">#734</a>.</li>
 </ul>
 </section>
-<section id="section-6" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-6">0.11.0</h2>
+<section id="section-7" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-7">0.11.0</h2>
 <p>Breaking changes:</p>
 <ul>
 <li><code>type</code> column is replaced by <code>type</code> attribute.</li>
@@ -711,8 +711,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-6">0.11.0</h2>
 <li><code>ordinal::clm</code>: Support <code>cum.prob</code> and <code>linear.predictor</code> prediction types. Thanks to <a href="https://github.com/MrJerryTAO">@MrJerryTAO</a> for report <a href="https://github.com/vincentarelbundock/marginaleffects/issues/717">#717</a>.</li>
 </ul>
 </section>
-<section id="section-7" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-7">0.10.0</h2>
+<section id="section-8" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-8">0.10.0</h2>
 <p>Performance:</p>
 <ul>
 <li>2-4x faster execution for many calls. Thanks to Etienne Bacher.</li>
@@ -766,8 +766,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-7">0.10.0</h2>
 <li><code>hypotheses()</code> now works on lists and in calls to <code>lapply()</code>, <code>purrr::map()</code>, etc. Issue <a href="https://github.com/vincentarelbundock/marginaleffects/issues/660">#660</a>.</li>
 </ul>
 </section>
-<section id="section-8" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-8">0.9.0</h2>
+<section id="section-9" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-9">0.9.0</h2>
 <p>Breaking changes:</p>
 <ul>
 <li>All functions return an <code>estimate</code> column instead of the function-specific <code>predicted</code>, <code>comparisons</code>, <code>dydx</code>, etc. This change only affects unit-level estimates, and not average estimates, which already used the <code>estimate</code> column name.</li>
@@ -819,8 +819,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-8">0.9.0</h2>
 <li>Removed major performance bottleneck for <code>slopes()</code></li>
 </ul>
 </section>
-<section id="section-9" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-9">0.8.1</h2>
+<section id="section-10" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-10">0.8.1</h2>
 <ul>
 <li><code>deltamethod()</code> can run hypothesis tests on objects produced by the <code>comparisons()</code>, <code>marginaleffects()</code>, <code>predictions()</code>, and <code>marginalmeans()</code> functions. This feature relies on <code>match.call()</code>, which means it may not always work when used programmatically, inside functions and nested environments. It is generally safer and more efficient to use the <code>hypothesis</code> argument.</li>
 <li><code>plot_cme()</code> and <code>plot_cco()</code> accept lists with user-specified values for the regressors, and can display nice labels for shortcut string-functions like “threenum” or “quartile”.</li>
@@ -835,8 +835,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-9">0.8.1</h2>
 <li>Missing values for some regressors in the <code>comparisons()</code> output for <code>brms</code> models.</li>
 </ul>
 </section>
-<section id="section-10" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-10">0.8.0</h2>
+<section id="section-11" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-11">0.8.0</h2>
 <p>Breaking change:</p>
 <ul>
 <li>The <code>interaction</code> argument is deprecated and replaced by the <code>cross</code> argument. This is to reduce ambiguity with respect to the <code>interaction</code> argument in <code>emmeans</code>, which does something completely different, akin to the difference-in-differences illustrated in the Interactions vignette.</li>
@@ -877,8 +877,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-10">0.8.0</h2>
 <li>Fixed a bug in the <code>tidy()</code> function for <code>glmmTMB</code> models without random effects, which caused all t statistics to be identical.</li>
 </ul>
 </section>
-<section id="section-11" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-11">0.7.1</h2>
+<section id="section-12" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-12">0.7.1</h2>
 <ul>
 <li>New supported model class: <code>gamlss</code>. Thanks to Marcio Augusto Diniz.</li>
 <li><code>marginalmeans()</code> accepts a <code>wts</code> argument with values: “equal”, “proportional”, “cells”.</li>
@@ -906,8 +906,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-11">0.7.1</h2>
 </ul></li>
 </ul>
 </section>
-<section id="section-12" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-12">0.7.0</h2>
+<section id="section-13" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-13">0.7.0</h2>
 <p>Breaking changes:</p>
 <ul>
 <li><code>by</code> is deprecated in <code>summary()</code> and <code>tidy()</code>. Use the same <code>by</code> argument in the main functions instead: <code>comparisons()</code>, <code>marginaleffects()</code>, <code>predictions()</code></li>
@@ -941,8 +941,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-12">0.7.0</h2>
 <li>documentation improvements</li>
 </ul>
 </section>
-<section id="section-13" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-13">0.6.0</h2>
+<section id="section-14" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-14">0.6.0</h2>
 <p>New supported packages and models:</p>
 <ul>
 <li><code>tidymodels</code> objects of class <code>tidy_model</code> are supported if the fit engine is supported by <code>marginaleffects</code>.</li>
@@ -979,8 +979,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-13">0.6.0</h2>
 <li><code>tidy.predictions()</code> computes standard errors and confidence intervals for linear models or GLM on the link scale.</li>
 </ul>
 </section>
-<section id="section-14" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-14">0.5.0</h2>
+<section id="section-15" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-15">0.5.0</h2>
 <p>Breaking changes:</p>
 <ul>
 <li><code>type</code> no longer accepts a character vector. Must be a single string.</li>
@@ -1040,8 +1040,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-14">0.5.0</h2>
 </ul></li>
 </ul>
 </section>
-<section id="section-15" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-15">0.4.1</h2>
+<section id="section-16" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-16">0.4.1</h2>
 <p>New supported packages and models:</p>
 <ul>
 <li><code>stats::loess</code></li>
@@ -1056,8 +1056,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-15">0.4.1</h2>
 <li>Bug fixes</li>
 </ul>
 </section>
-<section id="section-16" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-16">0.4.0</h2>
+<section id="section-17" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-17">0.4.0</h2>
 <p>New function:</p>
 <ul>
 <li><code>comparisons()</code> computes contrasts</li>
@@ -1070,14 +1070,14 @@ <h2 class="unnumbered anchored" data-anchor-id="section-16">0.4.0</h2>
 <li>Do not overwrite user’s <code>ggplot2::theme_set()</code> call</li>
 </ul>
 </section>
-<section id="section-17" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-17">0.3.4</h2>
+<section id="section-18" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-18">0.3.4</h2>
 <ul>
 <li>Bug fixes</li>
 </ul>
 </section>
-<section id="section-18" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-18">0.3.3</h2>
+<section id="section-19" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-19">0.3.3</h2>
 <p>New supported models:</p>
 <ul>
 <li><code>mclogit::mclogit</code></li>
@@ -1093,8 +1093,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-18">0.3.3</h2>
 <li>CRAN test fixes</li>
 </ul>
 </section>
-<section id="section-19" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-19">0.3.2</h2>
+<section id="section-20" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-20">0.3.2</h2>
 <p>Support for new models and packages:</p>
 <ul>
 <li><code>brglm2::bracl</code></li>
@@ -1110,12 +1110,12 @@ <h2 class="unnumbered anchored" data-anchor-id="section-19">0.3.2</h2>
 <li>CRAN tests</li>
 </ul>
 </section>
-<section id="section-20" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-20">0.3.1</h2>
+<section id="section-21" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-21">0.3.1</h2>
 <p>documentation bugfix</p>
 </section>
-<section id="section-21" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-21">0.3.0</h2>
+<section id="section-22" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-22">0.3.0</h2>
 <p>Breaking changes:</p>
 <ul>
 <li><code>predictions</code> returns predictions for every observation in the original dataset instead of <code>newdata=datagrid()</code>.</li>
@@ -1145,8 +1145,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-21">0.3.0</h2>
 <li>Many new tests, including several against <code>emmeans</code></li>
 </ul>
 </section>
-<section id="section-22" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-22">0.2.0</h2>
+<section id="section-23" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-23">0.2.0</h2>
 <p>Breaking change:</p>
 <ul>
 <li><code>data</code> argument becomes <code>newdata</code> in all functions.</li>
@@ -1175,8 +1175,8 @@ <h2 class="unnumbered anchored" data-anchor-id="section-22">0.2.0</h2>
 <li>New validity checks and tests</li>
 </ul>
 </section>
-<section id="section-23" class="level2 unnumbered">
-<h2 class="unnumbered anchored" data-anchor-id="section-23">0.1.0</h2>
+<section id="section-24" class="level2 unnumbered">
+<h2 class="unnumbered anchored" data-anchor-id="section-24">0.1.0</h2>
 <p>First release. Bravo!</p>
 <p>Thanks to Marco Avina Mendoza, Resul Umit, and all those who offered comments and suggestions.</p>
 
diff --git a/articles/alternative_software.html b/articles/alternative_software.html
index b081a70d5..e5dbb168f 100644
--- a/articles/alternative_software.html
+++ b/articles/alternative_software.html
@@ -5,7 +5,7 @@
 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 22&nbsp; Alternative Software</title>
+<title>The Marginal Effects Zoo (0.15.1) - 22&nbsp; Alternative Software</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -104,7 +104,7 @@
 gtag('js', new Date());
 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script><script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 22&nbsp; Alternative Software">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 22&nbsp; Alternative Software">
 <meta name="twitter:description" content="">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
 <meta name="twitter:card" content="summary_large_image">
@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
diff --git a/articles/bootstrap.html b/articles/bootstrap.html
index 20de6be2b..ae9996c48 100644
--- a/articles/bootstrap.html
+++ b/articles/bootstrap.html
@@ -5,7 +5,7 @@
 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 9&nbsp; Bootstrap &amp; Simulation</title>
+<title>The Marginal Effects Zoo (0.15.1) - 9&nbsp; Bootstrap &amp; Simulation</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -104,7 +104,7 @@
 gtag('js', new Date());
 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 9&nbsp; Bootstrap &amp; Simulation">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 9&nbsp; Bootstrap &amp; Simulation">
 <meta name="twitter:description" content="">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
 <meta name="twitter:card" content="summary_large_image">
@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -552,9 +552,9 @@ <h1 class="title">
 <span>  <span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/inferences.html">inferences</a></span><span class="op">(</span>method <span class="op">=</span> <span class="st">"boot"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.266 -0.634  0.459</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.156 -0.337  0.291</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216      0.174 -0.333  0.364</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.274 -0.640  0.398</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.159 -0.329  0.319</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216      0.175 -0.323  0.388</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, std.error, conf.low, conf.high </span></span>
 <span><span class="co">#&gt; Type:  response</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -566,9 +566,9 @@ <h1 class="title">
 <span><span class="va">est</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.265 -0.565  0.468</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.159 -0.295  0.312</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216      0.183 -0.318  0.417</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.268 -0.646  0.468</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.157 -0.316  0.294</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216      0.182 -0.357  0.368</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, std.error, conf.low, conf.high </span></span>
 <span><span class="co">#&gt; Type:  response</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -588,21 +588,21 @@ <h1 class="title">
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Bootstrap Statistics :</span></span>
-<span><span class="co">#&gt;        original       bias    std. error</span></span>
-<span><span class="co">#&gt; t1* -0.11025325  0.007095463   0.2647118</span></span>
-<span><span class="co">#&gt; t2* -0.02006005  0.001977153   0.1588222</span></span>
-<span><span class="co">#&gt; t3*  0.02158742 -0.000386270   0.1826585</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span><span class="co">#&gt;        original        bias    std. error</span></span>
+<span><span class="co">#&gt; t1* -0.11025325  3.620023e-03   0.2682605</span></span>
+<span><span class="co">#&gt; t2* -0.02006005  5.498175e-05   0.1569108</span></span>
+<span><span class="co">#&gt; t3*  0.02158742 -1.591206e-03   0.1821860</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 <p>Or we can extract the individual draws with the <a href="reference/posterior_draws.html"><code>posterior_draws()</code></a> function:</p>
 <div class="cell">
 <div class="sourceCode" id="cb7"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/posterior_draws.html">posterior_draws</a></span><span class="op">(</span><span class="va">est</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/r/utils/head.html">head</a></span><span class="op">(</span><span class="op">)</span></span>
 <span><span class="co">#&gt;   drawid        draw        term contrast    Species    estimate predicted_lo predicted_hi predicted std.error   conf.low conf.high</span></span>
-<span><span class="co">#&gt; 1      1  0.34291638 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2647118 -0.5647467 0.4675544</span></span>
-<span><span class="co">#&gt; 2      1  0.11507292 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1588222 -0.2951916 0.3121828</span></span>
-<span><span class="co">#&gt; 3      1  0.00986429 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1826585 -0.3182129 0.4167165</span></span>
-<span><span class="co">#&gt; 4      2 -0.17769879 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2647118 -0.5647467 0.4675544</span></span>
-<span><span class="co">#&gt; 5      2 -0.08218494 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1588222 -0.2951916 0.3121828</span></span>
-<span><span class="co">#&gt; 6      2 -0.03808062 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1826585 -0.3182129 0.4167165</span></span>
+<span><span class="co">#&gt; 1      1 -0.20715923 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2682605 -0.6455666 0.4681477</span></span>
+<span><span class="co">#&gt; 2      1  0.11246218 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1569108 -0.3156269 0.2944124</span></span>
+<span><span class="co">#&gt; 3      1  0.26005004 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1821860 -0.3565338 0.3684294</span></span>
+<span><span class="co">#&gt; 4      2 -0.02019376 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2682605 -0.6455666 0.4681477</span></span>
+<span><span class="co">#&gt; 5      2 -0.12237572 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1569108 -0.3156269 0.2944124</span></span>
+<span><span class="co">#&gt; 6      2 -0.16955910 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1821860 -0.3565338 0.3684294</span></span>
 <span></span>
 <span><span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/posterior_draws.html">posterior_draws</a></span><span class="op">(</span><span class="va">est</span>, shape <span class="op">=</span> <span class="st">"DxP"</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/r/base/dim.html">dim</a></span><span class="op">(</span><span class="op">)</span></span>
 <span><span class="co">#&gt; [1] 500   3</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -617,9 +617,9 @@ <h1 class="title">
 <span><span class="va">est</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;         Term Contrast    Species Estimate  2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103 -0.524  0.386</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201 -0.277  0.260</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216 -0.300  0.362</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103 -0.619  0.581</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201 -0.355  0.265</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216 -0.356  0.399</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, conf.low, conf.high </span></span>
 <span><span class="co">#&gt; Type:  response</span></span>
@@ -629,28 +629,28 @@ <h1 class="title">
 <span><span class="co">#&gt; # A tibble: 101 × 3</span></span>
 <span><span class="co">#&gt;    splits           id           estimates       </span></span>
 <span><span class="co">#&gt;    &lt;list&gt;           &lt;chr&gt;        &lt;list&gt;          </span></span>
-<span><span class="co">#&gt;  1 &lt;split [150/53]&gt; Bootstrap001 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  2 &lt;split [150/53]&gt; Bootstrap002 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  3 &lt;split [150/57]&gt; Bootstrap003 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  4 &lt;split [150/52]&gt; Bootstrap004 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  5 &lt;split [150/62]&gt; Bootstrap005 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  6 &lt;split [150/47]&gt; Bootstrap006 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  7 &lt;split [150/56]&gt; Bootstrap007 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  8 &lt;split [150/51]&gt; Bootstrap008 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt;  9 &lt;split [150/54]&gt; Bootstrap009 &lt;tibble [3 × 7]&gt;</span></span>
-<span><span class="co">#&gt; 10 &lt;split [150/55]&gt; Bootstrap010 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  1 &lt;split [150/48]&gt; Bootstrap001 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  2 &lt;split [150/55]&gt; Bootstrap002 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  3 &lt;split [150/50]&gt; Bootstrap003 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  4 &lt;split [150/50]&gt; Bootstrap004 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  5 &lt;split [150/50]&gt; Bootstrap005 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  6 &lt;split [150/53]&gt; Bootstrap006 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  7 &lt;split [150/58]&gt; Bootstrap007 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  8 &lt;split [150/58]&gt; Bootstrap008 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt;  9 &lt;split [150/55]&gt; Bootstrap009 &lt;tibble [3 × 7]&gt;</span></span>
+<span><span class="co">#&gt; 10 &lt;split [150/57]&gt; Bootstrap010 &lt;tibble [3 × 7]&gt;</span></span>
 <span><span class="co">#&gt; # ℹ 91 more rows</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 <p>Or we can extract the individual draws with the <a href="reference/posterior_draws.html"><code>posterior_draws()</code></a> function:</p>
 <div class="cell">
 <div class="sourceCode" id="cb9"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/posterior_draws.html">posterior_draws</a></span><span class="op">(</span><span class="va">est</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/r/utils/head.html">head</a></span><span class="op">(</span><span class="op">)</span></span>
-<span><span class="co">#&gt;   drawid         draw        term contrast    Species    estimate predicted_lo predicted_hi predicted   conf.low conf.high</span></span>
-<span><span class="co">#&gt; 1      1 -0.239618402 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.5241231 0.3859502</span></span>
-<span><span class="co">#&gt; 2      1 -0.072908264 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.2771160 0.2599758</span></span>
-<span><span class="co">#&gt; 3      1  0.004071542 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3000476 0.3619014</span></span>
-<span><span class="co">#&gt; 4      2 -0.216492088 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.5241231 0.3859502</span></span>
-<span><span class="co">#&gt; 5      2  0.003450908 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.2771160 0.2599758</span></span>
-<span><span class="co">#&gt; 6      2  0.105011435 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3000476 0.3619014</span></span>
+<span><span class="co">#&gt;   drawid        draw        term contrast    Species    estimate predicted_lo predicted_hi predicted   conf.low conf.high</span></span>
+<span><span class="co">#&gt; 1      1 -0.06230329 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.6185060 0.5805186</span></span>
+<span><span class="co">#&gt; 2      1 -0.11938081 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.3551267 0.2650783</span></span>
+<span><span class="co">#&gt; 3      1 -0.14573683 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3555306 0.3991535</span></span>
+<span><span class="co">#&gt; 4      2  0.12473364 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.6185060 0.5805186</span></span>
+<span><span class="co">#&gt; 5      2  0.09889677 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.3551267 0.2650783</span></span>
+<span><span class="co">#&gt; 6      2  0.08696638 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3555306 0.3991535</span></span>
 <span></span>
 <span><span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/posterior_draws.html">posterior_draws</a></span><span class="op">(</span><span class="va">est</span>, shape <span class="op">=</span> <span class="st">"PxD"</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/r/base/dim.html">dim</a></span><span class="op">(</span><span class="op">)</span></span>
 <span><span class="co">#&gt; [1]   3 100</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -666,10 +666,10 @@ <h1 class="title">
 <div class="sourceCode" id="cb10"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/comparisons.html">avg_comparisons</a></span><span class="op">(</span><span class="va">mod</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/inferences.html">inferences</a></span><span class="op">(</span>method <span class="op">=</span> <span class="st">"fwb"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;          Term            Contrast Estimate Std. Error  2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  Petal.Length +1                    0.8929     0.0784  0.737  1.046</span></span>
-<span><span class="co">#&gt;  Petal.Width  +1                   -0.0362     0.1644 -0.351  0.293</span></span>
-<span><span class="co">#&gt;  Species      versicolor - setosa  -1.4629     0.3265 -2.118 -0.851</span></span>
-<span><span class="co">#&gt;  Species      virginica - setosa   -1.9842     0.3892 -2.800 -1.257</span></span>
+<span><span class="co">#&gt;  Petal.Length +1                    0.8929     0.0807  0.736  1.052</span></span>
+<span><span class="co">#&gt;  Petal.Width  +1                   -0.0362     0.1551 -0.341  0.279</span></span>
+<span><span class="co">#&gt;  Species      versicolor - setosa  -1.4629     0.3221 -2.099 -0.813</span></span>
+<span><span class="co">#&gt;  Species      virginica - setosa   -1.9842     0.3781 -2.724 -1.221</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, contrast, estimate, std.error, conf.low, conf.high </span></span>
 <span><span class="co">#&gt; Type:  response</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -691,9 +691,9 @@ <h1 class="title">
 <span>  <span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/inferences.html">inferences</a></span><span class="op">(</span>method <span class="op">=</span> <span class="st">"simulation"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.291 -0.741  0.435</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.164 -0.347  0.299</span></span>
-<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216      0.168 -0.306  0.341</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.283 -0.662  0.437</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.151 -0.324  0.268</span></span>
+<span><span class="co">#&gt;  Petal.Width mean(+1) virginica    0.0216      0.159 -0.290  0.313</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, contrast, Species, estimate, std.error, conf.low, conf.high, predicted_lo, predicted_hi, predicted, tmp_idx </span></span>
 <span><span class="co">#&gt; Type:  response</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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diff --git a/articles/brms.html b/articles/brms.html
index 15de55e29..645d0fe60 100644
--- a/articles/brms.html
+++ b/articles/brms.html
@@ -5,7 +5,7 @@
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-<title>The Marginal Effects Zoo (0.15.0) - 8&nbsp; Bayes</title>
+<title>The Marginal Effects Zoo (0.15.1) - 8&nbsp; Bayes</title>
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@@ -104,7 +104,7 @@
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 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 8&nbsp; Bayes">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 8&nbsp; Bayes">
 <meta name="twitter:description" content="">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
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@@ -115,7 +115,7 @@
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -1066,8 +1066,8 @@ <h1 class="title">
 <span>    by <span class="op">=</span> <span class="st">"x"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;  x Estimate 2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  0    0.454 0.339  0.567</span></span>
-<span><span class="co">#&gt;  1    0.553 0.433  0.667</span></span>
+<span><span class="co">#&gt;  0    0.452 0.339  0.564</span></span>
+<span><span class="co">#&gt;  1    0.551 0.431  0.660</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: x, estimate, conf.low, conf.high </span></span>
 <span><span class="co">#&gt; Type:  response</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -1081,7 +1081,7 @@ <h1 class="title">
 <span>    sample_new_levels <span class="op">=</span> <span class="st">"gaussian"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;  Term Contrast Estimate  2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;     x    1 - 0    0.096 0.0471  0.161</span></span>
+<span><span class="co">#&gt;     x    1 - 0   0.0968 0.0484  0.161</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, contrast, estimate, conf.low, conf.high </span></span>
 <span><span class="co">#&gt; Type:  response</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
@@ -1099,7 +1099,7 @@ <h1 class="title">
 <span>  effects <span class="op">=</span> <span class="st">"integrateoutRE"</span><span class="op">)</span></span>
 <span><span class="va">bm</span><span class="op">$</span><span class="va">ContrastSummary</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span><span class="op">(</span><span class="op">)</span></span>
 <span><span class="co">#&gt;            M        Mdn         LL        UL PercentROPE PercentMID   CI CIType ROPE  MID Label</span></span>
-<span><span class="co">#&gt; 1 0.09920292 0.09745593 0.04806257 0.1629364          NA         NA 0.95    ETI &lt;NA&gt; &lt;NA&gt; AME x</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<span><span class="co">#&gt; 1 0.09850989 0.09628107 0.04727819 0.1630572          NA         NA 0.95    ETI &lt;NA&gt; &lt;NA&gt; AME x</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 <p>See the <a href="alternative_software.html">alternative software vignette</a> for more information on <code>brmsmargins</code>.</p>
 </section></section><section id="multinomial-logit" class="level2" data-number="8.3"><h2 data-number="8.3" class="anchored" data-anchor-id="multinomial-logit">
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diff --git a/articles/categorical.html b/articles/categorical.html
index 6c1c4839d..221bd3d65 100644
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@@ -5,7 +5,7 @@
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 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 10&nbsp; Categorical outcomes</title>
+<title>The Marginal Effects Zoo (0.15.1) - 10&nbsp; Categorical outcomes</title>
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+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 10&nbsp; Categorical outcomes">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
diff --git a/articles/comparisons.html b/articles/comparisons.html
index 5332058fa..0bd53e758 100644
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-<title>The Marginal Effects Zoo (0.15.0) - 3&nbsp; Comparisons</title>
+<title>The Marginal Effects Zoo (0.15.1) - 3&nbsp; Comparisons</title>
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 3&nbsp; Comparisons">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 3&nbsp; Comparisons">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
diff --git a/articles/conformal.html b/articles/conformal.html
index 5361b1f92..a0f316a35 100644
--- a/articles/conformal.html
+++ b/articles/conformal.html
@@ -5,7 +5,7 @@
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 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 11&nbsp; Conformal prediction</title>
+<title>The Marginal Effects Zoo (0.15.1) - 11&nbsp; Conformal prediction</title>
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@@ -123,7 +123,7 @@
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 11&nbsp; Conformal prediction">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 11&nbsp; Conformal prediction">
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@@ -134,7 +134,7 @@
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
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diff --git a/articles/elasticity.html b/articles/elasticity.html
index 5d876eed3..9eb15e1f5 100644
--- a/articles/elasticity.html
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 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 13&nbsp; Elasticity</title>
+<title>The Marginal Effects Zoo (0.15.1) - 13&nbsp; Elasticity</title>
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 13&nbsp; Elasticity">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 13&nbsp; Elasticity">
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@@ -115,7 +115,7 @@
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
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diff --git a/articles/equivalence.html b/articles/equivalence.html
index 4a6b9885c..11c2f4961 100644
--- a/articles/equivalence.html
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 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 14&nbsp; Equivalence Tests</title>
+<title>The Marginal Effects Zoo (0.15.1) - 14&nbsp; Equivalence Tests</title>
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@@ -104,7 +104,7 @@
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 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 14&nbsp; Equivalence Tests">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 14&nbsp; Equivalence Tests">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
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diff --git a/articles/experiments.html b/articles/experiments.html
index d3a0d6ffd..ce4980d5e 100644
--- a/articles/experiments.html
+++ b/articles/experiments.html
@@ -5,7 +5,7 @@
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 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 15&nbsp; Experiments</title>
+<title>The Marginal Effects Zoo (0.15.1) - 15&nbsp; Experiments</title>
 <style>
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@@ -105,7 +105,7 @@
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 15&nbsp; Experiments">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 15&nbsp; Experiments">
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@@ -116,7 +116,7 @@
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -543,7 +543,7 @@ <h1 class="title">
 <span class="header-section-number">15.1.2</span> Evaluating Effects From The Model Summary</h3>
 <p>Since this model is fairly simple the estimated differences between any of the four possible combinations of <code>vs</code> and <code>am</code> can be read from the regression table, which we create using the <code>modelsummary</code> package:</p>
 <div class="cell">
-<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://modelsummary.com/reference/modelsummary.html">modelsummary</a></span><span class="op">(</span><span class="va">fit</span>, gof_map <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span><span class="op">(</span><span class="st">"r.squared"</span>, <span class="st">"nobs"</span><span class="op">)</span><span class="op">)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb3"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/modelsummary/man/modelsummary.html">modelsummary</a></span><span class="op">(</span><span class="va">fit</span>, gof_map <span class="op">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span><span class="op">(</span><span class="st">"r.squared"</span>, <span class="st">"nobs"</span><span class="op">)</span><span class="op">)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output-display">
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diff --git a/articles/extensions.html b/articles/extensions.html
index e83b136ae..b0b68ae7a 100644
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 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 23&nbsp; Extensions</title>
+<title>The Marginal Effects Zoo (0.15.1) - 23&nbsp; Extensions</title>
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 23&nbsp; Extensions">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 23&nbsp; Extensions">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/faq.html b/articles/faq.html
index 59ba2a3ad..25df7b9df 100644
--- a/articles/faq.html
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 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 30&nbsp; FAQ</title>
+<title>The Marginal Effects Zoo (0.15.1) - 30&nbsp; FAQ</title>
 <style>
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@@ -70,7 +70,7 @@
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 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 30&nbsp; FAQ">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 30&nbsp; FAQ">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/functions.html b/articles/functions.html
index b3d817744..1edc02248 100644
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@@ -5,7 +5,7 @@
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 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
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-<title>The Marginal Effects Zoo (0.15.0) - Functions</title>
+<title>The Marginal Effects Zoo (0.15.1) - Functions</title>
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@@ -89,7 +89,7 @@
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - Functions">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - Functions">
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 </head>
@@ -99,7 +99,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/gam.html b/articles/gam.html
index 2b85b6f12..7583827c0 100644
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-<title>The Marginal Effects Zoo (0.15.0) - 16&nbsp; GAM</title>
+<title>The Marginal Effects Zoo (0.15.1) - 16&nbsp; GAM</title>
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 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 16&nbsp; GAM">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 16&nbsp; GAM">
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@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/gformula.html b/articles/gformula.html
index 998d9458f..490e1a9ad 100644
--- a/articles/gformula.html
+++ b/articles/gformula.html
@@ -5,7 +5,7 @@
 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 12&nbsp; Causal Inference</title>
+<title>The Marginal Effects Zoo (0.15.1) - 12&nbsp; Causal Inference</title>
 <style>
 code{white-space: pre-wrap;}
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@@ -104,7 +104,7 @@
 gtag('js', new Date());
 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script><script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 12&nbsp; Causal Inference">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 12&nbsp; Causal Inference">
 <meta name="twitter:description" content="">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
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@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -719,7 +719,7 @@ <h1 class="title">
 <span><span class="va">bootstrap</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>                           estimate std.error conf.low conf.high
-1 Treatment - No Treatment 3.517374 0.4849327 2.566924  4.467825</code></pre>
+1 Treatment - No Treatment 3.517374 0.4786991 2.579141  4.455607</code></pre>
 </div>
 </div>
 <p>The results are close to those that we obtained with <code><a href="https://rdrr.io/pkg/marginaleffects/man/comparisons.html">comparisons()</a></code>, but the confidence interval differs slightly because of the difference between bootstrapping and the delta method.</p>
diff --git a/articles/hypothesis.html b/articles/hypothesis.html
index 8d2b4fd9e..a59e32044 100644
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 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 7&nbsp; Hypothesis Tests</title>
+<title>The Marginal Effects Zoo (0.15.1) - 7&nbsp; Hypothesis Tests</title>
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 </script><script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 7&nbsp; Hypothesis Tests">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 7&nbsp; Hypothesis Tests">
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@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -732,8 +732,8 @@ <h1 class="title">
 <span>  hypothesis <span class="op">=</span> <span class="st">"median = mean"</span></span>
 <span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
-<span><span class="co">#&gt;           Term Estimate Std. Error   z Pr(&gt;|z|)   S   2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  median = mean  0.00176    0.00881 0.2    0.841 0.2 -0.0155  0.019</span></span>
+<span><span class="co">#&gt;           Term Estimate Std. Error      z Pr(&gt;|z|)   S   2.5 % 97.5 %</span></span>
+<span><span class="co">#&gt;  median = mean -0.00261    0.00701 -0.372     0.71 0.5 -0.0163 0.0111</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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diff --git a/articles/links.html b/articles/links.html
index 0221fd2a4..4852ae0e5 100644
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@@ -8,7 +8,7 @@
 
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
 
-<title>The Marginal Effects Zoo (0.15.0) - 24&nbsp; Links</title>
+<title>The Marginal Effects Zoo (0.15.1) - 24&nbsp; Links</title>
 <style>
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@@ -80,7 +80,7 @@
 </script>
 
 
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 24&nbsp; Links">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 24&nbsp; Links">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
 <meta name="twitter:card" content="summary_large_image">
 </head>
@@ -93,7 +93,7 @@
       <div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/lme4.html b/articles/lme4.html
index 5c524e87f..ece7a705a 100644
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-<title>The Marginal Effects Zoo (0.15.0) - 18&nbsp; Mixed Effects</title>
+<title>The Marginal Effects Zoo (0.15.1) - 18&nbsp; Mixed Effects</title>
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 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 18&nbsp; Mixed Effects">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 18&nbsp; Mixed Effects">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/logit.html b/articles/logit.html
index bacac96b4..d68a0b166 100644
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-<title>The Marginal Effects Zoo (0.15.0) - 17&nbsp; Logit</title>
+<title>The Marginal Effects Zoo (0.15.1) - 17&nbsp; Logit</title>
 <style>
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@@ -104,7 +104,7 @@
 gtag('js', new Date());
 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script><script src="../site_libs/kePrint-0.0.1/kePrint.js"></script><link href="../site_libs/lightable-0.0.1/lightable.css" rel="stylesheet">
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 17&nbsp; Logit">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 17&nbsp; Logit">
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@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
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             <div id="quarto-search" class="" title="Search"></div>
@@ -537,7 +537,7 @@ <h1 class="title">
 <span class="header-section-number">17.1.1</span> One-Number Summaries</h3>
 <p>As usual, we can produce a one-number summary of the relationship of interest by exponentiating the coefficients, which yields an Odds Ratio (OR):</p>
 <div class="cell">
-<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://modelsummary.com/reference/modelsummary.html">modelsummary</a></span><span class="op">(</span><span class="va">mod</span>, exponentiate <span class="op">=</span> <span class="cn">TRUE</span>, coef_omit <span class="op">=</span> <span class="st">"^(?!txSK)"</span><span class="op">)</span> </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode" id="cb2"><pre class="downlit sourceCode r code-with-copy"><code class="sourceCode R"><span><span class="fu"><a href="https://rdrr.io/pkg/modelsummary/man/modelsummary.html">modelsummary</a></span><span class="op">(</span><span class="va">mod</span>, exponentiate <span class="op">=</span> <span class="cn">TRUE</span>, coef_omit <span class="op">=</span> <span class="st">"^(?!txSK)"</span><span class="op">)</span> </span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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diff --git a/articles/marginaleffects.html b/articles/marginaleffects.html
index fce14d85f..bd062261a 100644
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 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 1&nbsp; Get Started</title>
+<title>The Marginal Effects Zoo (0.15.1) - 1&nbsp; Get Started</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -104,7 +104,7 @@
 gtag('js', new Date());
 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 1&nbsp; Get Started">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 1&nbsp; Get Started">
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 <meta name="twitter:image" content="fig/marginaleffects_card.png">
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@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -710,21 +710,21 @@ <h1 class="title">
 <div class="sourceCode cell-code" id="cb24"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="bu">cmp</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>shape: (96, 9)
-┌──────┬──────────┬──────────┬───────────┬───┬──────────┬───────┬───────┬───────┐
-│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|)  ┆ S     ┆ 2.5%  ┆ 97.5% │
-│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---   ┆ ---   ┆ ---   │
-│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str      ┆ str   ┆ str   ┆ str   │
-╞══════╪══════════╪══════════╪═══════════╪═══╪══════════╪═══════╪═══════╪═══════╡
-│ wt   ┆ +1       ┆ -6.61    ┆ 1.87      ┆ … ┆ 0.00165  ┆ 9.24  ┆ -10.5 ┆ -2.76 │
-│ wt   ┆ +1       ┆ -6.61    ┆ 1.87      ┆ … ┆ 0.00165  ┆ 9.24  ┆ -10.5 ┆ -2.76 │
-│ wt   ┆ +1       ┆ -7.16    ┆ 1.8       ┆ … ┆ 0.000554 ┆ 10.8  ┆ -10.9 ┆ -3.45 │
-│ wt   ┆ +1       ┆ -3.21    ┆ 2.01      ┆ … ┆ 0.124    ┆ 3.02  ┆ -7.35 ┆ 0.939 │
-│ …    ┆ …        ┆ …        ┆ …         ┆ … ┆ …        ┆ …     ┆ …     ┆ …     │
-│ am   ┆ 1 - 0    ┆ 2.11     ┆ 2.29      ┆ … ┆ 0.367    ┆ 1.45  ┆ -2.62 ┆ 6.83  │
-│ am   ┆ 1 - 0    ┆ 0.895    ┆ 1.64      ┆ … ┆ 0.591    ┆ 0.758 ┆ -2.5  ┆ 4.29  │
-│ am   ┆ 1 - 0    ┆ 4.03     ┆ 3.24      ┆ … ┆ 0.226    ┆ 2.15  ┆ -2.66 ┆ 10.7  │
-│ am   ┆ 1 - 0    ┆ -0.237   ┆ 1.59      ┆ … ┆ 0.883    ┆ 0.18  ┆ -3.51 ┆ 3.04  │
-└──────┴──────────┴──────────┴───────────┴───┴──────────┴───────┴───────┴───────┘
+┌──────┬──────────┬──────────┬───────────┬───┬─────────┬───────┬─────────┬─────────┐
+│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%    ┆ 97.5%   │
+│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---     │
+│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str     ┆ str     │
+╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═════════╪═════════╡
+│ hp   ┆ +1       ┆ -0.0369  ┆ 0.0185    ┆ … ┆ 0.0575  ┆ 4.12  ┆ -0.0751 ┆ 0.00128 │
+│ hp   ┆ +1       ┆ -0.0287  ┆ 0.0156    ┆ … ┆ 0.0788  ┆ 3.67  ┆ -0.0609 ┆ 0.00357 │
+│ hp   ┆ +1       ┆ -0.0466  ┆ 0.0226    ┆ … ┆ 0.0502  ┆ 4.32  ┆ -0.0932 ┆ 4.6e-05 │
+│ hp   ┆ +1       ┆ -0.0423  ┆ 0.0133    ┆ … ┆ 0.00401 ┆ 7.96  ┆ -0.0697 ┆ -0.0149 │
+│ …    ┆ …        ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …       ┆ …       │
+│ am   ┆ 1 - 0    ┆ 2.11     ┆ 2.29      ┆ … ┆ 0.367   ┆ 1.45  ┆ -2.62   ┆ 6.83    │
+│ am   ┆ 1 - 0    ┆ 0.895    ┆ 1.64      ┆ … ┆ 0.591   ┆ 0.758 ┆ -2.5    ┆ 4.29    │
+│ am   ┆ 1 - 0    ┆ 4.03     ┆ 3.24      ┆ … ┆ 0.226   ┆ 2.15  ┆ -2.66   ┆ 10.7    │
+│ am   ┆ 1 - 0    ┆ -0.237   ┆ 1.59      ┆ … ┆ 0.883   ┆ 0.18  ┆ -3.51   ┆ 3.04    │
+└──────┴──────────┴──────────┴───────────┴───┴─────────┴───────┴─────────┴─────────┘
 
 Columns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb</code></pre>
 </div>
@@ -939,21 +939,21 @@ <h1 class="title">
 <div class="sourceCode cell-code" id="cb44"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(mfx)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>shape: (96, 9)
-┌──────┬──────────┬──────────┬───────────┬───┬─────────┬───────┬─────────┬──────────┐
-│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%    ┆ 97.5%    │
-│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---      │
-│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str     ┆ str      │
-╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═════════╪══════════╡
-│ hp   ┆ dY/dX    ┆ -0.0369  ┆ 0.0175    ┆ … ┆ 0.0453  ┆ 4.46  ┆ -0.073  ┆ -0.00083 │
-│ hp   ┆ dY/dX    ┆ -0.0287  ┆ 0.0176    ┆ … ┆ 0.116   ┆ 3.11  ┆ -0.065  ┆ 0.00763  │
-│ hp   ┆ dY/dX    ┆ -0.0466  ┆ 0.0226    ┆ … ┆ 0.0506  ┆ 4.3   ┆ -0.0933 ┆ 0.00014  │
-│ hp   ┆ dY/dX    ┆ -0.0423  ┆ 0.0135    ┆ … ┆ 0.00459 ┆ 7.77  ┆ -0.0702 ┆ -0.0144  │
-│ …    ┆ …        ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …       ┆ …        │
-│ wt   ┆ dY/dX    ┆ -1.65    ┆ 3.74      ┆ … ┆ 0.663   ┆ 0.593 ┆ -9.38   ┆ 6.07     │
-│ wt   ┆ dY/dX    ┆ -4.52    ┆ 2.47      ┆ … ┆ 0.0801  ┆ 3.64  ┆ -9.63   ┆ 0.586    │
-│ wt   ┆ dY/dX    ┆ 0.635    ┆ 4.88      ┆ … ┆ 0.898   ┆ 0.156 ┆ -9.45   ┆ 10.7     │
-│ wt   ┆ dY/dX    ┆ -6.65    ┆ 1.86      ┆ … ┆ 0.00154 ┆ 9.34  ┆ -10.5   ┆ -2.81    │
-└──────┴──────────┴──────────┴───────────┴───┴─────────┴───────┴─────────┴──────────┘
+┌──────┬──────────┬───────────┬───────────┬───┬─────────┬───────┬───────────┬──────────┐
+│ Term ┆ Contrast ┆ Estimate  ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%      ┆ 97.5%    │
+│ ---  ┆ ---      ┆ ---       ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---       ┆ ---      │
+│ str  ┆ str      ┆ str       ┆ str       ┆   ┆ str     ┆ str   ┆ str       ┆ str      │
+╞══════╪══════════╪═══════════╪═══════════╪═══╪═════════╪═══════╪═══════════╪══════════╡
+│ am   ┆ dY/dX    ┆ 3.25e+03  ┆ 1.68e+04  ┆ … ┆ 0.848   ┆ 0.237 ┆ -3.15e+04 ┆ 3.8e+04  │
+│ am   ┆ dY/dX    ┆ -5.44e+03 ┆ 1.57e+04  ┆ … ┆ 0.732   ┆ 0.451 ┆ -3.78e+04 ┆ 2.69e+04 │
+│ am   ┆ dY/dX    ┆ 1.2e+04   ┆ 2.35e+04  ┆ … ┆ 0.614   ┆ 0.704 ┆ -3.64e+04 ┆ 6.05e+04 │
+│ am   ┆ dY/dX    ┆ -1.7e+04  ┆ 1.87e+04  ┆ … ┆ 0.371   ┆ 1.43  ┆ -5.56e+04 ┆ 2.15e+04 │
+│ …    ┆ …        ┆ …         ┆ …         ┆ … ┆ …       ┆ …     ┆ …         ┆ …        │
+│ wt   ┆ dY/dX    ┆ -1.65     ┆ 3.74      ┆ … ┆ 0.663   ┆ 0.593 ┆ -9.38     ┆ 6.07     │
+│ wt   ┆ dY/dX    ┆ -4.52     ┆ 2.47      ┆ … ┆ 0.0801  ┆ 3.64  ┆ -9.63     ┆ 0.586    │
+│ wt   ┆ dY/dX    ┆ 0.635     ┆ 4.88      ┆ … ┆ 0.898   ┆ 0.156 ┆ -9.45     ┆ 10.7     │
+│ wt   ┆ dY/dX    ┆ -6.65     ┆ 1.86      ┆ … ┆ 0.00154 ┆ 9.34  ┆ -10.5     ┆ -2.81    │
+└──────┴──────────┴───────────┴───────────┴───┴─────────┴───────┴───────────┴──────────┘
 
 Columns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb</code></pre>
 </div>
@@ -1259,12 +1259,12 @@ <h1 class="title">
 │ --- ┆ ---  ┆ ---               ┆ ---      ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---      │
 │ str ┆ str  ┆ str               ┆ str      ┆   ┆ str     ┆ str   ┆ str     ┆ str      │
 ╞═════╪══════╪═══════════════════╪══════════╪═══╪═════════╪═══════╪═════════╪══════════╡
+│ 1   ┆ hp   ┆ +1                ┆ -0.0436  ┆ … ┆ 0.0515  ┆ 4.28  ┆ -0.0876 ┆ 0.0003   │
+│ 0   ┆ hp   ┆ +1                ┆ -0.0343  ┆ … ┆ 0.041   ┆ 4.61  ┆ -0.067  ┆ -0.00152 │
 │ 1   ┆ am   ┆ mean(1) - mean(0) ┆ 1.9      ┆ … ┆ 0.418   ┆ 1.26  ┆ -2.86   ┆ 6.67     │
 │ 0   ┆ am   ┆ mean(1) - mean(0) ┆ -1.38    ┆ … ┆ 0.589   ┆ 0.764 ┆ -6.59   ┆ 3.83     │
 │ 1   ┆ wt   ┆ +1                ┆ -6.07    ┆ … ┆ 0.00522 ┆ 7.58  ┆ -10.2   ┆ -1.99    │
 │ 0   ┆ wt   ┆ +1                ┆ -2.48    ┆ … ┆ 0.0554  ┆ 4.17  ┆ -5.02   ┆ 0.0621   │
-│ 1   ┆ hp   ┆ +1                ┆ -0.0436  ┆ … ┆ 0.0515  ┆ 4.28  ┆ -0.0876 ┆ 0.0003   │
-│ 0   ┆ hp   ┆ +1                ┆ -0.0343  ┆ … ┆ 0.041   ┆ 4.61  ┆ -0.067  ┆ -0.00152 │
 └─────┴──────┴───────────────────┴──────────┴───┴─────────┴───────┴─────────┴──────────┘
 
 Columns: am, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high</code></pre>
@@ -1583,8 +1583,8 @@ <h1 class="title">
 │ ---  ┆ ---         ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │
 │ str  ┆ str         ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │
 ╞══════╪═════════════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡
-│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.25e-05 ┆ 16.3 ┆ 4.43  ┆ 10    │
 │ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.432     ┆ … ┆ 0.0147   ┆ 6.09 ┆ 0.239 ┆ 2.01  │
+│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.25e-05 ┆ 16.3 ┆ 4.43  ┆ 10    │
 └──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘
 
 Columns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high</code></pre>
diff --git a/articles/marginalmeans.html b/articles/marginalmeans.html
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-<title>The Marginal Effects Zoo (0.15.0) - 5&nbsp; Marginal Means</title>
+<title>The Marginal Effects Zoo (0.15.1) - 5&nbsp; Marginal Means</title>
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+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 5&nbsp; Marginal Means">
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+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/mrp.html b/articles/mrp.html
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+++ b/articles/mrp.html
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-<title>The Marginal Effects Zoo (0.15.0) - 20&nbsp; MrP</title>
+<title>The Marginal Effects Zoo (0.15.1) - 20&nbsp; MrP</title>
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 20&nbsp; MrP">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 20&nbsp; MrP">
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-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
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diff --git a/articles/multiple_imputation.html b/articles/multiple_imputation.html
index 0152dd5e2..8d01805f7 100644
--- a/articles/multiple_imputation.html
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-<title>The Marginal Effects Zoo (0.15.0) - 19&nbsp; Missing Data</title>
+<title>The Marginal Effects Zoo (0.15.1) - 19&nbsp; Missing Data</title>
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 19&nbsp; Missing Data">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 19&nbsp; Missing Data">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/performance.html b/articles/performance.html
index 67b3fc27a..9bdd61238 100644
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+++ b/articles/performance.html
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 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 25&nbsp; Performance</title>
+<title>The Marginal Effects Zoo (0.15.1) - 25&nbsp; Performance</title>
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+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 25&nbsp; Performance">
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       <div class="navbar-brand-container mx-auto">
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-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/plot.html b/articles/plot.html
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+<title>The Marginal Effects Zoo (0.15.1) - 6&nbsp; Plots</title>
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+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 6&nbsp; Plots">
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       <div class="navbar-brand-container mx-auto">
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-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/plot_files/figure-html/unnamed-chunk-18-1.png b/articles/plot_files/figure-html/unnamed-chunk-18-1.png
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 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 2&nbsp; Predictions</title>
+<title>The Marginal Effects Zoo (0.15.1) - 2&nbsp; Predictions</title>
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 2&nbsp; Predictions">
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+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
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diff --git a/articles/reference/predictions.html b/articles/reference/predictions.html
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 <span><span class="co">#&gt;  Estimate</span></span>
-<span><span class="co">#&gt;     -7.00</span></span>
+<span><span class="co">#&gt;     -6.99</span></span>
 <span><span class="co">#&gt;     -2.99</span></span>
 <span><span class="co">#&gt;      1.01</span></span>
 <span><span class="co">#&gt;      5.01</span></span>
diff --git a/articles/slopes_files/figure-html/unnamed-chunk-19-1.png b/articles/slopes_files/figure-html/unnamed-chunk-19-1.png
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diff --git a/articles/supported_models.html b/articles/supported_models.html
index cf82f2a84..6c600ddc7 100644
--- a/articles/supported_models.html
+++ b/articles/supported_models.html
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-<title>The Marginal Effects Zoo (0.15.0) - 28&nbsp; Supported Models</title>
+<title>The Marginal Effects Zoo (0.15.1) - 28&nbsp; Supported Models</title>
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+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 28&nbsp; Supported Models">
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       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
diff --git a/articles/svalues.html b/articles/svalues.html
index 846e56347..087fc63fb 100644
--- a/articles/svalues.html
+++ b/articles/svalues.html
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 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 21&nbsp; S Values</title>
+<title>The Marginal Effects Zoo (0.15.1) - 21&nbsp; S Values</title>
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-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 21&nbsp; S Values">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 21&nbsp; S Values">
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@@ -134,7 +134,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
diff --git a/articles/tables.html b/articles/tables.html
index c402d15f5..52dfa4b98 100644
--- a/articles/tables.html
+++ b/articles/tables.html
@@ -5,7 +5,7 @@
 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 29&nbsp; Tables</title>
+<title>The Marginal Effects Zoo (0.15.1) - 29&nbsp; Tables</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -104,7 +104,7 @@
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 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script><script src="../site_libs/kePrint-0.0.1/kePrint.js"></script><link href="../site_libs/lightable-0.0.1/lightable.css" rel="stylesheet">
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 29&nbsp; Tables">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 29&nbsp; Tables">
 <meta name="twitter:description" content="">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
 <meta name="twitter:card" content="summary_large_image">
@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
diff --git a/articles/uncertainty.html b/articles/uncertainty.html
index 1816bb3c4..b0c8d0ed5 100644
--- a/articles/uncertainty.html
+++ b/articles/uncertainty.html
@@ -5,7 +5,7 @@
 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0) - 27&nbsp; Standard Errors</title>
+<title>The Marginal Effects Zoo (0.15.1) - 27&nbsp; Standard Errors</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -104,7 +104,7 @@
 gtag('js', new Date());
 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script><script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0) - 27&nbsp; Standard Errors">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1) - 27&nbsp; Standard Errors">
 <meta name="twitter:description" content="">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
 <meta name="twitter:card" content="summary_large_image">
@@ -115,7 +115,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="../index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -611,27 +611,27 @@ <h1 class="title">
 <span><span class="va">mod</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/predictions.html">predictions</a></span><span class="op">(</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/inferences.html">inferences</a></span><span class="op">(</span>method <span class="op">=</span> <span class="st">"simulation"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt;  Estimate Std. Error    2.5 % 97.5 %</span></span>
-<span><span class="co">#&gt;  7.84e-01      0.188 2.86e-01  0.974</span></span>
-<span><span class="co">#&gt;  7.84e-01      0.161 3.68e-01  0.962</span></span>
-<span><span class="co">#&gt;  8.98e-01      0.138 4.74e-01  0.990</span></span>
-<span><span class="co">#&gt;  8.74e-01      0.228 1.80e-01  0.995</span></span>
-<span><span class="co">#&gt;  1.31e-02      0.179 5.19e-05  0.716</span></span>
+<span><span class="co">#&gt;  7.84e-01      0.197 2.54e-01  0.976</span></span>
+<span><span class="co">#&gt;  7.84e-01      0.166 3.47e-01  0.961</span></span>
+<span><span class="co">#&gt;  8.98e-01      0.146 4.64e-01  0.989</span></span>
+<span><span class="co">#&gt;  8.74e-01      0.240 1.71e-01  0.996</span></span>
+<span><span class="co">#&gt;  1.31e-02      0.180 8.50e-05  0.750</span></span>
 <span><span class="co">#&gt; --- 22 rows omitted. See ?avg_predictions and ?print.marginaleffects --- </span></span>
-<span><span class="co">#&gt;  3.83e-01      0.296 2.50e-02  0.944</span></span>
-<span><span class="co">#&gt;  1.21e-06      0.126 8.11e-13  0.346</span></span>
-<span><span class="co">#&gt;  6.89e-03      0.145 3.10e-05  0.565</span></span>
-<span><span class="co">#&gt;  8.07e-11      0.148 2.22e-16  0.600</span></span>
-<span><span class="co">#&gt;  7.95e-01      0.162 3.68e-01  0.969</span></span>
+<span><span class="co">#&gt;  3.83e-01      0.293 1.21e-02  0.957</span></span>
+<span><span class="co">#&gt;  1.21e-06      0.140 5.46e-12  0.447</span></span>
+<span><span class="co">#&gt;  6.89e-03      0.169 2.66e-05  0.700</span></span>
+<span><span class="co">#&gt;  8.07e-11      0.161 2.22e-16  0.787</span></span>
+<span><span class="co">#&gt;  7.95e-01      0.169 3.50e-01  0.968</span></span>
 <span><span class="co">#&gt; Columns: rowid, estimate, std.error, conf.low, conf.high, vs, hp, wt, gear </span></span>
 <span><span class="co">#&gt; Type:  response</span></span>
 <span></span>
 <span><span class="va">mod</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/slopes.html">avg_slopes</a></span><span class="op">(</span>vcov <span class="op">=</span> <span class="op">~</span><span class="va">gear</span><span class="op">)</span> <span class="op">|&gt;</span> <span class="fu"><a href="https://rdrr.io/pkg/marginaleffects/man/inferences.html">inferences</a></span><span class="op">(</span>method <span class="op">=</span> <span class="st">"simulation"</span><span class="op">)</span></span>
 <span><span class="co">#&gt; </span></span>
-<span><span class="co">#&gt;  Term Contrast  Estimate Std. Error   2.5 %  97.5 %</span></span>
-<span><span class="co">#&gt;  gear    4 - 3 -3.92e-02    0.05506 -0.0914 0.12902</span></span>
-<span><span class="co">#&gt;  gear    5 - 3 -1.93e-01    0.26716 -0.4888 0.33333</span></span>
-<span><span class="co">#&gt;  hp      dY/dX -5.02e-03    0.00439 -0.0115 0.00378</span></span>
-<span><span class="co">#&gt;  wt      dY/dX -4.06e-05    0.30024 -0.5342 0.71972</span></span>
+<span><span class="co">#&gt;  Term Contrast  Estimate Std. Error   2.5 % 97.5 %</span></span>
+<span><span class="co">#&gt;  gear    4 - 3 -3.92e-02    0.05652 -0.0900 0.1301</span></span>
+<span><span class="co">#&gt;  gear    5 - 3 -1.93e-01    0.27567 -0.4868 0.3329</span></span>
+<span><span class="co">#&gt;  hp      dY/dX -5.02e-03    0.00464 -0.0114 0.0053</span></span>
+<span><span class="co">#&gt;  wt      dY/dX -4.06e-05    0.32928 -0.6744 0.7151</span></span>
 <span><span class="co">#&gt; </span></span>
 <span><span class="co">#&gt; Columns: term, contrast, estimate, std.error, conf.low, conf.high </span></span>
 <span><span class="co">#&gt; Type:  response</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
diff --git a/articles/uncertainty_files/figure-html/unnamed-chunk-6-1.png b/articles/uncertainty_files/figure-html/unnamed-chunk-6-1.png
index ce81671bf..9cf34f672 100644
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diff --git a/index.html b/index.html
index 87322c007..52b4957ec 100644
--- a/index.html
+++ b/index.html
@@ -5,7 +5,7 @@
 <meta name="generator" content="quarto-1.4.151">
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 <meta name="author" content="Vincent Arel-Bundock and contributors.">
-<title>The Marginal Effects Zoo (0.15.0)</title>
+<title>The Marginal Effects Zoo (0.15.1)</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
@@ -69,7 +69,7 @@
 gtag('js', new Date());
 gtag('config', 'G-V6MC2RPFXR', { 'anonymize_ip': true});
 </script>
-<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.0)">
+<meta name="twitter:title" content="The Marginal Effects Zoo (0.15.1)">
 <meta name="twitter:description" content="">
 <meta name="twitter:image" content="fig/marginaleffects_card.png">
 <meta name="twitter:card" content="summary_large_image">
@@ -80,7 +80,7 @@
   <header id="quarto-header" class="headroom fixed-top"><nav class="navbar navbar-expand-lg navbar-dark "><div class="navbar-container container-fluid">
       <div class="navbar-brand-container mx-auto">
     <a class="navbar-brand" href="./index.html">
-    <span class="navbar-title">The Marginal Effects Zoo (0.15.0)</span>
+    <span class="navbar-title">The Marginal Effects Zoo (0.15.1)</span>
     </a>
   </div>
             <div id="quarto-search" class="" title="Search"></div>
@@ -448,7 +448,7 @@
     </div>
 <!-- main -->
 <main class="content" id="quarto-document-content"><header id="title-block-header" class="quarto-title-block default"><div class="quarto-title">
-<h1 class="title">The Marginal Effects Zoo (0.15.0)</h1>
+<h1 class="title">The Marginal Effects Zoo (0.15.1)</h1>
 </div>
 
 
@@ -611,7 +611,7 @@ <h1 class="title">The Marginal Effects Zoo (0.15.0)</h1>
 
 To cite package 'marginaleffects' in publications use:
 
-  Arel-Bundock V (2023). _marginaleffects: Predictions, Comparisons, Slopes, Marginal Means, and Hypothesis Tests_. R package version 0.15.0.9002, &lt;https://marginaleffects.com/&gt;.
+  Arel-Bundock V (2023). _marginaleffects: Predictions, Comparisons, Slopes, Marginal Means, and Hypothesis Tests_. R package version 0.15.1, &lt;https://marginaleffects.com/&gt;.
 
 A BibTeX entry for LaTeX users is
 
@@ -620,7 +620,7 @@ <h1 class="title">The Marginal Effects Zoo (0.15.0)</h1>
 Tests},
     author = {Vincent Arel-Bundock},
     year = {2023},
-    note = {R package version 0.15.0.9002},
+    note = {R package version 0.15.1},
     url = {https://marginaleffects.com/},
   }</code></pre>
 </div>
diff --git a/search.json b/search.json
index c1d360ba7..4eb9d6c82 100644
--- a/search.json
+++ b/search.json
@@ -2,9 +2,9 @@
   {
     "objectID": "index.html",
     "href": "index.html",
-    "title": "The Marginal Effects Zoo (0.15.0)",
+    "title": "The Marginal Effects Zoo (0.15.1)",
     "section": "",
-    "text": "🚨🚨🚨 This November, the marginaleffects author (Vincent) will be giving an online seminar called: “Interpreting and Communicating Statistical Results with R.” If you want to become a marginaleffects expert and support the development of the package, sign up at Code Horizons! 🚨🚨🚨\n\n1 Why?\nInterpreting the parameters estimated by complex statistical models is often challenging. Many applied researchers are keen to report simple quantities that carry clear scientific meaning but, in doing so, they face three primary obstacles:\n\nIntuitive estimands—and their standard errors—are often tedious to compute.\nThe terminology to describe these estimands is not standardized, and varies tremendously across disciplines.\nModeling packages in R and Python produce inconsistent objects which require users to write custom (and error-prone) code to interpret statistical results.\n\nThe “Marginal Effects Zoo” book and the marginaleffects packages for R and Python are designed to help analysts overcome these challenges. The free online book provides a unified framework to describe and compute a wide range of estimands. The marginaleffects package implements this framework and offers a consistent interface to interpret the estimates from over 86 classes of statistical models.\n\n2 What?\nThe marginaleffects package allows R and Python users to compute and plot three principal quantities of interest: (1) predictions, (2) comparisons (contrasts, risk ratios, odds, lift, etc.), and (3) slopes. In addition, the package includes a convenience function to compute a fourth estimand, “marginal means”, which is a special case of averaged predictions. marginaleffects can also average (or “marginalize”) unit-level (or “conditional”) estimates of all those quantities. Finally, marginaleffects can also conduct hypothesis and eequivalence tests on coefficient estimates and on any of the quantities generated by the package.\nPredictions:\n\nThe outcome predicted by a fitted model on a specified scale for a given combination of values of the predictor variables, such as their observed values, their means, or factor levels. a.k.a. Fitted values, adjusted predictions. predictions(), avg_predictions(), plot_predictions().\n\nComparisons:\n\nCompare the predictions made by a model for different regressor values (e.g., college graduates vs. others): contrasts, differences, risk ratios, odds, lift, etc. comparisons(), avg_comparisons(), plot_comparisons().\n\nSlopes:\n\nPartial derivative of the regression equation with respect to a regressor of interest. a.k.a. Marginal effects, trends. slopes(), avg_slopes(), plot_slopes().\n\nMarginal Means:\n\nPredictions of a model, averaged across a “reference grid” of categorical predictors. marginalmeans().\n\nHypothesis and Equivalence Tests:\n\nHypothesis and equivalence tests can be conducted on linear or non-linear functions of model coefficients, or on any of the quantities computed by the marginaleffects packages (predictions, slopes, comparisons, marginal means, etc.). Uncertainy estimates can be obtained via the delta method (with or without robust standard errors), bootstrap, or simulation.\n\n\n\n\n\nGoal\nFunction\n\n\n\nPredictions\npredictions()\n\n\n\navg_predictions()\n\n\n\nplot_predictions()\n\n\nComparisons\ncomparisons()\n\n\n\navg_comparisons()\n\n\n\nplot_comparisons()\n\n\nSlopes\nslopes()\n\n\n\navg_slopes()\n\n\n\nplot_slopes()\n\n\nMarginal Means\nmarginal_means()\n\n\nGrids\ndatagrid()\n\n\n\ndatagridcf()\n\n\nHypothesis & Equivalence\nhypotheses()\n\n\nBayes, Bootstrap, Simulation\nposterior_draws()\n\n\n\ninferences()\n\n\n\n\n\n\n3 Benefits\nThe advantages of marginaleffects include:\n\n\nPowerful: It can compute predictions, comparisons (contrasts, risk ratios, etc.), slopes, and conduct hypothesis tests for 86 different classes of models in R and Python.\n\nSimple: All functions share a simple and unified interface.\n\nDocumented: Each function is thoroughly documented with abundant examples. The website includes 20,000+ words of vignettes and case studies.\n\nEfficient: Some operations are orders of magnitude faster than with the margins package, and the memory footprint is much smaller.\n\nThin: Few dependencies.\n\nStandards-compliant: marginaleffects follows “tidy” principles and returns objects that work with standard functions like summary(), head(), tidy(), and glance(). These objects are easy to program with and feed to other packages like modelsummary.\n\n\nValid: When possible, numerical results are checked against alternative software like Stata or other R packages. Unfortunately, it is not possible to test every model type, so users are still strongly encouraged to cross-check their results.\n\nExtensible: Adding support for new models is very easy, often requiring less than 10 lines of new code. Please submit feature requests on Github.\n\n\nActive development: Bugs are fixed promptly.\n\n4 License and Citation\nThe marginaleffects package is licensed under the GNU General Public License v3.0. The content of this website/book is licensed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0).\n\nWarning in citation(\"marginaleffects\"): no date field in DESCRIPTION file of package 'marginaleffects'\n\nTo cite package 'marginaleffects' in publications use:\n\n  Arel-Bundock V (2023). _marginaleffects: Predictions, Comparisons, Slopes, Marginal Means, and Hypothesis Tests_. R package version 0.15.0.9002, &lt;https://marginaleffects.com/&gt;.\n\nA BibTeX entry for LaTeX users is\n\n  @Manual{,\n    title = {marginaleffects: Predictions, Comparisons, Slopes, Marginal Means, and Hypothesis\nTests},\n    author = {Vincent Arel-Bundock},\n    year = {2023},\n    note = {R package version 0.15.0.9002},\n    url = {https://marginaleffects.com/},\n  }"
+    "text": "🚨🚨🚨 This November, the marginaleffects author (Vincent) will be giving an online seminar called: “Interpreting and Communicating Statistical Results with R.” If you want to become a marginaleffects expert and support the development of the package, sign up at Code Horizons! 🚨🚨🚨\n\n1 Why?\nInterpreting the parameters estimated by complex statistical models is often challenging. Many applied researchers are keen to report simple quantities that carry clear scientific meaning but, in doing so, they face three primary obstacles:\n\nIntuitive estimands—and their standard errors—are often tedious to compute.\nThe terminology to describe these estimands is not standardized, and varies tremendously across disciplines.\nModeling packages in R and Python produce inconsistent objects which require users to write custom (and error-prone) code to interpret statistical results.\n\nThe “Marginal Effects Zoo” book and the marginaleffects packages for R and Python are designed to help analysts overcome these challenges. The free online book provides a unified framework to describe and compute a wide range of estimands. The marginaleffects package implements this framework and offers a consistent interface to interpret the estimates from over 86 classes of statistical models.\n\n2 What?\nThe marginaleffects package allows R and Python users to compute and plot three principal quantities of interest: (1) predictions, (2) comparisons (contrasts, risk ratios, odds, lift, etc.), and (3) slopes. In addition, the package includes a convenience function to compute a fourth estimand, “marginal means”, which is a special case of averaged predictions. marginaleffects can also average (or “marginalize”) unit-level (or “conditional”) estimates of all those quantities. Finally, marginaleffects can also conduct hypothesis and eequivalence tests on coefficient estimates and on any of the quantities generated by the package.\nPredictions:\n\nThe outcome predicted by a fitted model on a specified scale for a given combination of values of the predictor variables, such as their observed values, their means, or factor levels. a.k.a. Fitted values, adjusted predictions. predictions(), avg_predictions(), plot_predictions().\n\nComparisons:\n\nCompare the predictions made by a model for different regressor values (e.g., college graduates vs. others): contrasts, differences, risk ratios, odds, lift, etc. comparisons(), avg_comparisons(), plot_comparisons().\n\nSlopes:\n\nPartial derivative of the regression equation with respect to a regressor of interest. a.k.a. Marginal effects, trends. slopes(), avg_slopes(), plot_slopes().\n\nMarginal Means:\n\nPredictions of a model, averaged across a “reference grid” of categorical predictors. marginalmeans().\n\nHypothesis and Equivalence Tests:\n\nHypothesis and equivalence tests can be conducted on linear or non-linear functions of model coefficients, or on any of the quantities computed by the marginaleffects packages (predictions, slopes, comparisons, marginal means, etc.). Uncertainy estimates can be obtained via the delta method (with or without robust standard errors), bootstrap, or simulation.\n\n\n\n\n\nGoal\nFunction\n\n\n\nPredictions\npredictions()\n\n\n\navg_predictions()\n\n\n\nplot_predictions()\n\n\nComparisons\ncomparisons()\n\n\n\navg_comparisons()\n\n\n\nplot_comparisons()\n\n\nSlopes\nslopes()\n\n\n\navg_slopes()\n\n\n\nplot_slopes()\n\n\nMarginal Means\nmarginal_means()\n\n\nGrids\ndatagrid()\n\n\n\ndatagridcf()\n\n\nHypothesis & Equivalence\nhypotheses()\n\n\nBayes, Bootstrap, Simulation\nposterior_draws()\n\n\n\ninferences()\n\n\n\n\n\n\n3 Benefits\nThe advantages of marginaleffects include:\n\n\nPowerful: It can compute predictions, comparisons (contrasts, risk ratios, etc.), slopes, and conduct hypothesis tests for 86 different classes of models in R and Python.\n\nSimple: All functions share a simple and unified interface.\n\nDocumented: Each function is thoroughly documented with abundant examples. The website includes 20,000+ words of vignettes and case studies.\n\nEfficient: Some operations are orders of magnitude faster than with the margins package, and the memory footprint is much smaller.\n\nThin: Few dependencies.\n\nStandards-compliant: marginaleffects follows “tidy” principles and returns objects that work with standard functions like summary(), head(), tidy(), and glance(). These objects are easy to program with and feed to other packages like modelsummary.\n\n\nValid: When possible, numerical results are checked against alternative software like Stata or other R packages. Unfortunately, it is not possible to test every model type, so users are still strongly encouraged to cross-check their results.\n\nExtensible: Adding support for new models is very easy, often requiring less than 10 lines of new code. Please submit feature requests on Github.\n\n\nActive development: Bugs are fixed promptly.\n\n4 License and Citation\nThe marginaleffects package is licensed under the GNU General Public License v3.0. The content of this website/book is licensed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0).\n\nWarning in citation(\"marginaleffects\"): no date field in DESCRIPTION file of package 'marginaleffects'\n\nTo cite package 'marginaleffects' in publications use:\n\n  Arel-Bundock V (2023). _marginaleffects: Predictions, Comparisons, Slopes, Marginal Means, and Hypothesis Tests_. R package version 0.15.1, &lt;https://marginaleffects.com/&gt;.\n\nA BibTeX entry for LaTeX users is\n\n  @Manual{,\n    title = {marginaleffects: Predictions, Comparisons, Slopes, Marginal Means, and Hypothesis\nTests},\n    author = {Vincent Arel-Bundock},\n    year = {2023},\n    note = {R package version 0.15.1},\n    url = {https://marginaleffects.com/},\n  }"
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     "title": "\n1  Get Started\n",
     "section": "\n1.2 Estimands: Predictions, Comparisons, and Slopes",
-    "text": "1.2 Estimands: Predictions, Comparisons, and Slopes\nThe marginaleffects package allows R users to compute and plot three principal quantities of interest: (1) predictions, (2) comparisons, and (3) slopes. In addition, the package includes a convenience function to compute a fourth estimand, “marginal means”, which is a special case of averaged predictions. marginaleffects can also average (or “marginalize”) unit-level (or “conditional”) estimates of all those quantities, and conduct hypothesis tests on them.\nPredictions:\n\nThe outcome predicted by a fitted model on a specified scale for a given combination of values of the predictor variables, such as their observed values, their means, or factor levels. a.k.a. Fitted values, adjusted predictions. predictions(), avg_predictions(), plot_predictions().\n\nComparisons:\n\nCompare the predictions made by a model for different regressor values (e.g., college graduates vs. others): contrasts, differences, risk ratios, odds, etc. comparisons(), avg_comparisons(), plot_comparisons().\n\nSlopes:\n\nPartial derivative of the regression equation with respect to a regressor of interest. a.k.a. Marginal effects, trends. slopes(), avg_slopes(), plot_slopes().\n\nMarginal Means:\n\nPredictions of a model, averaged across a “reference grid” of categorical predictors. marginalmeans().\n\nHypothesis and Equivalence Tests:\n\nHypothesis and equivalence tests can be conducted on linear or non-linear functions of model coefficients, or on any of the quantities computed by the marginaleffects packages (predictions, slopes, comparisons, marginal means, etc.). Uncertainy estimates can be obtained via the delta method (with or without robust standard errors), bootstrap, or simulation.\n\nPredictions, comparisons, and slopes are fundamentally unit-level (or “conditional”) quantities. Except in the simplest linear case, estimates will typically vary based on the values of all the regressors in a model. Each of the observations in a dataset is thus associated with its own prediction, comparison, and slope estimates. Below, we will see that it can be useful to marginalize (or “average over”) unit-level estimates to report an “average prediction”, “average comparison”, or “average slope”.\nOne ambiguous aspect of the definitions above is that the word “marginal” comes up in two different and opposite ways:\n\nIn “marginal effects,” we refer to the effect of a tiny (marginal) change in the regressor on the outcome. This is a slope, or derivative.\nIn “marginal means,” we refer to the process of marginalizing across rows of a prediction grid. This is an average, or integral.\n\nOn this website and in this package, we reserve the expression “marginal effect” to mean a “slope” or “partial derivative”.\nThe marginaleffects package includes functions to estimate, average, plot, and summarize all of the estimands described above. The objects produced by marginaleffects are “tidy”: they produce simple data frames in “long” format. They are also “standards-compliant” and work seamlessly with standard functions like summary(), head(), tidy(), and glance(), as well with external packages like modelsummary or ggplot2.\nWe now apply marginaleffects functions to compute each of the estimands described above. First, we fit a linear regression model with multiplicative interactions:\n\n\nR\nPython\n\n\n\n\nlibrary(marginaleffects)\n\nmod &lt;- lm(mpg ~ hp * wt * am, data = mtcars)\n\n\n\n\nimport polars as pl\nimport numpy as np\nimport statsmodels.formula.api as smf\nfrom marginaleffects import *\n\nmtcars = pl.read_csv(\"https://vincentarelbundock.github.io/Rdatasets/csv/datasets/mtcars.csv\")\n\nmod = smf.ols(\"mpg ~ hp * wt * am\", data = mtcars).fit()\n\n\n\n\nThen, we call the predictions() function. As noted above, predictions are unit-level estimates, so there is one specific prediction per observation. By default, the predictions() function makes one prediction per observation in the dataset that was used to fit the original model. Since mtcars has 32 rows, the predictions() outcome also has 32 rows:\n\n\nR\nPython\n\n\n\n\npre &lt;- predictions(mod)\n\nnrow(mtcars)\n\n[1] 32\n\nnrow(pre)\n\n[1] 32\n\npre\n\n\n Estimate Std. Error     z Pr(&gt;|z|)     S 2.5 % 97.5 %\n     22.5      0.884 25.44   &lt;0.001 471.7  20.8   24.2\n     20.8      1.194 17.42   &lt;0.001 223.3  18.5   23.1\n     25.3      0.709 35.66   &lt;0.001 922.7  23.9   26.7\n     20.3      0.704 28.75   &lt;0.001 601.5  18.9   21.6\n     17.0      0.712 23.88   &lt;0.001 416.2  15.6   18.4\n--- 22 rows omitted. See ?avg_predictions and ?print.marginaleffects --- \n     29.6      1.874 15.80   &lt;0.001 184.3  25.9   33.3\n     15.9      1.311 12.13   &lt;0.001 110.0  13.3   18.5\n     19.4      1.145 16.95   &lt;0.001 211.6  17.2   21.7\n     14.8      2.017  7.33   &lt;0.001  42.0  10.8   18.7\n     21.5      1.072 20.02   &lt;0.001 293.8  19.4   23.6\nColumns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, mpg, hp, wt, am \nType:  response \n\n\n\n\n\npre = predictions(mod)\n\nmtcars.shape\n\n(32, 12)\n\npre.shape\n\n(32, 20)\n\nprint(pre)\n\nshape: (32, 7)\n┌──────────┬───────────┬──────┬──────────┬──────┬──────┬───────┐\n│ Estimate ┆ Std.Error ┆ z    ┆ P(&gt;|z|)  ┆ S    ┆ 2.5% ┆ 97.5% │\n│ ---      ┆ ---       ┆ ---  ┆ ---      ┆ ---  ┆ ---  ┆ ---   │\n│ str      ┆ str       ┆ str  ┆ str      ┆ str  ┆ str  ┆ str   │\n╞══════════╪═══════════╪══════╪══════════╪══════╪══════╪═══════╡\n│ 22.5     ┆ 0.884     ┆ 25.4 ┆ 0        ┆ inf  ┆ 20.7 ┆ 24.3  │\n│ 20.8     ┆ 1.19      ┆ 17.4 ┆ 4e-15    ┆ 47.8 ┆ 18.3 ┆ 23.3  │\n│ 25.3     ┆ 0.709     ┆ 35.7 ┆ 0        ┆ inf  ┆ 23.8 ┆ 26.7  │\n│ 20.3     ┆ 0.704     ┆ 28.8 ┆ 0        ┆ inf  ┆ 18.8 ┆ 21.7  │\n│ …        ┆ …         ┆ …    ┆ …        ┆ …    ┆ …    ┆ …     │\n│ 15.9     ┆ 1.31      ┆ 12.1 ┆ 1e-11    ┆ 36.5 ┆ 13.2 ┆ 18.6  │\n│ 19.4     ┆ 1.15      ┆ 16.9 ┆ 7.33e-15 ┆ 47   ┆ 17   ┆ 21.8  │\n│ 14.8     ┆ 2.02      ┆ 7.33 ┆ 1.43e-07 ┆ 22.7 ┆ 10.6 ┆ 19    │\n│ 21.5     ┆ 1.07      ┆ 20   ┆ 2.22e-16 ┆ 52   ┆ 19.2 ┆ 23.7  │\n└──────────┴───────────┴──────┴──────────┴──────┴──────┴───────┘\n\nColumns: rowid, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nNow, we use the comparisons() function to compute the difference in predicted outcome when each of the predictors is incremented by 1 unit (one predictor at a time, holding all others constant). Once again, comparisons are unit-level quantities. And since there are 3 predictors in the model and our data has 32 rows, we obtain 96 comparisons:\n\n\nR\nPython\n\n\n\n\ncmp &lt;- comparisons(mod)\n\nnrow(cmp)\n\n[1] 96\n\ncmp\n\n\n Term Contrast Estimate Std. Error      z Pr(&gt;|z|)   S  2.5 % 97.5 %\n   am    1 - 0    0.325       1.68  0.193   0.8467 0.2  -2.97  3.622\n   am    1 - 0   -0.544       1.57 -0.347   0.7287 0.5  -3.62  2.530\n   am    1 - 0    1.201       2.35  0.511   0.6090 0.7  -3.40  5.802\n   am    1 - 0   -1.703       1.87 -0.912   0.3618 1.5  -5.36  1.957\n   am    1 - 0   -0.615       1.68 -0.366   0.7146 0.5  -3.91  2.680\n--- 86 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- \n   wt    +1      -6.518       1.88 -3.462   &lt;0.001 10.9 -10.21 -2.828\n   wt    +1      -1.653       3.74 -0.442   0.6588  0.6  -8.99  5.683\n   wt    +1      -4.520       2.47 -1.830   0.0672  3.9  -9.36  0.321\n   wt    +1       0.635       4.89  0.130   0.8966  0.2  -8.95 10.216\n   wt    +1      -6.647       1.86 -3.572   &lt;0.001 11.5 -10.29 -2.999\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\ncmp = comparisons(mod)\n\ncmp.shape\n\n(96, 25)\n\nprint(cmp)\n\nshape: (96, 9)\n┌──────┬──────────┬──────────┬───────────┬───┬──────────┬───────┬───────┬───────┐\n│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|)  ┆ S     ┆ 2.5%  ┆ 97.5% │\n│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---   ┆ ---   ┆ ---   │\n│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str      ┆ str   ┆ str   ┆ str   │\n╞══════╪══════════╪══════════╪═══════════╪═══╪══════════╪═══════╪═══════╪═══════╡\n│ wt   ┆ +1       ┆ -6.61    ┆ 1.87      ┆ … ┆ 0.00165  ┆ 9.24  ┆ -10.5 ┆ -2.76 │\n│ wt   ┆ +1       ┆ -6.61    ┆ 1.87      ┆ … ┆ 0.00165  ┆ 9.24  ┆ -10.5 ┆ -2.76 │\n│ wt   ┆ +1       ┆ -7.16    ┆ 1.8       ┆ … ┆ 0.000554 ┆ 10.8  ┆ -10.9 ┆ -3.45 │\n│ wt   ┆ +1       ┆ -3.21    ┆ 2.01      ┆ … ┆ 0.124    ┆ 3.02  ┆ -7.35 ┆ 0.939 │\n│ …    ┆ …        ┆ …        ┆ …         ┆ … ┆ …        ┆ …     ┆ …     ┆ …     │\n│ am   ┆ 1 - 0    ┆ 2.11     ┆ 2.29      ┆ … ┆ 0.367    ┆ 1.45  ┆ -2.62 ┆ 6.83  │\n│ am   ┆ 1 - 0    ┆ 0.895    ┆ 1.64      ┆ … ┆ 0.591    ┆ 0.758 ┆ -2.5  ┆ 4.29  │\n│ am   ┆ 1 - 0    ┆ 4.03     ┆ 3.24      ┆ … ┆ 0.226    ┆ 2.15  ┆ -2.66 ┆ 10.7  │\n│ am   ┆ 1 - 0    ┆ -0.237   ┆ 1.59      ┆ … ┆ 0.883    ┆ 0.18  ┆ -3.51 ┆ 3.04  │\n└──────┴──────────┴──────────┴───────────┴───┴──────────┴───────┴───────┴───────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nThe comparisons() function allows customized queries. For example, what happens to the predicted outcome when the hp variable increases from 100 to 120?\n\n\nR\nPython\n\n\n\n\ncomparisons(mod, variables = list(hp = c(120, 100)))\n\n\n Term  Contrast Estimate Std. Error      z Pr(&gt;|z|)   S  2.5 %  97.5 %\n   hp 120 - 100   -0.738      0.370 -1.995  0.04607 4.4 -1.463 -0.0129\n   hp 120 - 100   -0.574      0.313 -1.836  0.06640 3.9 -1.186  0.0388\n   hp 120 - 100   -0.931      0.452 -2.062  0.03922 4.7 -1.817 -0.0460\n   hp 120 - 100   -0.845      0.266 -3.182  0.00146 9.4 -1.366 -0.3248\n   hp 120 - 100   -0.780      0.268 -2.909  0.00362 8.1 -1.306 -0.2547\n--- 22 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- \n   hp 120 - 100   -1.451      0.705 -2.058  0.03958 4.7 -2.834 -0.0692\n   hp 120 - 100   -0.384      0.270 -1.422  0.15498 2.7 -0.912  0.1451\n   hp 120 - 100   -0.641      0.334 -1.918  0.05513 4.2 -1.297  0.0141\n   hp 120 - 100   -0.126      0.272 -0.463  0.64360 0.6 -0.659  0.4075\n   hp 120 - 100   -0.635      0.332 -1.911  0.05598 4.2 -1.286  0.0162\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\ncmp = comparisons(mod, variables = {\"hp\": [120, 100]})\nprint(cmp)\n\nshape: (32, 9)\n┌──────┬───────────┬──────────┬───────────┬───┬─────────┬───────┬───────────┬───────┐\n│ Term ┆ Contrast  ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%      ┆ 97.5% │\n│ ---  ┆ ---       ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---       ┆ ---   │\n│ str  ┆ str       ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str       ┆ str   │\n╞══════╪═══════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═══════════╪═══════╡\n│ hp   ┆ 100 - 120 ┆ 0.738    ┆ 0.37      ┆ … ┆ 0.0576  ┆ 4.12  ┆ -0.0256   ┆ 1.5   │\n│ hp   ┆ 100 - 120 ┆ 0.574    ┆ 0.313     ┆ … ┆ 0.0788  ┆ 3.67  ┆ -0.0713   ┆ 1.22  │\n│ hp   ┆ 100 - 120 ┆ 0.931    ┆ 0.452     ┆ … ┆ 0.0502  ┆ 4.32  ┆ -0.000918 ┆ 1.86  │\n│ hp   ┆ 100 - 120 ┆ 0.845    ┆ 0.266     ┆ … ┆ 0.00401 ┆ 7.96  ┆ 0.297     ┆ 1.39  │\n│ …    ┆ …         ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …         ┆ …     │\n│ hp   ┆ 100 - 120 ┆ 0.384    ┆ 0.27      ┆ … ┆ 0.168   ┆ 2.57  ┆ -0.173    ┆ 0.941 │\n│ hp   ┆ 100 - 120 ┆ 0.641    ┆ 0.334     ┆ … ┆ 0.0671  ┆ 3.9   ┆ -0.0488   ┆ 1.33  │\n│ hp   ┆ 100 - 120 ┆ 0.126    ┆ 0.272     ┆ … ┆ 0.648   ┆ 0.626 ┆ -0.436    ┆ 0.688 │\n│ hp   ┆ 100 - 120 ┆ 0.635    ┆ 0.332     ┆ … ┆ 0.068   ┆ 3.88  ┆ -0.0507   ┆ 1.32  │\n└──────┴───────────┴──────────┴───────────┴───┴─────────┴───────┴───────────┴───────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nWhat happens to the predicted outcome when the wt variable increases by 1 standard deviation about its mean?\n\n\nR\nPython\n\n\n\n\ncomparisons(mod, variables = list(hp = \"sd\"))\n\n\n Term                Contrast Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 %  97.5 %\n   hp (x + sd/2) - (x - sd/2)   -2.530      1.269 -1.995  0.04607 4.4 -5.02 -0.0441\n   hp (x + sd/2) - (x - sd/2)   -1.967      1.072 -1.836  0.06640 3.9 -4.07  0.1332\n   hp (x + sd/2) - (x - sd/2)   -3.193      1.549 -2.062  0.03922 4.7 -6.23 -0.1578\n   hp (x + sd/2) - (x - sd/2)   -2.898      0.911 -3.182  0.00146 9.4 -4.68 -1.1133\n   hp (x + sd/2) - (x - sd/2)   -2.675      0.919 -2.909  0.00362 8.1 -4.48 -0.8731\n--- 22 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- \n   hp (x + sd/2) - (x - sd/2)   -4.976      2.418 -2.058  0.03958 4.7 -9.71 -0.2373\n   hp (x + sd/2) - (x - sd/2)   -1.315      0.925 -1.422  0.15498 2.7 -3.13  0.4974\n   hp (x + sd/2) - (x - sd/2)   -2.199      1.147 -1.918  0.05513 4.2 -4.45  0.0483\n   hp (x + sd/2) - (x - sd/2)   -0.432      0.933 -0.463  0.64360 0.6 -2.26  1.3970\n   hp (x + sd/2) - (x - sd/2)   -2.177      1.139 -1.911  0.05598 4.2 -4.41  0.0556\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\ncmp = comparisons(mod, variables = {\"hp\": \"sd\"})\nprint(cmp)\n\nshape: (32, 9)\n┌──────┬────────────────────┬──────────┬───────────┬───┬─────────┬───────┬───────┬─────────┐\n│ Term ┆ Contrast           ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%  ┆ 97.5%   │\n│ ---  ┆ ---                ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---   ┆ ---     │\n│ str  ┆ str                ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str   ┆ str     │\n╞══════╪════════════════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═══════╪═════════╡\n│ hp   ┆ +68.56286848932059 ┆ -2.53    ┆ 1.27      ┆ … ┆ 0.0576  ┆ 4.12  ┆ -5.15 ┆ 0.0878  │\n│ hp   ┆ +68.56286848932059 ┆ -1.97    ┆ 1.07      ┆ … ┆ 0.0788  ┆ 3.67  ┆ -4.18 ┆ 0.245   │\n│ hp   ┆ +68.56286848932059 ┆ -3.19    ┆ 1.55      ┆ … ┆ 0.0502  ┆ 4.32  ┆ -6.39 ┆ 0.00315 │\n│ hp   ┆ +68.56286848932059 ┆ -2.9     ┆ 0.911     ┆ … ┆ 0.00401 ┆ 7.96  ┆ -4.78 ┆ -1.02   │\n│ …    ┆ …                  ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …     ┆ …       │\n│ hp   ┆ +68.56286848932059 ┆ -1.32    ┆ 0.925     ┆ … ┆ 0.168   ┆ 2.57  ┆ -3.22 ┆ 0.594   │\n│ hp   ┆ +68.56286848932059 ┆ -2.2     ┆ 1.15      ┆ … ┆ 0.0671  ┆ 3.9   ┆ -4.57 ┆ 0.167   │\n│ hp   ┆ +68.56286848932059 ┆ -0.432   ┆ 0.933     ┆ … ┆ 0.648   ┆ 0.626 ┆ -2.36 ┆ 1.49    │\n│ hp   ┆ +68.56286848932059 ┆ -2.18    ┆ 1.14      ┆ … ┆ 0.068   ┆ 3.88  ┆ -4.53 ┆ 0.174   │\n└──────┴────────────────────┴──────────┴───────────┴───┴─────────┴───────┴───────┴─────────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nThe comparisons() function also allows users to specify arbitrary functions of predictions, with the comparison argument. For example, what is the average ratio between predicted Miles per Gallon after an increase of 50 units in Horsepower?\n\n\nR\nPython\n\n\n\n\ncomparisons(\n  mod,\n  variables = list(hp = 50),\n  comparison = \"ratioavg\")\n\n\n Term  Contrast Estimate Std. Error    z Pr(&gt;|z|)     S 2.5 % 97.5 %\n   hp mean(+50)     0.91     0.0291 31.3   &lt;0.001 711.9 0.853  0.966\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted \nType:  response \n\n\n\n\n\ncmp = comparisons(\n  mod,\n  variables = {\"hp\": 50},\n  comparison = \"ratioavg\")\nprint(cmp)\n\nshape: (1, 9)\n┌──────┬──────────┬──────────┬───────────┬───┬─────────┬─────┬──────┬───────┐\n│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ --- ┆ ---  ┆ ---   │\n│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str ┆ str  ┆ str   │\n╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪═════╪══════╪═══════╡\n│ hp   ┆ +50      ┆ 0.91     ┆ 0.0291    ┆ … ┆ 0       ┆ inf ┆ 0.85 ┆ 0.97  │\n└──────┴──────────┴──────────┴───────────┴───┴─────────┴─────┴──────┴───────┘\n\nColumns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nSee the Comparisons vignette for detailed explanations and more options.\nThe slopes() function allows us to compute the partial derivative of the outcome equation with respect to each of the predictors. Once again, we obtain a data frame with 96 rows:\n\n\nR\nPython\n\n\n\n\nmfx &lt;- slopes(mod)\n\nnrow(mfx)\n\n[1] 96\n\nmfx\n\n\n Term Contrast Estimate Std. Error      z Pr(&gt;|z|)   S  2.5 % 97.5 %\n   am    1 - 0    0.325       1.68  0.193   0.8467 0.2  -2.97  3.622\n   am    1 - 0   -0.544       1.57 -0.347   0.7287 0.5  -3.62  2.530\n   am    1 - 0    1.201       2.35  0.511   0.6090 0.7  -3.40  5.802\n   am    1 - 0   -1.703       1.87 -0.912   0.3618 1.5  -5.36  1.957\n   am    1 - 0   -0.615       1.68 -0.366   0.7146 0.5  -3.91  2.680\n--- 86 rows omitted. See ?avg_slopes and ?print.marginaleffects --- \n   wt    dY/dX   -6.518       1.88 -3.462   &lt;0.001 10.9 -10.21 -2.827\n   wt    dY/dX   -1.653       3.74 -0.442   0.6588  0.6  -8.99  5.683\n   wt    dY/dX   -4.520       2.47 -1.830   0.0673  3.9  -9.36  0.321\n   wt    dY/dX    0.635       4.89  0.130   0.8966  0.2  -8.95 10.215\n   wt    dY/dX   -6.647       1.86 -3.571   &lt;0.001 11.5 -10.29 -2.999\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\nmfx = slopes(mod)\n\nmfx.shape\n\n(96, 25)\n\nprint(mfx)\n\nshape: (96, 9)\n┌──────┬──────────┬──────────┬───────────┬───┬─────────┬───────┬─────────┬──────────┐\n│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%    ┆ 97.5%    │\n│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---      │\n│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str     ┆ str      │\n╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═════════╪══════════╡\n│ hp   ┆ dY/dX    ┆ -0.0369  ┆ 0.0175    ┆ … ┆ 0.0453  ┆ 4.46  ┆ -0.073  ┆ -0.00083 │\n│ hp   ┆ dY/dX    ┆ -0.0287  ┆ 0.0176    ┆ … ┆ 0.116   ┆ 3.11  ┆ -0.065  ┆ 0.00763  │\n│ hp   ┆ dY/dX    ┆ -0.0466  ┆ 0.0226    ┆ … ┆ 0.0506  ┆ 4.3   ┆ -0.0933 ┆ 0.00014  │\n│ hp   ┆ dY/dX    ┆ -0.0423  ┆ 0.0135    ┆ … ┆ 0.00459 ┆ 7.77  ┆ -0.0702 ┆ -0.0144  │\n│ …    ┆ …        ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …       ┆ …        │\n│ wt   ┆ dY/dX    ┆ -1.65    ┆ 3.74      ┆ … ┆ 0.663   ┆ 0.593 ┆ -9.38   ┆ 6.07     │\n│ wt   ┆ dY/dX    ┆ -4.52    ┆ 2.47      ┆ … ┆ 0.0801  ┆ 3.64  ┆ -9.63   ┆ 0.586    │\n│ wt   ┆ dY/dX    ┆ 0.635    ┆ 4.88      ┆ … ┆ 0.898   ┆ 0.156 ┆ -9.45   ┆ 10.7     │\n│ wt   ┆ dY/dX    ┆ -6.65    ┆ 1.86      ┆ … ┆ 0.00154 ┆ 9.34  ┆ -10.5   ┆ -2.81    │\n└──────┴──────────┴──────────┴───────────┴───┴─────────┴───────┴─────────┴──────────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb"
+    "text": "1.2 Estimands: Predictions, Comparisons, and Slopes\nThe marginaleffects package allows R users to compute and plot three principal quantities of interest: (1) predictions, (2) comparisons, and (3) slopes. In addition, the package includes a convenience function to compute a fourth estimand, “marginal means”, which is a special case of averaged predictions. marginaleffects can also average (or “marginalize”) unit-level (or “conditional”) estimates of all those quantities, and conduct hypothesis tests on them.\nPredictions:\n\nThe outcome predicted by a fitted model on a specified scale for a given combination of values of the predictor variables, such as their observed values, their means, or factor levels. a.k.a. Fitted values, adjusted predictions. predictions(), avg_predictions(), plot_predictions().\n\nComparisons:\n\nCompare the predictions made by a model for different regressor values (e.g., college graduates vs. others): contrasts, differences, risk ratios, odds, etc. comparisons(), avg_comparisons(), plot_comparisons().\n\nSlopes:\n\nPartial derivative of the regression equation with respect to a regressor of interest. a.k.a. Marginal effects, trends. slopes(), avg_slopes(), plot_slopes().\n\nMarginal Means:\n\nPredictions of a model, averaged across a “reference grid” of categorical predictors. marginalmeans().\n\nHypothesis and Equivalence Tests:\n\nHypothesis and equivalence tests can be conducted on linear or non-linear functions of model coefficients, or on any of the quantities computed by the marginaleffects packages (predictions, slopes, comparisons, marginal means, etc.). Uncertainy estimates can be obtained via the delta method (with or without robust standard errors), bootstrap, or simulation.\n\nPredictions, comparisons, and slopes are fundamentally unit-level (or “conditional”) quantities. Except in the simplest linear case, estimates will typically vary based on the values of all the regressors in a model. Each of the observations in a dataset is thus associated with its own prediction, comparison, and slope estimates. Below, we will see that it can be useful to marginalize (or “average over”) unit-level estimates to report an “average prediction”, “average comparison”, or “average slope”.\nOne ambiguous aspect of the definitions above is that the word “marginal” comes up in two different and opposite ways:\n\nIn “marginal effects,” we refer to the effect of a tiny (marginal) change in the regressor on the outcome. This is a slope, or derivative.\nIn “marginal means,” we refer to the process of marginalizing across rows of a prediction grid. This is an average, or integral.\n\nOn this website and in this package, we reserve the expression “marginal effect” to mean a “slope” or “partial derivative”.\nThe marginaleffects package includes functions to estimate, average, plot, and summarize all of the estimands described above. The objects produced by marginaleffects are “tidy”: they produce simple data frames in “long” format. They are also “standards-compliant” and work seamlessly with standard functions like summary(), head(), tidy(), and glance(), as well with external packages like modelsummary or ggplot2.\nWe now apply marginaleffects functions to compute each of the estimands described above. First, we fit a linear regression model with multiplicative interactions:\n\n\nR\nPython\n\n\n\n\nlibrary(marginaleffects)\n\nmod &lt;- lm(mpg ~ hp * wt * am, data = mtcars)\n\n\n\n\nimport polars as pl\nimport numpy as np\nimport statsmodels.formula.api as smf\nfrom marginaleffects import *\n\nmtcars = pl.read_csv(\"https://vincentarelbundock.github.io/Rdatasets/csv/datasets/mtcars.csv\")\n\nmod = smf.ols(\"mpg ~ hp * wt * am\", data = mtcars).fit()\n\n\n\n\nThen, we call the predictions() function. As noted above, predictions are unit-level estimates, so there is one specific prediction per observation. By default, the predictions() function makes one prediction per observation in the dataset that was used to fit the original model. Since mtcars has 32 rows, the predictions() outcome also has 32 rows:\n\n\nR\nPython\n\n\n\n\npre &lt;- predictions(mod)\n\nnrow(mtcars)\n\n[1] 32\n\nnrow(pre)\n\n[1] 32\n\npre\n\n\n Estimate Std. Error     z Pr(&gt;|z|)     S 2.5 % 97.5 %\n     22.5      0.884 25.44   &lt;0.001 471.7  20.8   24.2\n     20.8      1.194 17.42   &lt;0.001 223.3  18.5   23.1\n     25.3      0.709 35.66   &lt;0.001 922.7  23.9   26.7\n     20.3      0.704 28.75   &lt;0.001 601.5  18.9   21.6\n     17.0      0.712 23.88   &lt;0.001 416.2  15.6   18.4\n--- 22 rows omitted. See ?avg_predictions and ?print.marginaleffects --- \n     29.6      1.874 15.80   &lt;0.001 184.3  25.9   33.3\n     15.9      1.311 12.13   &lt;0.001 110.0  13.3   18.5\n     19.4      1.145 16.95   &lt;0.001 211.6  17.2   21.7\n     14.8      2.017  7.33   &lt;0.001  42.0  10.8   18.7\n     21.5      1.072 20.02   &lt;0.001 293.8  19.4   23.6\nColumns: rowid, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, mpg, hp, wt, am \nType:  response \n\n\n\n\n\npre = predictions(mod)\n\nmtcars.shape\n\n(32, 12)\n\npre.shape\n\n(32, 20)\n\nprint(pre)\n\nshape: (32, 7)\n┌──────────┬───────────┬──────┬──────────┬──────┬──────┬───────┐\n│ Estimate ┆ Std.Error ┆ z    ┆ P(&gt;|z|)  ┆ S    ┆ 2.5% ┆ 97.5% │\n│ ---      ┆ ---       ┆ ---  ┆ ---      ┆ ---  ┆ ---  ┆ ---   │\n│ str      ┆ str       ┆ str  ┆ str      ┆ str  ┆ str  ┆ str   │\n╞══════════╪═══════════╪══════╪══════════╪══════╪══════╪═══════╡\n│ 22.5     ┆ 0.884     ┆ 25.4 ┆ 0        ┆ inf  ┆ 20.7 ┆ 24.3  │\n│ 20.8     ┆ 1.19      ┆ 17.4 ┆ 4e-15    ┆ 47.8 ┆ 18.3 ┆ 23.3  │\n│ 25.3     ┆ 0.709     ┆ 35.7 ┆ 0        ┆ inf  ┆ 23.8 ┆ 26.7  │\n│ 20.3     ┆ 0.704     ┆ 28.8 ┆ 0        ┆ inf  ┆ 18.8 ┆ 21.7  │\n│ …        ┆ …         ┆ …    ┆ …        ┆ …    ┆ …    ┆ …     │\n│ 15.9     ┆ 1.31      ┆ 12.1 ┆ 1e-11    ┆ 36.5 ┆ 13.2 ┆ 18.6  │\n│ 19.4     ┆ 1.15      ┆ 16.9 ┆ 7.33e-15 ┆ 47   ┆ 17   ┆ 21.8  │\n│ 14.8     ┆ 2.02      ┆ 7.33 ┆ 1.43e-07 ┆ 22.7 ┆ 10.6 ┆ 19    │\n│ 21.5     ┆ 1.07      ┆ 20   ┆ 2.22e-16 ┆ 52   ┆ 19.2 ┆ 23.7  │\n└──────────┴───────────┴──────┴──────────┴──────┴──────┴───────┘\n\nColumns: rowid, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nNow, we use the comparisons() function to compute the difference in predicted outcome when each of the predictors is incremented by 1 unit (one predictor at a time, holding all others constant). Once again, comparisons are unit-level quantities. And since there are 3 predictors in the model and our data has 32 rows, we obtain 96 comparisons:\n\n\nR\nPython\n\n\n\n\ncmp &lt;- comparisons(mod)\n\nnrow(cmp)\n\n[1] 96\n\ncmp\n\n\n Term Contrast Estimate Std. Error      z Pr(&gt;|z|)   S  2.5 % 97.5 %\n   am    1 - 0    0.325       1.68  0.193   0.8467 0.2  -2.97  3.622\n   am    1 - 0   -0.544       1.57 -0.347   0.7287 0.5  -3.62  2.530\n   am    1 - 0    1.201       2.35  0.511   0.6090 0.7  -3.40  5.802\n   am    1 - 0   -1.703       1.87 -0.912   0.3618 1.5  -5.36  1.957\n   am    1 - 0   -0.615       1.68 -0.366   0.7146 0.5  -3.91  2.680\n--- 86 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- \n   wt    +1      -6.518       1.88 -3.462   &lt;0.001 10.9 -10.21 -2.828\n   wt    +1      -1.653       3.74 -0.442   0.6588  0.6  -8.99  5.683\n   wt    +1      -4.520       2.47 -1.830   0.0672  3.9  -9.36  0.321\n   wt    +1       0.635       4.89  0.130   0.8966  0.2  -8.95 10.216\n   wt    +1      -6.647       1.86 -3.572   &lt;0.001 11.5 -10.29 -2.999\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\ncmp = comparisons(mod)\n\ncmp.shape\n\n(96, 25)\n\nprint(cmp)\n\nshape: (96, 9)\n┌──────┬──────────┬──────────┬───────────┬───┬─────────┬───────┬─────────┬─────────┐\n│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%    ┆ 97.5%   │\n│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---     │\n│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str     ┆ str     │\n╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═════════╪═════════╡\n│ hp   ┆ +1       ┆ -0.0369  ┆ 0.0185    ┆ … ┆ 0.0575  ┆ 4.12  ┆ -0.0751 ┆ 0.00128 │\n│ hp   ┆ +1       ┆ -0.0287  ┆ 0.0156    ┆ … ┆ 0.0788  ┆ 3.67  ┆ -0.0609 ┆ 0.00357 │\n│ hp   ┆ +1       ┆ -0.0466  ┆ 0.0226    ┆ … ┆ 0.0502  ┆ 4.32  ┆ -0.0932 ┆ 4.6e-05 │\n│ hp   ┆ +1       ┆ -0.0423  ┆ 0.0133    ┆ … ┆ 0.00401 ┆ 7.96  ┆ -0.0697 ┆ -0.0149 │\n│ …    ┆ …        ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …       ┆ …       │\n│ am   ┆ 1 - 0    ┆ 2.11     ┆ 2.29      ┆ … ┆ 0.367   ┆ 1.45  ┆ -2.62   ┆ 6.83    │\n│ am   ┆ 1 - 0    ┆ 0.895    ┆ 1.64      ┆ … ┆ 0.591   ┆ 0.758 ┆ -2.5    ┆ 4.29    │\n│ am   ┆ 1 - 0    ┆ 4.03     ┆ 3.24      ┆ … ┆ 0.226   ┆ 2.15  ┆ -2.66   ┆ 10.7    │\n│ am   ┆ 1 - 0    ┆ -0.237   ┆ 1.59      ┆ … ┆ 0.883   ┆ 0.18  ┆ -3.51   ┆ 3.04    │\n└──────┴──────────┴──────────┴───────────┴───┴─────────┴───────┴─────────┴─────────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nThe comparisons() function allows customized queries. For example, what happens to the predicted outcome when the hp variable increases from 100 to 120?\n\n\nR\nPython\n\n\n\n\ncomparisons(mod, variables = list(hp = c(120, 100)))\n\n\n Term  Contrast Estimate Std. Error      z Pr(&gt;|z|)   S  2.5 %  97.5 %\n   hp 120 - 100   -0.738      0.370 -1.995  0.04607 4.4 -1.463 -0.0129\n   hp 120 - 100   -0.574      0.313 -1.836  0.06640 3.9 -1.186  0.0388\n   hp 120 - 100   -0.931      0.452 -2.062  0.03922 4.7 -1.817 -0.0460\n   hp 120 - 100   -0.845      0.266 -3.182  0.00146 9.4 -1.366 -0.3248\n   hp 120 - 100   -0.780      0.268 -2.909  0.00362 8.1 -1.306 -0.2547\n--- 22 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- \n   hp 120 - 100   -1.451      0.705 -2.058  0.03958 4.7 -2.834 -0.0692\n   hp 120 - 100   -0.384      0.270 -1.422  0.15498 2.7 -0.912  0.1451\n   hp 120 - 100   -0.641      0.334 -1.918  0.05513 4.2 -1.297  0.0141\n   hp 120 - 100   -0.126      0.272 -0.463  0.64360 0.6 -0.659  0.4075\n   hp 120 - 100   -0.635      0.332 -1.911  0.05598 4.2 -1.286  0.0162\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\ncmp = comparisons(mod, variables = {\"hp\": [120, 100]})\nprint(cmp)\n\nshape: (32, 9)\n┌──────┬───────────┬──────────┬───────────┬───┬─────────┬───────┬───────────┬───────┐\n│ Term ┆ Contrast  ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%      ┆ 97.5% │\n│ ---  ┆ ---       ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---       ┆ ---   │\n│ str  ┆ str       ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str       ┆ str   │\n╞══════╪═══════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═══════════╪═══════╡\n│ hp   ┆ 100 - 120 ┆ 0.738    ┆ 0.37      ┆ … ┆ 0.0576  ┆ 4.12  ┆ -0.0256   ┆ 1.5   │\n│ hp   ┆ 100 - 120 ┆ 0.574    ┆ 0.313     ┆ … ┆ 0.0788  ┆ 3.67  ┆ -0.0713   ┆ 1.22  │\n│ hp   ┆ 100 - 120 ┆ 0.931    ┆ 0.452     ┆ … ┆ 0.0502  ┆ 4.32  ┆ -0.000918 ┆ 1.86  │\n│ hp   ┆ 100 - 120 ┆ 0.845    ┆ 0.266     ┆ … ┆ 0.00401 ┆ 7.96  ┆ 0.297     ┆ 1.39  │\n│ …    ┆ …         ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …         ┆ …     │\n│ hp   ┆ 100 - 120 ┆ 0.384    ┆ 0.27      ┆ … ┆ 0.168   ┆ 2.57  ┆ -0.173    ┆ 0.941 │\n│ hp   ┆ 100 - 120 ┆ 0.641    ┆ 0.334     ┆ … ┆ 0.0671  ┆ 3.9   ┆ -0.0488   ┆ 1.33  │\n│ hp   ┆ 100 - 120 ┆ 0.126    ┆ 0.272     ┆ … ┆ 0.648   ┆ 0.626 ┆ -0.436    ┆ 0.688 │\n│ hp   ┆ 100 - 120 ┆ 0.635    ┆ 0.332     ┆ … ┆ 0.068   ┆ 3.88  ┆ -0.0507   ┆ 1.32  │\n└──────┴───────────┴──────────┴───────────┴───┴─────────┴───────┴───────────┴───────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nWhat happens to the predicted outcome when the wt variable increases by 1 standard deviation about its mean?\n\n\nR\nPython\n\n\n\n\ncomparisons(mod, variables = list(hp = \"sd\"))\n\n\n Term                Contrast Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 %  97.5 %\n   hp (x + sd/2) - (x - sd/2)   -2.530      1.269 -1.995  0.04607 4.4 -5.02 -0.0441\n   hp (x + sd/2) - (x - sd/2)   -1.967      1.072 -1.836  0.06640 3.9 -4.07  0.1332\n   hp (x + sd/2) - (x - sd/2)   -3.193      1.549 -2.062  0.03922 4.7 -6.23 -0.1578\n   hp (x + sd/2) - (x - sd/2)   -2.898      0.911 -3.182  0.00146 9.4 -4.68 -1.1133\n   hp (x + sd/2) - (x - sd/2)   -2.675      0.919 -2.909  0.00362 8.1 -4.48 -0.8731\n--- 22 rows omitted. See ?avg_comparisons and ?print.marginaleffects --- \n   hp (x + sd/2) - (x - sd/2)   -4.976      2.418 -2.058  0.03958 4.7 -9.71 -0.2373\n   hp (x + sd/2) - (x - sd/2)   -1.315      0.925 -1.422  0.15498 2.7 -3.13  0.4974\n   hp (x + sd/2) - (x - sd/2)   -2.199      1.147 -1.918  0.05513 4.2 -4.45  0.0483\n   hp (x + sd/2) - (x - sd/2)   -0.432      0.933 -0.463  0.64360 0.6 -2.26  1.3970\n   hp (x + sd/2) - (x - sd/2)   -2.177      1.139 -1.911  0.05598 4.2 -4.41  0.0556\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\ncmp = comparisons(mod, variables = {\"hp\": \"sd\"})\nprint(cmp)\n\nshape: (32, 9)\n┌──────┬────────────────────┬──────────┬───────────┬───┬─────────┬───────┬───────┬─────────┐\n│ Term ┆ Contrast           ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%  ┆ 97.5%   │\n│ ---  ┆ ---                ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---   ┆ ---     │\n│ str  ┆ str                ┆ str      ┆ str       ┆   ┆ str     ┆ str   ┆ str   ┆ str     │\n╞══════╪════════════════════╪══════════╪═══════════╪═══╪═════════╪═══════╪═══════╪═════════╡\n│ hp   ┆ +68.56286848932059 ┆ -2.53    ┆ 1.27      ┆ … ┆ 0.0576  ┆ 4.12  ┆ -5.15 ┆ 0.0878  │\n│ hp   ┆ +68.56286848932059 ┆ -1.97    ┆ 1.07      ┆ … ┆ 0.0788  ┆ 3.67  ┆ -4.18 ┆ 0.245   │\n│ hp   ┆ +68.56286848932059 ┆ -3.19    ┆ 1.55      ┆ … ┆ 0.0502  ┆ 4.32  ┆ -6.39 ┆ 0.00315 │\n│ hp   ┆ +68.56286848932059 ┆ -2.9     ┆ 0.911     ┆ … ┆ 0.00401 ┆ 7.96  ┆ -4.78 ┆ -1.02   │\n│ …    ┆ …                  ┆ …        ┆ …         ┆ … ┆ …       ┆ …     ┆ …     ┆ …       │\n│ hp   ┆ +68.56286848932059 ┆ -1.32    ┆ 0.925     ┆ … ┆ 0.168   ┆ 2.57  ┆ -3.22 ┆ 0.594   │\n│ hp   ┆ +68.56286848932059 ┆ -2.2     ┆ 1.15      ┆ … ┆ 0.0671  ┆ 3.9   ┆ -4.57 ┆ 0.167   │\n│ hp   ┆ +68.56286848932059 ┆ -0.432   ┆ 0.933     ┆ … ┆ 0.648   ┆ 0.626 ┆ -2.36 ┆ 1.49    │\n│ hp   ┆ +68.56286848932059 ┆ -2.18    ┆ 1.14      ┆ … ┆ 0.068   ┆ 3.88  ┆ -4.53 ┆ 0.174   │\n└──────┴────────────────────┴──────────┴───────────┴───┴─────────┴───────┴───────┴─────────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb\n\n\n\n\n\nThe comparisons() function also allows users to specify arbitrary functions of predictions, with the comparison argument. For example, what is the average ratio between predicted Miles per Gallon after an increase of 50 units in Horsepower?\n\n\nR\nPython\n\n\n\n\ncomparisons(\n  mod,\n  variables = list(hp = 50),\n  comparison = \"ratioavg\")\n\n\n Term  Contrast Estimate Std. Error    z Pr(&gt;|z|)     S 2.5 % 97.5 %\n   hp mean(+50)     0.91     0.0291 31.3   &lt;0.001 711.9 0.853  0.966\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted \nType:  response \n\n\n\n\n\ncmp = comparisons(\n  mod,\n  variables = {\"hp\": 50},\n  comparison = \"ratioavg\")\nprint(cmp)\n\nshape: (1, 9)\n┌──────┬──────────┬──────────┬───────────┬───┬─────────┬─────┬──────┬───────┐\n│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ --- ┆ ---  ┆ ---   │\n│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str ┆ str  ┆ str   │\n╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪═════╪══════╪═══════╡\n│ hp   ┆ +50      ┆ 0.91     ┆ 0.0291    ┆ … ┆ 0       ┆ inf ┆ 0.85 ┆ 0.97  │\n└──────┴──────────┴──────────┴───────────┴───┴─────────┴─────┴──────┴───────┘\n\nColumns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nSee the Comparisons vignette for detailed explanations and more options.\nThe slopes() function allows us to compute the partial derivative of the outcome equation with respect to each of the predictors. Once again, we obtain a data frame with 96 rows:\n\n\nR\nPython\n\n\n\n\nmfx &lt;- slopes(mod)\n\nnrow(mfx)\n\n[1] 96\n\nmfx\n\n\n Term Contrast Estimate Std. Error      z Pr(&gt;|z|)   S  2.5 % 97.5 %\n   am    1 - 0    0.325       1.68  0.193   0.8467 0.2  -2.97  3.622\n   am    1 - 0   -0.544       1.57 -0.347   0.7287 0.5  -3.62  2.530\n   am    1 - 0    1.201       2.35  0.511   0.6090 0.7  -3.40  5.802\n   am    1 - 0   -1.703       1.87 -0.912   0.3618 1.5  -5.36  1.957\n   am    1 - 0   -0.615       1.68 -0.366   0.7146 0.5  -3.91  2.680\n--- 86 rows omitted. See ?avg_slopes and ?print.marginaleffects --- \n   wt    dY/dX   -6.518       1.88 -3.462   &lt;0.001 10.9 -10.21 -2.827\n   wt    dY/dX   -1.653       3.74 -0.442   0.6588  0.6  -8.99  5.683\n   wt    dY/dX   -4.520       2.47 -1.830   0.0673  3.9  -9.36  0.321\n   wt    dY/dX    0.635       4.89  0.130   0.8966  0.2  -8.95 10.215\n   wt    dY/dX   -6.647       1.86 -3.571   &lt;0.001 11.5 -10.29 -2.999\nColumns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted, mpg, hp, wt, am \nType:  response \n\n\n\n\n\nmfx = slopes(mod)\n\nmfx.shape\n\n(96, 25)\n\nprint(mfx)\n\nshape: (96, 9)\n┌──────┬──────────┬───────────┬───────────┬───┬─────────┬───────┬───────────┬──────────┐\n│ Term ┆ Contrast ┆ Estimate  ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%      ┆ 97.5%    │\n│ ---  ┆ ---      ┆ ---       ┆ ---       ┆   ┆ ---     ┆ ---   ┆ ---       ┆ ---      │\n│ str  ┆ str      ┆ str       ┆ str       ┆   ┆ str     ┆ str   ┆ str       ┆ str      │\n╞══════╪══════════╪═══════════╪═══════════╪═══╪═════════╪═══════╪═══════════╪══════════╡\n│ am   ┆ dY/dX    ┆ 3.25e+03  ┆ 1.68e+04  ┆ … ┆ 0.848   ┆ 0.237 ┆ -3.15e+04 ┆ 3.8e+04  │\n│ am   ┆ dY/dX    ┆ -5.44e+03 ┆ 1.57e+04  ┆ … ┆ 0.732   ┆ 0.451 ┆ -3.78e+04 ┆ 2.69e+04 │\n│ am   ┆ dY/dX    ┆ 1.2e+04   ┆ 2.35e+04  ┆ … ┆ 0.614   ┆ 0.704 ┆ -3.64e+04 ┆ 6.05e+04 │\n│ am   ┆ dY/dX    ┆ -1.7e+04  ┆ 1.87e+04  ┆ … ┆ 0.371   ┆ 1.43  ┆ -5.56e+04 ┆ 2.15e+04 │\n│ …    ┆ …        ┆ …         ┆ …         ┆ … ┆ …       ┆ …     ┆ …         ┆ …        │\n│ wt   ┆ dY/dX    ┆ -1.65     ┆ 3.74      ┆ … ┆ 0.663   ┆ 0.593 ┆ -9.38     ┆ 6.07     │\n│ wt   ┆ dY/dX    ┆ -4.52     ┆ 2.47      ┆ … ┆ 0.0801  ┆ 3.64  ┆ -9.63     ┆ 0.586    │\n│ wt   ┆ dY/dX    ┆ 0.635     ┆ 4.88      ┆ … ┆ 0.898   ┆ 0.156 ┆ -9.45     ┆ 10.7     │\n│ wt   ┆ dY/dX    ┆ -6.65     ┆ 1.86      ┆ … ┆ 0.00154 ┆ 9.34  ┆ -10.5     ┆ -2.81    │\n└──────┴──────────┴───────────┴───────────┴───┴─────────┴───────┴───────────┴──────────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, rownames, mpg, cyl, disp, hp, drat, wt, qsec, vs, am, gear, carb"
   },
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     "objectID": "articles/marginaleffects.html#grid",
@@ -32,14 +32,14 @@
     "href": "articles/marginaleffects.html#averaging",
     "title": "\n1  Get Started\n",
     "section": "\n1.4 Averaging",
-    "text": "1.4 Averaging\nSince predictions, comparisons, and slopes are conditional quantities, they can be a bit unwieldy. Often, it can be useful to report a one-number summary instead of one estimate per observation. Instead of presenting “conditional” estimates, some methodologists recommend reporting “marginal” estimates, that is, an average of unit-level estimates.\n(This use of the word “marginal” as “averaging” should not be confused with the term “marginal effect” which, in the econometrics tradition, corresponds to a partial derivative, or the effect of a “small/marginal” change.)\nTo marginalize (average over) our unit-level estimates, we can use the by argument or the one of the convenience functions: avg_predictions(), avg_comparisons(), or avg_slopes(). For example, both of these commands give us the same result: the average predicted outcome in the mtcars dataset:\n\n\nR\nPython\n\n\n\n\navg_predictions(mod)\n\n\n Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n     20.1       0.39 51.5   &lt;0.001 Inf  19.3   20.9\n\nColumns: estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\np = avg_predictions(mod)\nprint(p)\n\nshape: (1, 7)\n┌──────────┬───────────┬──────┬─────────┬─────┬──────┬───────┐\n│ Estimate ┆ Std.Error ┆ z    ┆ P(&gt;|z|) ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---      ┆ ---       ┆ ---  ┆ ---     ┆ --- ┆ ---  ┆ ---   │\n│ str      ┆ str       ┆ str  ┆ str     ┆ str ┆ str  ┆ str   │\n╞══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡\n│ 20.1     ┆ 0.39      ┆ 51.5 ┆ 0       ┆ inf ┆ 19.3 ┆ 20.9  │\n└──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘\n\nColumns: estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nThis is equivalent to manual computation by:\n\n\nR\nPython\n\n\n\n\nmean(predict(mod))\n\n[1] 20.09062\n\n\n\n\n\nnp.mean(mod.predict())\n\n20.090624999999992\n\n\n\n\n\nThe main marginaleffects functions all include a by argument, which allows us to marginalize within sub-groups of the data. For example,\n\n\nR\nPython\n\n\n\n\navg_comparisons(mod, by = \"am\")\n\n\n Term          Contrast am Estimate Std. Error      z Pr(&gt;|z|)   S   2.5 %   97.5 %\n   am mean(1) - mean(0)  0  -1.3830     2.5250 -0.548  0.58388 0.8 -6.3319  3.56589\n   am mean(1) - mean(0)  1   1.9029     2.3086  0.824  0.40980 1.3 -2.6219  6.42773\n   hp mean(+1)           0  -0.0343     0.0159 -2.160  0.03079 5.0 -0.0654 -0.00317\n   hp mean(+1)           1  -0.0436     0.0213 -2.050  0.04039 4.6 -0.0854 -0.00191\n   wt mean(+1)           0  -2.4799     1.2316 -2.014  0.04406 4.5 -4.8939 -0.06595\n   wt mean(+1)           1  -6.0718     1.9762 -3.072  0.00212 8.9 -9.9451 -2.19846\n\nColumns: term, contrast, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted \nType:  response \n\n\n\n\n\ncmp = avg_comparisons(mod, by = \"am\")\nprint(cmp)\n\nshape: (6, 10)\n┌─────┬──────┬───────────────────┬──────────┬───┬─────────┬───────┬─────────┬──────────┐\n│ am  ┆ Term ┆ Contrast          ┆ Estimate ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%    ┆ 97.5%    │\n│ --- ┆ ---  ┆ ---               ┆ ---      ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---      │\n│ str ┆ str  ┆ str               ┆ str      ┆   ┆ str     ┆ str   ┆ str     ┆ str      │\n╞═════╪══════╪═══════════════════╪══════════╪═══╪═════════╪═══════╪═════════╪══════════╡\n│ 1   ┆ am   ┆ mean(1) - mean(0) ┆ 1.9      ┆ … ┆ 0.418   ┆ 1.26  ┆ -2.86   ┆ 6.67     │\n│ 0   ┆ am   ┆ mean(1) - mean(0) ┆ -1.38    ┆ … ┆ 0.589   ┆ 0.764 ┆ -6.59   ┆ 3.83     │\n│ 1   ┆ wt   ┆ +1                ┆ -6.07    ┆ … ┆ 0.00522 ┆ 7.58  ┆ -10.2   ┆ -1.99    │\n│ 0   ┆ wt   ┆ +1                ┆ -2.48    ┆ … ┆ 0.0554  ┆ 4.17  ┆ -5.02   ┆ 0.0621   │\n│ 1   ┆ hp   ┆ +1                ┆ -0.0436  ┆ … ┆ 0.0515  ┆ 4.28  ┆ -0.0876 ┆ 0.0003   │\n│ 0   ┆ hp   ┆ +1                ┆ -0.0343  ┆ … ┆ 0.041   ┆ 4.61  ┆ -0.067  ┆ -0.00152 │\n└─────┴──────┴───────────────────┴──────────┴───┴─────────┴───────┴─────────┴──────────┘\n\nColumns: am, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nMarginal Means are a special case of predictions, which are marginalized (or averaged) across a balanced grid of categorical predictors. To illustrate, we estimate a new model with categorical predictors:\n\n\nR\nPython\n\n\n\n\ndat &lt;- mtcars\ndat$am &lt;- as.logical(dat$am)\ndat$cyl &lt;- as.factor(dat$cyl)\nmod_cat &lt;- lm(mpg ~ am + cyl + hp, data = dat)\n\n\n\n\ndat = mtcars \\\n  .with_columns(pl.col(\"am\").cast(pl.Boolean),\n                pl.col(\"cyl\").cast(pl.Utf8))\nmod_cat = smf.ols('mpg ~ am + cyl + hp', data=dat).fit()\n\n\n\n\nWe can compute marginal means manually using the functions already described:\n\n\nR\nPython\n\n\n\n\navg_predictions(\n  mod_cat,\n  newdata = datagrid(cyl = unique, am = unique),\n  by = \"am\")\n\n\n    am Estimate Std. Error    z Pr(&gt;|z|)     S 2.5 % 97.5 %\n FALSE     18.3      0.785 23.3   &lt;0.001 397.4  16.8   19.9\n  TRUE     22.5      0.834 26.9   &lt;0.001 528.6  20.8   24.1\n\nColumns: am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\np = avg_predictions(\n  mod_cat,\n  newdata = datagrid(mod_cat, cyl = dat[\"cyl\"].unique(), am = dat[\"am\"].unique()),\n  by = \"am\")\nprint(p)\n\nshape: (2, 8)\n┌───────┬──────────┬───────────┬──────┬─────────┬─────┬──────┬───────┐\n│ am    ┆ Estimate ┆ Std.Error ┆ z    ┆ P(&gt;|z|) ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---   ┆ ---      ┆ ---       ┆ ---  ┆ ---     ┆ --- ┆ ---  ┆ ---   │\n│ bool  ┆ str      ┆ str       ┆ str  ┆ str     ┆ str ┆ str  ┆ str   │\n╞═══════╪══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡\n│ true  ┆ 22.5     ┆ 0.834     ┆ 26.9 ┆ 0       ┆ inf ┆ 20.8 ┆ 24.2  │\n│ false ┆ 18.3     ┆ 0.785     ┆ 23.3 ┆ 0       ┆ inf ┆ 16.7 ┆ 19.9  │\n└───────┴──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘\n\nColumns: am, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nFor convenience, the marginaleffects package for R also includes a marginal_means() function:\n\nmarginal_means(mod_cat, variables = \"am\")\n\n\n Term Value Mean Std. Error    z Pr(&gt;|z|)     S 2.5 % 97.5 %\n   am FALSE 18.3      0.785 23.3   &lt;0.001 397.4  16.8   19.9\n   am  TRUE 22.5      0.834 26.9   &lt;0.001 528.6  20.8   24.1\n\nResults averaged over levels of: cyl, am \nColumns: term, value, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\nThe Marginal Means vignette offers more detail."
+    "text": "1.4 Averaging\nSince predictions, comparisons, and slopes are conditional quantities, they can be a bit unwieldy. Often, it can be useful to report a one-number summary instead of one estimate per observation. Instead of presenting “conditional” estimates, some methodologists recommend reporting “marginal” estimates, that is, an average of unit-level estimates.\n(This use of the word “marginal” as “averaging” should not be confused with the term “marginal effect” which, in the econometrics tradition, corresponds to a partial derivative, or the effect of a “small/marginal” change.)\nTo marginalize (average over) our unit-level estimates, we can use the by argument or the one of the convenience functions: avg_predictions(), avg_comparisons(), or avg_slopes(). For example, both of these commands give us the same result: the average predicted outcome in the mtcars dataset:\n\n\nR\nPython\n\n\n\n\navg_predictions(mod)\n\n\n Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n     20.1       0.39 51.5   &lt;0.001 Inf  19.3   20.9\n\nColumns: estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\np = avg_predictions(mod)\nprint(p)\n\nshape: (1, 7)\n┌──────────┬───────────┬──────┬─────────┬─────┬──────┬───────┐\n│ Estimate ┆ Std.Error ┆ z    ┆ P(&gt;|z|) ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---      ┆ ---       ┆ ---  ┆ ---     ┆ --- ┆ ---  ┆ ---   │\n│ str      ┆ str       ┆ str  ┆ str     ┆ str ┆ str  ┆ str   │\n╞══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡\n│ 20.1     ┆ 0.39      ┆ 51.5 ┆ 0       ┆ inf ┆ 19.3 ┆ 20.9  │\n└──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘\n\nColumns: estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nThis is equivalent to manual computation by:\n\n\nR\nPython\n\n\n\n\nmean(predict(mod))\n\n[1] 20.09062\n\n\n\n\n\nnp.mean(mod.predict())\n\n20.090624999999992\n\n\n\n\n\nThe main marginaleffects functions all include a by argument, which allows us to marginalize within sub-groups of the data. For example,\n\n\nR\nPython\n\n\n\n\navg_comparisons(mod, by = \"am\")\n\n\n Term          Contrast am Estimate Std. Error      z Pr(&gt;|z|)   S   2.5 %   97.5 %\n   am mean(1) - mean(0)  0  -1.3830     2.5250 -0.548  0.58388 0.8 -6.3319  3.56589\n   am mean(1) - mean(0)  1   1.9029     2.3086  0.824  0.40980 1.3 -2.6219  6.42773\n   hp mean(+1)           0  -0.0343     0.0159 -2.160  0.03079 5.0 -0.0654 -0.00317\n   hp mean(+1)           1  -0.0436     0.0213 -2.050  0.04039 4.6 -0.0854 -0.00191\n   wt mean(+1)           0  -2.4799     1.2316 -2.014  0.04406 4.5 -4.8939 -0.06595\n   wt mean(+1)           1  -6.0718     1.9762 -3.072  0.00212 8.9 -9.9451 -2.19846\n\nColumns: term, contrast, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted_lo, predicted_hi, predicted \nType:  response \n\n\n\n\n\ncmp = avg_comparisons(mod, by = \"am\")\nprint(cmp)\n\nshape: (6, 10)\n┌─────┬──────┬───────────────────┬──────────┬───┬─────────┬───────┬─────────┬──────────┐\n│ am  ┆ Term ┆ Contrast          ┆ Estimate ┆ … ┆ P(&gt;|z|) ┆ S     ┆ 2.5%    ┆ 97.5%    │\n│ --- ┆ ---  ┆ ---               ┆ ---      ┆   ┆ ---     ┆ ---   ┆ ---     ┆ ---      │\n│ str ┆ str  ┆ str               ┆ str      ┆   ┆ str     ┆ str   ┆ str     ┆ str      │\n╞═════╪══════╪═══════════════════╪══════════╪═══╪═════════╪═══════╪═════════╪══════════╡\n│ 1   ┆ hp   ┆ +1                ┆ -0.0436  ┆ … ┆ 0.0515  ┆ 4.28  ┆ -0.0876 ┆ 0.0003   │\n│ 0   ┆ hp   ┆ +1                ┆ -0.0343  ┆ … ┆ 0.041   ┆ 4.61  ┆ -0.067  ┆ -0.00152 │\n│ 1   ┆ am   ┆ mean(1) - mean(0) ┆ 1.9      ┆ … ┆ 0.418   ┆ 1.26  ┆ -2.86   ┆ 6.67     │\n│ 0   ┆ am   ┆ mean(1) - mean(0) ┆ -1.38    ┆ … ┆ 0.589   ┆ 0.764 ┆ -6.59   ┆ 3.83     │\n│ 1   ┆ wt   ┆ +1                ┆ -6.07    ┆ … ┆ 0.00522 ┆ 7.58  ┆ -10.2   ┆ -1.99    │\n│ 0   ┆ wt   ┆ +1                ┆ -2.48    ┆ … ┆ 0.0554  ┆ 4.17  ┆ -5.02   ┆ 0.0621   │\n└─────┴──────┴───────────────────┴──────────┴───┴─────────┴───────┴─────────┴──────────┘\n\nColumns: am, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nMarginal Means are a special case of predictions, which are marginalized (or averaged) across a balanced grid of categorical predictors. To illustrate, we estimate a new model with categorical predictors:\n\n\nR\nPython\n\n\n\n\ndat &lt;- mtcars\ndat$am &lt;- as.logical(dat$am)\ndat$cyl &lt;- as.factor(dat$cyl)\nmod_cat &lt;- lm(mpg ~ am + cyl + hp, data = dat)\n\n\n\n\ndat = mtcars \\\n  .with_columns(pl.col(\"am\").cast(pl.Boolean),\n                pl.col(\"cyl\").cast(pl.Utf8))\nmod_cat = smf.ols('mpg ~ am + cyl + hp', data=dat).fit()\n\n\n\n\nWe can compute marginal means manually using the functions already described:\n\n\nR\nPython\n\n\n\n\navg_predictions(\n  mod_cat,\n  newdata = datagrid(cyl = unique, am = unique),\n  by = \"am\")\n\n\n    am Estimate Std. Error    z Pr(&gt;|z|)     S 2.5 % 97.5 %\n FALSE     18.3      0.785 23.3   &lt;0.001 397.4  16.8   19.9\n  TRUE     22.5      0.834 26.9   &lt;0.001 528.6  20.8   24.1\n\nColumns: am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\np = avg_predictions(\n  mod_cat,\n  newdata = datagrid(mod_cat, cyl = dat[\"cyl\"].unique(), am = dat[\"am\"].unique()),\n  by = \"am\")\nprint(p)\n\nshape: (2, 8)\n┌───────┬──────────┬───────────┬──────┬─────────┬─────┬──────┬───────┐\n│ am    ┆ Estimate ┆ Std.Error ┆ z    ┆ P(&gt;|z|) ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---   ┆ ---      ┆ ---       ┆ ---  ┆ ---     ┆ --- ┆ ---  ┆ ---   │\n│ bool  ┆ str      ┆ str       ┆ str  ┆ str     ┆ str ┆ str  ┆ str   │\n╞═══════╪══════════╪═══════════╪══════╪═════════╪═════╪══════╪═══════╡\n│ true  ┆ 22.5     ┆ 0.834     ┆ 26.9 ┆ 0       ┆ inf ┆ 20.8 ┆ 24.2  │\n│ false ┆ 18.3     ┆ 0.785     ┆ 23.3 ┆ 0       ┆ inf ┆ 16.7 ┆ 19.9  │\n└───────┴──────────┴───────────┴──────┴─────────┴─────┴──────┴───────┘\n\nColumns: am, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nFor convenience, the marginaleffects package for R also includes a marginal_means() function:\n\nmarginal_means(mod_cat, variables = \"am\")\n\n\n Term Value Mean Std. Error    z Pr(&gt;|z|)     S 2.5 % 97.5 %\n   am FALSE 18.3      0.785 23.3   &lt;0.001 397.4  16.8   19.9\n   am  TRUE 22.5      0.834 26.9   &lt;0.001 528.6  20.8   24.1\n\nResults averaged over levels of: cyl, am \nColumns: term, value, am, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\nThe Marginal Means vignette offers more detail."
   },
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     "objectID": "articles/marginaleffects.html#hypothesis-and-equivalence-tests",
     "href": "articles/marginaleffects.html#hypothesis-and-equivalence-tests",
     "title": "\n1  Get Started\n",
     "section": "\n1.5 Hypothesis and equivalence tests",
-    "text": "1.5 Hypothesis and equivalence tests\nThe hypotheses() function and the hypothesis argument can be used to conduct linear and non-linear hypothesis tests on model coefficients, or on any of the quantities computed by the functions introduced above.\nConsider this model:\n\n\nR\nPython\n\n\n\n\nmod &lt;- lm(mpg ~ qsec * drat, data = mtcars)\ncoef(mod)\n\n(Intercept)        qsec        drat   qsec:drat \n 12.3371987  -1.0241183  -3.4371461   0.5973153 \n\n\n\n\n\nmod = smf.ols('mpg ~ qsec * drat', data=mtcars).fit()\nprint(mod.params)\n\nIntercept    12.337199\nqsec         -1.024118\ndrat         -3.437146\nqsec:drat     0.597315\ndtype: float64\n\n\n\n\n\nCan we reject the null hypothesis that the drat coefficient is 2 times the size of the qsec coefficient?\n\nhypotheses(mod, \"drat = 2 * qsec\")\n\n\n            Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n drat = 2 * qsec    -1.39       10.8 -0.129    0.897 0.2 -22.5   19.7\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \n\n\nWe can ask the same question but refer to parameters by position, with indices b1, b2, b3, etc.:\n\n\nR\nPython\n\n\n\n\nhypotheses(mod, \"b3 = 2 * b2\")\n\n\n        Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n b3 = 2 * b2    -1.39       10.8 -0.129    0.897 0.2 -22.5   19.7\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \n\n\n\n\n\nh = hypotheses(mod, \"b3 = 2 * b2\")\nprint(h)\n\nshape: (1, 8)\n┌─────────┬──────────┬───────────┬────────┬─────────┬───────┬───────┬───────┐\n│ Term    ┆ Estimate ┆ Std.Error ┆ z      ┆ P(&gt;|z|) ┆ S     ┆ 2.5%  ┆ 97.5% │\n│ ---     ┆ ---      ┆ ---       ┆ ---    ┆ ---     ┆ ---   ┆ ---   ┆ ---   │\n│ str     ┆ str      ┆ str       ┆ str    ┆ str     ┆ str   ┆ str   ┆ str   │\n╞═════════╪══════════╪═══════════╪════════╪═════════╪═══════╪═══════╪═══════╡\n│ b3=2*b2 ┆ -1.39    ┆ 10.8      ┆ -0.129 ┆ 0.898   ┆ 0.155 ┆ -23.5 ┆ 20.7  │\n└─────────┴──────────┴───────────┴────────┴─────────┴───────┴───────┴───────┘\n\nColumns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nThe main functions in marginaleffects all have a hypothesis argument, which means that we can do complex model testing. For example, consider two slope estimates:\n\n\nR\nPython\n\n\n\n\nslopes(\n  mod,\n  variables = \"drat\",\n  newdata = datagrid(qsec = range))\n\n\n Term qsec Estimate Std. Error    z Pr(&gt;|z|)   S  2.5 % 97.5 %\n drat 14.5     5.22       3.80 1.38   0.1690 2.6 -2.221   12.7\n drat 22.9    10.24       5.15 1.99   0.0469 4.4  0.142   20.3\n\nColumns: rowid, term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, qsec, predicted_lo, predicted_hi, predicted, mpg, drat \nType:  response \n\n\n\n\n\ns = slopes(\n  mod,\n  variables = \"drat\",\n  newdata = datagrid(mod, qsec = [mtcars[\"qsec\"].min(), mtcars[\"qsec\"].max()]))\nprint(s)\n\nshape: (2, 9)\n┌──────┬──────────┬──────────┬───────────┬───┬─────────┬──────┬────────┬───────┐\n│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S    ┆ 2.5%   ┆ 97.5% │\n│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---  ┆ ---    ┆ ---   │\n│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str  ┆ str    ┆ str   │\n╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪══════╪════════╪═══════╡\n│ drat ┆ dY/dX    ┆ 5.22     ┆ 3.8       ┆ … ┆ 0.18    ┆ 2.47 ┆ -2.56  ┆ 13    │\n│ drat ┆ dY/dX    ┆ 10.2     ┆ 5.16      ┆ … ┆ 0.0573  ┆ 4.13 ┆ -0.338 ┆ 20.8  │\n└──────┴──────────┴──────────┴───────────┴───┴─────────┴──────┴────────┴───────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, qsec, rownames, mpg, cyl, disp, hp, drat, wt, vs, am, gear, carb\n\n\n\n\n\nAre these two slopes significantly different from one another? To test this, we can use the hypothesis argument:\n\n\nR\nPython\n\n\n\n\nslopes(\n  mod,\n  hypothesis = \"b1 = b2\",\n  variables = \"drat\",\n  newdata = datagrid(qsec = range))\n\n\n  Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n b1=b2    -5.02       8.52 -0.589    0.556 0.8 -21.7   11.7\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\ns = slopes(\n  mod,\n  hypothesis = \"b1 = b2\",\n  variables = \"drat\",\n  newdata = datagrid(mod, qsec = [mtcars[\"qsec\"].min(), mtcars[\"qsec\"].max()]))\nprint(s)\n\nshape: (1, 8)\n┌───────┬──────────┬───────────┬────────┬─────────┬───────┬───────┬───────┐\n│ Term  ┆ Estimate ┆ Std.Error ┆ z      ┆ P(&gt;|z|) ┆ S     ┆ 2.5%  ┆ 97.5% │\n│ ---   ┆ ---      ┆ ---       ┆ ---    ┆ ---     ┆ ---   ┆ ---   ┆ ---   │\n│ str   ┆ str      ┆ str       ┆ str    ┆ str     ┆ str   ┆ str   ┆ str   │\n╞═══════╪══════════╪═══════════╪════════╪═════════╪═══════╪═══════╪═══════╡\n│ b1=b2 ┆ -5.02    ┆ 8.53      ┆ -0.588 ┆ 0.561   ┆ 0.834 ┆ -22.5 ┆ 12.5  │\n└───────┴──────────┴───────────┴────────┴─────────┴───────┴───────┴───────┘\n\nColumns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nAlternatively, we can also refer to values with term names (when they are unique):\n\n\nR\nPython\n\n\n\n\navg_slopes(mod)\n\n\n Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n drat     7.22      1.365 5.29  &lt; 0.001 23.0 4.549   9.90\n qsec     1.12      0.433 2.60  0.00942  6.7 0.276   1.97\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\navg_slopes(mod, hypothesis = \"drat = qsec\")\n\n\n      Term Estimate Std. Error   z Pr(&gt;|z|)    S 2.5 % 97.5 %\n drat=qsec      6.1       1.45 4.2   &lt;0.001 15.2  3.25   8.95\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\ns = avg_slopes(mod)\nprint(s)\n\nshape: (2, 9)\n┌──────┬─────────────┬──────────┬───────────┬───┬──────────┬──────┬───────┬───────┐\n│ Term ┆ Contrast    ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │\n│ ---  ┆ ---         ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │\n│ str  ┆ str         ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │\n╞══════╪═════════════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡\n│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.25e-05 ┆ 16.3 ┆ 4.43  ┆ 10    │\n│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.432     ┆ … ┆ 0.0147   ┆ 6.09 ┆ 0.239 ┆ 2.01  │\n└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘\n\nColumns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\ns = avg_slopes(mod, hypothesis = \"drat = qsec\")\nprint(s)\n\nshape: (1, 8)\n┌───────────┬──────────┬───────────┬─────┬──────────┬─────┬──────┬───────┐\n│ Term      ┆ Estimate ┆ Std.Error ┆ z   ┆ P(&gt;|z|)  ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---       ┆ ---      ┆ ---       ┆ --- ┆ ---      ┆ --- ┆ ---  ┆ ---   │\n│ str       ┆ str      ┆ str       ┆ str ┆ str      ┆ str ┆ str  ┆ str   │\n╞═══════════╪══════════╪═══════════╪═════╪══════════╪═════╪══════╪═══════╡\n│ drat=qsec ┆ 6.1      ┆ 1.45      ┆ 4.2 ┆ 0.000245 ┆ 12  ┆ 3.13 ┆ 9.07  │\n└───────────┴──────────┴───────────┴─────┴──────────┴─────┴──────┴───────┘\n\nColumns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nNow, imagine that for theoretical (or substantive or clinical) reasons, we only care about slopes larger than 2. We can use the equivalence argument to conduct an equivalence test:\n\n\nR\nPython\n\n\n\n\navg_slopes(mod, equivalence = c(-2, 2))\n\n\n Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 % p (NonSup) p (NonInf) p (Equiv)\n drat     7.22      1.365 5.29  &lt; 0.001 23.0 4.549   9.90     0.9999     &lt;0.001    0.9999\n qsec     1.12      0.433 2.60  0.00942  6.7 0.276   1.97     0.0215     &lt;0.001    0.0215\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, statistic.noninf, statistic.nonsup, p.value.noninf, p.value.nonsup, p.value.equiv \nType:  response \n\n\n\n\n\ns = avg_slopes(mod, equivalence = [-2., 2.])\nprint(s)\n\nshape: (2, 9)\n┌──────┬─────────────┬──────────┬───────────┬───┬──────────┬──────┬───────┬───────┐\n│ Term ┆ Contrast    ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │\n│ ---  ┆ ---         ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │\n│ str  ┆ str         ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │\n╞══════╪═════════════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡\n│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.432     ┆ … ┆ 0.0147   ┆ 6.09 ┆ 0.239 ┆ 2.01  │\n│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.25e-05 ┆ 16.3 ┆ 4.43  ┆ 10    │\n└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘\n\nColumns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, statistic_noninf, statistic_nonsup, p_value_noninf, p_value_nonsup, p_value_equiv\n\n\n\n\n\nSee the Hypothesis Tests and Custom Contrasts vignette for background, details, and for instructions on how to conduct hypothesis tests in more complex situations."
+    "text": "1.5 Hypothesis and equivalence tests\nThe hypotheses() function and the hypothesis argument can be used to conduct linear and non-linear hypothesis tests on model coefficients, or on any of the quantities computed by the functions introduced above.\nConsider this model:\n\n\nR\nPython\n\n\n\n\nmod &lt;- lm(mpg ~ qsec * drat, data = mtcars)\ncoef(mod)\n\n(Intercept)        qsec        drat   qsec:drat \n 12.3371987  -1.0241183  -3.4371461   0.5973153 \n\n\n\n\n\nmod = smf.ols('mpg ~ qsec * drat', data=mtcars).fit()\nprint(mod.params)\n\nIntercept    12.337199\nqsec         -1.024118\ndrat         -3.437146\nqsec:drat     0.597315\ndtype: float64\n\n\n\n\n\nCan we reject the null hypothesis that the drat coefficient is 2 times the size of the qsec coefficient?\n\nhypotheses(mod, \"drat = 2 * qsec\")\n\n\n            Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n drat = 2 * qsec    -1.39       10.8 -0.129    0.897 0.2 -22.5   19.7\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \n\n\nWe can ask the same question but refer to parameters by position, with indices b1, b2, b3, etc.:\n\n\nR\nPython\n\n\n\n\nhypotheses(mod, \"b3 = 2 * b2\")\n\n\n        Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n b3 = 2 * b2    -1.39       10.8 -0.129    0.897 0.2 -22.5   19.7\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \n\n\n\n\n\nh = hypotheses(mod, \"b3 = 2 * b2\")\nprint(h)\n\nshape: (1, 8)\n┌─────────┬──────────┬───────────┬────────┬─────────┬───────┬───────┬───────┐\n│ Term    ┆ Estimate ┆ Std.Error ┆ z      ┆ P(&gt;|z|) ┆ S     ┆ 2.5%  ┆ 97.5% │\n│ ---     ┆ ---      ┆ ---       ┆ ---    ┆ ---     ┆ ---   ┆ ---   ┆ ---   │\n│ str     ┆ str      ┆ str       ┆ str    ┆ str     ┆ str   ┆ str   ┆ str   │\n╞═════════╪══════════╪═══════════╪════════╪═════════╪═══════╪═══════╪═══════╡\n│ b3=2*b2 ┆ -1.39    ┆ 10.8      ┆ -0.129 ┆ 0.898   ┆ 0.155 ┆ -23.5 ┆ 20.7  │\n└─────────┴──────────┴───────────┴────────┴─────────┴───────┴───────┴───────┘\n\nColumns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nThe main functions in marginaleffects all have a hypothesis argument, which means that we can do complex model testing. For example, consider two slope estimates:\n\n\nR\nPython\n\n\n\n\nslopes(\n  mod,\n  variables = \"drat\",\n  newdata = datagrid(qsec = range))\n\n\n Term qsec Estimate Std. Error    z Pr(&gt;|z|)   S  2.5 % 97.5 %\n drat 14.5     5.22       3.80 1.38   0.1690 2.6 -2.221   12.7\n drat 22.9    10.24       5.15 1.99   0.0469 4.4  0.142   20.3\n\nColumns: rowid, term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, qsec, predicted_lo, predicted_hi, predicted, mpg, drat \nType:  response \n\n\n\n\n\ns = slopes(\n  mod,\n  variables = \"drat\",\n  newdata = datagrid(mod, qsec = [mtcars[\"qsec\"].min(), mtcars[\"qsec\"].max()]))\nprint(s)\n\nshape: (2, 9)\n┌──────┬──────────┬──────────┬───────────┬───┬─────────┬──────┬────────┬───────┐\n│ Term ┆ Contrast ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|) ┆ S    ┆ 2.5%   ┆ 97.5% │\n│ ---  ┆ ---      ┆ ---      ┆ ---       ┆   ┆ ---     ┆ ---  ┆ ---    ┆ ---   │\n│ str  ┆ str      ┆ str      ┆ str       ┆   ┆ str     ┆ str  ┆ str    ┆ str   │\n╞══════╪══════════╪══════════╪═══════════╪═══╪═════════╪══════╪════════╪═══════╡\n│ drat ┆ dY/dX    ┆ 5.22     ┆ 3.8       ┆ … ┆ 0.18    ┆ 2.47 ┆ -2.56  ┆ 13    │\n│ drat ┆ dY/dX    ┆ 10.2     ┆ 5.16      ┆ … ┆ 0.0573  ┆ 4.13 ┆ -0.338 ┆ 20.8  │\n└──────┴──────────┴──────────┴───────────┴───┴─────────┴──────┴────────┴───────┘\n\nColumns: rowid, term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, predicted, predicted_lo, predicted_hi, qsec, rownames, mpg, cyl, disp, hp, drat, wt, vs, am, gear, carb\n\n\n\n\n\nAre these two slopes significantly different from one another? To test this, we can use the hypothesis argument:\n\n\nR\nPython\n\n\n\n\nslopes(\n  mod,\n  hypothesis = \"b1 = b2\",\n  variables = \"drat\",\n  newdata = datagrid(qsec = range))\n\n\n  Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n b1=b2    -5.02       8.52 -0.589    0.556 0.8 -21.7   11.7\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\ns = slopes(\n  mod,\n  hypothesis = \"b1 = b2\",\n  variables = \"drat\",\n  newdata = datagrid(mod, qsec = [mtcars[\"qsec\"].min(), mtcars[\"qsec\"].max()]))\nprint(s)\n\nshape: (1, 8)\n┌───────┬──────────┬───────────┬────────┬─────────┬───────┬───────┬───────┐\n│ Term  ┆ Estimate ┆ Std.Error ┆ z      ┆ P(&gt;|z|) ┆ S     ┆ 2.5%  ┆ 97.5% │\n│ ---   ┆ ---      ┆ ---       ┆ ---    ┆ ---     ┆ ---   ┆ ---   ┆ ---   │\n│ str   ┆ str      ┆ str       ┆ str    ┆ str     ┆ str   ┆ str   ┆ str   │\n╞═══════╪══════════╪═══════════╪════════╪═════════╪═══════╪═══════╪═══════╡\n│ b1=b2 ┆ -5.02    ┆ 8.53      ┆ -0.588 ┆ 0.561   ┆ 0.834 ┆ -22.5 ┆ 12.5  │\n└───────┴──────────┴───────────┴────────┴─────────┴───────┴───────┴───────┘\n\nColumns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nAlternatively, we can also refer to values with term names (when they are unique):\n\n\nR\nPython\n\n\n\n\navg_slopes(mod)\n\n\n Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n drat     7.22      1.365 5.29  &lt; 0.001 23.0 4.549   9.90\n qsec     1.12      0.433 2.60  0.00942  6.7 0.276   1.97\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\navg_slopes(mod, hypothesis = \"drat = qsec\")\n\n\n      Term Estimate Std. Error   z Pr(&gt;|z|)    S 2.5 % 97.5 %\n drat=qsec      6.1       1.45 4.2   &lt;0.001 15.2  3.25   8.95\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\n\n\n\ns = avg_slopes(mod)\nprint(s)\n\nshape: (2, 9)\n┌──────┬─────────────┬──────────┬───────────┬───┬──────────┬──────┬───────┬───────┐\n│ Term ┆ Contrast    ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │\n│ ---  ┆ ---         ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │\n│ str  ┆ str         ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │\n╞══════╪═════════════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡\n│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.432     ┆ … ┆ 0.0147   ┆ 6.09 ┆ 0.239 ┆ 2.01  │\n│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.25e-05 ┆ 16.3 ┆ 4.43  ┆ 10    │\n└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘\n\nColumns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\ns = avg_slopes(mod, hypothesis = \"drat = qsec\")\nprint(s)\n\nshape: (1, 8)\n┌───────────┬──────────┬───────────┬─────┬──────────┬─────┬──────┬───────┐\n│ Term      ┆ Estimate ┆ Std.Error ┆ z   ┆ P(&gt;|z|)  ┆ S   ┆ 2.5% ┆ 97.5% │\n│ ---       ┆ ---      ┆ ---       ┆ --- ┆ ---      ┆ --- ┆ ---  ┆ ---   │\n│ str       ┆ str      ┆ str       ┆ str ┆ str      ┆ str ┆ str  ┆ str   │\n╞═══════════╪══════════╪═══════════╪═════╪══════════╪═════╪══════╪═══════╡\n│ drat=qsec ┆ 6.1      ┆ 1.45      ┆ 4.2 ┆ 0.000245 ┆ 12  ┆ 3.13 ┆ 9.07  │\n└───────────┴──────────┴───────────┴─────┴──────────┴─────┴──────┴───────┘\n\nColumns: term, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high\n\n\n\n\n\nNow, imagine that for theoretical (or substantive or clinical) reasons, we only care about slopes larger than 2. We can use the equivalence argument to conduct an equivalence test:\n\n\nR\nPython\n\n\n\n\navg_slopes(mod, equivalence = c(-2, 2))\n\n\n Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 % p (NonSup) p (NonInf) p (Equiv)\n drat     7.22      1.365 5.29  &lt; 0.001 23.0 4.549   9.90     0.9999     &lt;0.001    0.9999\n qsec     1.12      0.433 2.60  0.00942  6.7 0.276   1.97     0.0215     &lt;0.001    0.0215\n\nColumns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, statistic.noninf, statistic.nonsup, p.value.noninf, p.value.nonsup, p.value.equiv \nType:  response \n\n\n\n\n\ns = avg_slopes(mod, equivalence = [-2., 2.])\nprint(s)\n\nshape: (2, 9)\n┌──────┬─────────────┬──────────┬───────────┬───┬──────────┬──────┬───────┬───────┐\n│ Term ┆ Contrast    ┆ Estimate ┆ Std.Error ┆ … ┆ P(&gt;|z|)  ┆ S    ┆ 2.5%  ┆ 97.5% │\n│ ---  ┆ ---         ┆ ---      ┆ ---       ┆   ┆ ---      ┆ ---  ┆ ---   ┆ ---   │\n│ str  ┆ str         ┆ str      ┆ str       ┆   ┆ str      ┆ str  ┆ str   ┆ str   │\n╞══════╪═════════════╪══════════╪═══════════╪═══╪══════════╪══════╪═══════╪═══════╡\n│ qsec ┆ mean(dY/dX) ┆ 1.12     ┆ 0.432     ┆ … ┆ 0.0147   ┆ 6.09 ┆ 0.239 ┆ 2.01  │\n│ drat ┆ mean(dY/dX) ┆ 7.22     ┆ 1.37      ┆ … ┆ 1.25e-05 ┆ 16.3 ┆ 4.43  ┆ 10    │\n└──────┴─────────────┴──────────┴───────────┴───┴──────────┴──────┴───────┴───────┘\n\nColumns: term, contrast, estimate, std_error, statistic, p_value, s_value, conf_low, conf_high, statistic_noninf, statistic_nonsup, p_value_noninf, p_value_nonsup, p_value_equiv\n\n\n\n\n\nSee the Hypothesis Tests and Custom Contrasts vignette for background, details, and for instructions on how to conduct hypothesis tests in more complex situations."
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@@ -263,7 +263,7 @@
     "href": "articles/slopes.html#example-quadratic",
     "title": "\n4  Slopes\n",
     "section": "\n4.10 Example: Quadratic",
-    "text": "4.10 Example: Quadratic\nIn the “Definition” section of this vignette, we considered how marginal effects can be computed analytically in a simple quadratic equation context. We can now use the slopes() function to replicate our analysis of the quadratic function in a regression application.\nSay you estimate a linear regression model with a quadratic term:\n\\[Y = \\beta_0 + \\beta_1 X^2 + \\varepsilon\\]\nand obtain estimates of \\(\\beta_0=1\\) and \\(\\beta_1=2\\). Taking the partial derivative with respect to \\(X\\) and plugging in our estimates gives us the marginal effect of \\(X\\) on \\(Y\\):\n\\[\\partial Y / \\partial X = \\beta_0 + 2 \\cdot \\beta_1 X\\] \\[\\partial Y / \\partial X = 1 + 4X\\]\nThis result suggests that the effect of a change in \\(X\\) on \\(Y\\) depends on the level of \\(X\\). When \\(X\\) is large and positive, an increase in \\(X\\) is associated to a large increase in \\(Y\\). When \\(X\\) is small and positive, an increase in \\(X\\) is associated to a small increase in \\(Y\\). When \\(X\\) is a large negative value, an increase in \\(X\\) is associated with a decrease in \\(Y\\).\nmarginaleffects arrives at the same conclusion in simulated data:\n\nlibrary(tidyverse)\nN &lt;- 1e5\nquad &lt;- data.frame(x = rnorm(N))\nquad$y &lt;- 1 + 1 * quad$x + 2 * quad$x^2 + rnorm(N)\nmod &lt;- lm(y ~ x + I(x^2), quad)\n\nslopes(mod, newdata = datagrid(x = -2:2))  |&gt;\n    mutate(truth = 1 + 4 * x) |&gt;\n    select(estimate, truth)\n#&gt; \n#&gt;  Estimate\n#&gt;     -7.00\n#&gt;     -2.99\n#&gt;      1.01\n#&gt;      5.01\n#&gt;      9.01\n#&gt; \n#&gt; Columns: estimate, truth\n\nWe can plot conditional adjusted predictions with plot_predictions() function:\n\nplot_predictions(mod, condition = \"x\")\n\n\n\n\nWe can plot conditional marginal effects with the plot_slopes() function (see section below):\n\nplot_slopes(mod, variables = \"x\", condition = \"x\")\n\n\n\n\nAgain, the conclusion is the same. When \\(x&lt;0\\), an increase in \\(x\\) is associated with an decrease in \\(y\\). When \\(x&gt;1/4\\), the marginal effect is positive, which suggests that an increase in \\(x\\) is associated with an increase in \\(y\\)."
+    "text": "4.10 Example: Quadratic\nIn the “Definition” section of this vignette, we considered how marginal effects can be computed analytically in a simple quadratic equation context. We can now use the slopes() function to replicate our analysis of the quadratic function in a regression application.\nSay you estimate a linear regression model with a quadratic term:\n\\[Y = \\beta_0 + \\beta_1 X^2 + \\varepsilon\\]\nand obtain estimates of \\(\\beta_0=1\\) and \\(\\beta_1=2\\). Taking the partial derivative with respect to \\(X\\) and plugging in our estimates gives us the marginal effect of \\(X\\) on \\(Y\\):\n\\[\\partial Y / \\partial X = \\beta_0 + 2 \\cdot \\beta_1 X\\] \\[\\partial Y / \\partial X = 1 + 4X\\]\nThis result suggests that the effect of a change in \\(X\\) on \\(Y\\) depends on the level of \\(X\\). When \\(X\\) is large and positive, an increase in \\(X\\) is associated to a large increase in \\(Y\\). When \\(X\\) is small and positive, an increase in \\(X\\) is associated to a small increase in \\(Y\\). When \\(X\\) is a large negative value, an increase in \\(X\\) is associated with a decrease in \\(Y\\).\nmarginaleffects arrives at the same conclusion in simulated data:\n\nlibrary(tidyverse)\nN &lt;- 1e5\nquad &lt;- data.frame(x = rnorm(N))\nquad$y &lt;- 1 + 1 * quad$x + 2 * quad$x^2 + rnorm(N)\nmod &lt;- lm(y ~ x + I(x^2), quad)\n\nslopes(mod, newdata = datagrid(x = -2:2))  |&gt;\n    mutate(truth = 1 + 4 * x) |&gt;\n    select(estimate, truth)\n#&gt; \n#&gt;  Estimate\n#&gt;     -6.99\n#&gt;     -2.99\n#&gt;      1.01\n#&gt;      5.01\n#&gt;      9.01\n#&gt; \n#&gt; Columns: estimate, truth\n\nWe can plot conditional adjusted predictions with plot_predictions() function:\n\nplot_predictions(mod, condition = \"x\")\n\n\n\n\nWe can plot conditional marginal effects with the plot_slopes() function (see section below):\n\nplot_slopes(mod, variables = \"x\", condition = \"x\")\n\n\n\n\nAgain, the conclusion is the same. When \\(x&lt;0\\), an increase in \\(x\\) is associated with an decrease in \\(y\\). When \\(x&gt;1/4\\), the marginal effect is positive, which suggests that an increase in \\(x\\) is associated with an increase in \\(y\\)."
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@@ -403,7 +403,7 @@
     "href": "articles/hypothesis.html#hypothesis-tests-with-the-delta-method",
     "title": "\n7  Hypothesis Tests\n",
     "section": "\n7.2 Hypothesis tests with the delta method",
-    "text": "7.2 Hypothesis tests with the delta method\nThe marginaleffects package includes a powerful function called hypotheses(). This function emulates the behavior of the well-established car::deltaMethod and car::linearHypothesis functions, but it supports more models, requires fewer dependencies, and offers some convenience features like shortcuts for robust standard errors.\nhypotheses() can be used to compute estimates and standard errors of arbitrary functions of model parameters. For example, it can be used to conduct tests of equality between coefficients, or to test the value of some linear or non-linear combination of quantities of interest. hypotheses() can also be used to conduct hypothesis tests on other functions of a model’s parameter, such as adjusted predictions or marginal effects.\nLet’s start by estimating a simple model:\n\nlibrary(marginaleffects)\nmod &lt;- lm(mpg ~ hp + wt + factor(cyl), data = mtcars)\n\nWhen the FUN and hypothesis arguments of hypotheses() equal NULL (the default), the function returns a data.frame of raw estimates:\n\nhypotheses(mod)\n#&gt; \n#&gt;          Term Estimate Std. Error     z Pr(&gt;|z|)     S   2.5 %    97.5 %\n#&gt;  (Intercept)   35.8460      2.041 17.56   &lt;0.001 227.0 31.8457 39.846319\n#&gt;  hp            -0.0231      0.012 -1.93   0.0531   4.2 -0.0465  0.000306\n#&gt;  wt            -3.1814      0.720 -4.42   &lt;0.001  16.6 -4.5918 -1.771012\n#&gt;  factor(cyl)6  -3.3590      1.402 -2.40   0.0166   5.9 -6.1062 -0.611803\n#&gt;  factor(cyl)8  -3.1859      2.170 -1.47   0.1422   2.8 -7.4399  1.068169\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nTest of equality between coefficients:\n\nhypotheses(mod, \"hp = wt\")\n#&gt; \n#&gt;     Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n#&gt;  hp = wt     3.16       0.72 4.39   &lt;0.001 16.4  1.75   4.57\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nNon-linear function of coefficients\n\nhypotheses(mod, \"exp(hp + wt) = 0.1\")\n#&gt; \n#&gt;                Term Estimate Std. Error     z Pr(&gt;|z|)   S  2.5 %  97.5 %\n#&gt;  exp(hp + wt) = 0.1  -0.0594     0.0292 -2.04   0.0418 4.6 -0.117 -0.0022\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nThe vcov argument behaves in the same was as in the slopes() function. It allows us to easily compute robust standard errors:\n\nhypotheses(mod, \"hp = wt\", vcov = \"HC3\")\n#&gt; \n#&gt;     Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n#&gt;  hp = wt     3.16      0.805 3.92   &lt;0.001 13.5  1.58   4.74\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nWe can use shortcuts like b1, b2, ... to identify the position of each parameter in the output of FUN. For example, b2=b3 is equivalent to hp=wt because those term names appear in the 2nd and 3rd row when we call hypotheses(mod).\n\nhypotheses(mod, \"b2 = b3\")\n#&gt; \n#&gt;     Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n#&gt;  b2 = b3     3.16       0.72 4.39   &lt;0.001 16.4  1.75   4.57\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\n\nhypotheses(mod, hypothesis = \"b* / b3 = 1\")\n#&gt; \n#&gt;         Term  Estimate Std. Error         z Pr(&gt;|z|)    S  2.5 % 97.5 %\n#&gt;  b1 / b3 = 1 -12.26735    2.07340   -5.9165   &lt;0.001 28.2 -16.33 -8.204\n#&gt;  b2 / b3 = 1  -0.99273    0.00413 -240.5539   &lt;0.001  Inf  -1.00 -0.985\n#&gt;  b3 / b3 = 1   0.00000         NA        NA       NA   NA     NA     NA\n#&gt;  b4 / b3 = 1   0.05583    0.58287    0.0958    0.924  0.1  -1.09  1.198\n#&gt;  b5 / b3 = 1   0.00141    0.82981    0.0017    0.999  0.0  -1.62  1.628\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nTerm names with special characters must be enclosed in backticks:\n\nhypotheses(mod, \"`factor(cyl)6` = `factor(cyl)8`\")\n#&gt; \n#&gt;                             Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  `factor(cyl)6` = `factor(cyl)8`   -0.173       1.65 -0.105    0.917 0.1 -3.41   3.07\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\n\n7.2.1 Arbitrary functions: FUN\n\nThe FUN argument can be used to compute standard errors for arbitrary functions of model parameters. This user-supplied function must accept a single model object, and return a numeric vector or a data.frame with two columns named term and estimate.\n\nmod &lt;- glm(am ~ hp + mpg, data = mtcars, family = binomial)\n\nf &lt;- function(x) {\n    out &lt;- x$coefficients[\"hp\"] + x$coefficients[\"mpg\"]\n    return(out)\n}\nhypotheses(mod, FUN = f)\n#&gt; \n#&gt;  Term Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;     1     1.31      0.593 2.22   0.0266 5.2 0.153   2.48\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nWith labels:\n\nf &lt;- function(x) {\n    out &lt;- data.frame(\n        term = \"Horsepower + Miles per Gallon\",\n        estimate = x$coefficients[\"hp\"] + x$coefficients[\"mpg\"]\n    )\n    return(out)\n}\nhypotheses(mod, FUN = f)\n#&gt; \n#&gt;                           Term Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  Horsepower + Miles per Gallon     1.31      0.593 2.22   0.0266 5.2 0.153   2.48\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nTest of equality between two predictions (row 2 vs row 3):\n\nf &lt;- function(x) predict(x, newdata = mtcars)\nhypotheses(mod, FUN = f, hypothesis = \"b2 = b3\")\n#&gt; \n#&gt;     Term Estimate Std. Error     z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  b2 = b3    -1.33      0.616 -2.16   0.0305 5.0 -2.54 -0.125\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nNote that we specified the newdata argument in the f function. This is because the predict() method associated with lm objects will automatically the original fitted values when newdata is NULL, instead of returning the slightly altered fitted values which we need to compute numerical derivatives in the delta method.\nWe can also use numeric vectors to specify linear combinations of parameters. For example, there are 3 coefficients in the last model we estimated. To test the null hypothesis that the sum of the 2nd and 3rd coefficients is equal to 0, we can do:\n\nhypotheses(mod, hypothesis = c(0, 1, 1))\n#&gt; \n#&gt;    Term Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  custom     1.31      0.593 2.22   0.0266 5.2 0.153   2.48\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nSee below for more example of how to use string formulas, numeric vectors, or matrices to calculate custom contrasts, linear combinations, and linear or non-linear hypothesis tests.\n\n7.2.2 Arbitrary quantities with data frames\nmarginaleffects can also compute uncertainty estimates for arbitrary quantities hosted in a data frame, as long as the user can supply a variance-covariance matrix. (Thanks to Kyle F Butts for this cool feature and example!)\nSay you run a monte-carlo simulation and you want to perform hypothesis of various quantities returned from each simulation. The quantities are correlated within each draw:\n\n# simulated means and medians\ndraw &lt;- function(i) { \n  x &lt;- rnorm(n = 10000, mean = 0, sd = 1)\n  out &lt;- data.frame(median = median(x), mean =  mean(x))\n  return(out)\n}\nsims &lt;- do.call(\"rbind\", lapply(1:25, draw))\n\n# average mean and average median \ncoeftable &lt;- data.frame(\n  term = c(\"median\", \"mean\"),\n  estimate = c(mean(sims$median), mean(sims$mean))\n)\n\n# variance-covariance\nvcov &lt;- cov(sims)\n\n# is the median equal to the mean?\nhypotheses(\n  coeftable,\n  vcov = vcov,\n  hypothesis = \"median = mean\"\n)\n#&gt; \n#&gt;           Term Estimate Std. Error   z Pr(&gt;|z|)   S   2.5 % 97.5 %\n#&gt;  median = mean  0.00176    0.00881 0.2    0.841 0.2 -0.0155  0.019\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high"
+    "text": "7.2 Hypothesis tests with the delta method\nThe marginaleffects package includes a powerful function called hypotheses(). This function emulates the behavior of the well-established car::deltaMethod and car::linearHypothesis functions, but it supports more models, requires fewer dependencies, and offers some convenience features like shortcuts for robust standard errors.\nhypotheses() can be used to compute estimates and standard errors of arbitrary functions of model parameters. For example, it can be used to conduct tests of equality between coefficients, or to test the value of some linear or non-linear combination of quantities of interest. hypotheses() can also be used to conduct hypothesis tests on other functions of a model’s parameter, such as adjusted predictions or marginal effects.\nLet’s start by estimating a simple model:\n\nlibrary(marginaleffects)\nmod &lt;- lm(mpg ~ hp + wt + factor(cyl), data = mtcars)\n\nWhen the FUN and hypothesis arguments of hypotheses() equal NULL (the default), the function returns a data.frame of raw estimates:\n\nhypotheses(mod)\n#&gt; \n#&gt;          Term Estimate Std. Error     z Pr(&gt;|z|)     S   2.5 %    97.5 %\n#&gt;  (Intercept)   35.8460      2.041 17.56   &lt;0.001 227.0 31.8457 39.846319\n#&gt;  hp            -0.0231      0.012 -1.93   0.0531   4.2 -0.0465  0.000306\n#&gt;  wt            -3.1814      0.720 -4.42   &lt;0.001  16.6 -4.5918 -1.771012\n#&gt;  factor(cyl)6  -3.3590      1.402 -2.40   0.0166   5.9 -6.1062 -0.611803\n#&gt;  factor(cyl)8  -3.1859      2.170 -1.47   0.1422   2.8 -7.4399  1.068169\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nTest of equality between coefficients:\n\nhypotheses(mod, \"hp = wt\")\n#&gt; \n#&gt;     Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n#&gt;  hp = wt     3.16       0.72 4.39   &lt;0.001 16.4  1.75   4.57\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nNon-linear function of coefficients\n\nhypotheses(mod, \"exp(hp + wt) = 0.1\")\n#&gt; \n#&gt;                Term Estimate Std. Error     z Pr(&gt;|z|)   S  2.5 %  97.5 %\n#&gt;  exp(hp + wt) = 0.1  -0.0594     0.0292 -2.04   0.0418 4.6 -0.117 -0.0022\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nThe vcov argument behaves in the same was as in the slopes() function. It allows us to easily compute robust standard errors:\n\nhypotheses(mod, \"hp = wt\", vcov = \"HC3\")\n#&gt; \n#&gt;     Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n#&gt;  hp = wt     3.16      0.805 3.92   &lt;0.001 13.5  1.58   4.74\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nWe can use shortcuts like b1, b2, ... to identify the position of each parameter in the output of FUN. For example, b2=b3 is equivalent to hp=wt because those term names appear in the 2nd and 3rd row when we call hypotheses(mod).\n\nhypotheses(mod, \"b2 = b3\")\n#&gt; \n#&gt;     Term Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n#&gt;  b2 = b3     3.16       0.72 4.39   &lt;0.001 16.4  1.75   4.57\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\n\nhypotheses(mod, hypothesis = \"b* / b3 = 1\")\n#&gt; \n#&gt;         Term  Estimate Std. Error         z Pr(&gt;|z|)    S  2.5 % 97.5 %\n#&gt;  b1 / b3 = 1 -12.26735    2.07340   -5.9165   &lt;0.001 28.2 -16.33 -8.204\n#&gt;  b2 / b3 = 1  -0.99273    0.00413 -240.5539   &lt;0.001  Inf  -1.00 -0.985\n#&gt;  b3 / b3 = 1   0.00000         NA        NA       NA   NA     NA     NA\n#&gt;  b4 / b3 = 1   0.05583    0.58287    0.0958    0.924  0.1  -1.09  1.198\n#&gt;  b5 / b3 = 1   0.00141    0.82981    0.0017    0.999  0.0  -1.62  1.628\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nTerm names with special characters must be enclosed in backticks:\n\nhypotheses(mod, \"`factor(cyl)6` = `factor(cyl)8`\")\n#&gt; \n#&gt;                             Term Estimate Std. Error      z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  `factor(cyl)6` = `factor(cyl)8`   -0.173       1.65 -0.105    0.917 0.1 -3.41   3.07\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\n\n7.2.1 Arbitrary functions: FUN\n\nThe FUN argument can be used to compute standard errors for arbitrary functions of model parameters. This user-supplied function must accept a single model object, and return a numeric vector or a data.frame with two columns named term and estimate.\n\nmod &lt;- glm(am ~ hp + mpg, data = mtcars, family = binomial)\n\nf &lt;- function(x) {\n    out &lt;- x$coefficients[\"hp\"] + x$coefficients[\"mpg\"]\n    return(out)\n}\nhypotheses(mod, FUN = f)\n#&gt; \n#&gt;  Term Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;     1     1.31      0.593 2.22   0.0266 5.2 0.153   2.48\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nWith labels:\n\nf &lt;- function(x) {\n    out &lt;- data.frame(\n        term = \"Horsepower + Miles per Gallon\",\n        estimate = x$coefficients[\"hp\"] + x$coefficients[\"mpg\"]\n    )\n    return(out)\n}\nhypotheses(mod, FUN = f)\n#&gt; \n#&gt;                           Term Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  Horsepower + Miles per Gallon     1.31      0.593 2.22   0.0266 5.2 0.153   2.48\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nTest of equality between two predictions (row 2 vs row 3):\n\nf &lt;- function(x) predict(x, newdata = mtcars)\nhypotheses(mod, FUN = f, hypothesis = \"b2 = b3\")\n#&gt; \n#&gt;     Term Estimate Std. Error     z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  b2 = b3    -1.33      0.616 -2.16   0.0305 5.0 -2.54 -0.125\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nNote that we specified the newdata argument in the f function. This is because the predict() method associated with lm objects will automatically the original fitted values when newdata is NULL, instead of returning the slightly altered fitted values which we need to compute numerical derivatives in the delta method.\nWe can also use numeric vectors to specify linear combinations of parameters. For example, there are 3 coefficients in the last model we estimated. To test the null hypothesis that the sum of the 2nd and 3rd coefficients is equal to 0, we can do:\n\nhypotheses(mod, hypothesis = c(0, 1, 1))\n#&gt; \n#&gt;    Term Estimate Std. Error    z Pr(&gt;|z|)   S 2.5 % 97.5 %\n#&gt;  custom     1.31      0.593 2.22   0.0266 5.2 0.153   2.48\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high\n\nSee below for more example of how to use string formulas, numeric vectors, or matrices to calculate custom contrasts, linear combinations, and linear or non-linear hypothesis tests.\n\n7.2.2 Arbitrary quantities with data frames\nmarginaleffects can also compute uncertainty estimates for arbitrary quantities hosted in a data frame, as long as the user can supply a variance-covariance matrix. (Thanks to Kyle F Butts for this cool feature and example!)\nSay you run a monte-carlo simulation and you want to perform hypothesis of various quantities returned from each simulation. The quantities are correlated within each draw:\n\n# simulated means and medians\ndraw &lt;- function(i) { \n  x &lt;- rnorm(n = 10000, mean = 0, sd = 1)\n  out &lt;- data.frame(median = median(x), mean =  mean(x))\n  return(out)\n}\nsims &lt;- do.call(\"rbind\", lapply(1:25, draw))\n\n# average mean and average median \ncoeftable &lt;- data.frame(\n  term = c(\"median\", \"mean\"),\n  estimate = c(mean(sims$median), mean(sims$mean))\n)\n\n# variance-covariance\nvcov &lt;- cov(sims)\n\n# is the median equal to the mean?\nhypotheses(\n  coeftable,\n  vcov = vcov,\n  hypothesis = \"median = mean\"\n)\n#&gt; \n#&gt;           Term Estimate Std. Error      z Pr(&gt;|z|)   S   2.5 % 97.5 %\n#&gt;  median = mean -0.00261    0.00701 -0.372     0.71 0.5 -0.0163 0.0111\n#&gt; \n#&gt; Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high"
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@@ -452,7 +452,7 @@
     "href": "articles/brms.html#random-effects-model",
     "title": "\n8  Bayes\n",
     "section": "\n8.2 Random effects model",
-    "text": "8.2 Random effects model\nThis section replicates some of the analyses of a random effects model published in Andrew Heiss’ blog post: “A guide to correctly calculating posterior predictions and average marginal effects with multilevel Bayesian models.” The objective is mainly to illustrate the use of marginaleffects. Please refer to the original post for a detailed discussion of the quantities computed below.\nLoad libraries and download data:\n\nlibrary(brms)\nlibrary(ggdist)\nlibrary(patchwork)\nlibrary(marginaleffects)\n\nvdem_2015 &lt;- read.csv(\"https://github.com/vincentarelbundock/marginaleffects/raw/main/data-raw/vdem_2015.csv\")\n\nhead(vdem_2015)\n#&gt;   country_name country_text_id year                           region media_index party_autonomy_ord polyarchy civil_liberties party_autonomy\n#&gt; 1       Mexico             MEX 2015  Latin America and the Caribbean       0.837                  3     0.631           0.704           TRUE\n#&gt; 2     Suriname             SUR 2015  Latin America and the Caribbean       0.883                  4     0.777           0.887           TRUE\n#&gt; 3       Sweden             SWE 2015 Western Europe and North America       0.956                  4     0.915           0.968           TRUE\n#&gt; 4  Switzerland             CHE 2015 Western Europe and North America       0.939                  4     0.901           0.960           TRUE\n#&gt; 5        Ghana             GHA 2015               Sub-Saharan Africa       0.858                  4     0.724           0.921           TRUE\n#&gt; 6 South Africa             ZAF 2015               Sub-Saharan Africa       0.898                  4     0.752           0.869           TRUE\n\nFit a basic model:\n\nmod &lt;- brm(\n  bf(media_index ~ party_autonomy + civil_liberties + (1 | region),\n     phi ~ (1 | region)),\n  data = vdem_2015,\n  family = Beta(),\n  control = list(adapt_delta = 0.9))\n\n\n8.2.1 Posterior predictions\nTo compute posterior predictions for specific values of the regressors, we use the newdata argument and the datagrid() function. We also use the type argument to compute two types of predictions: accounting for residual (observation-level) residual variance (prediction) or ignoring it (response).\n\nnd = datagrid(model = mod,\n              party_autonomy = c(TRUE, FALSE),\n              civil_liberties = .5,\n              region = \"Middle East and North Africa\")\np1 &lt;- predictions(mod, type = \"response\", newdata = nd) |&gt;\n    posterior_draws() |&gt;\n    transform(type = \"Response\")\np2 &lt;- predictions(mod, type = \"prediction\", newdata = nd) |&gt;\n    posterior_draws() |&gt;\n    transform(type = \"Prediction\")\npred &lt;- rbind(p1, p2)\n\nExtract posterior draws and plot them:\n\nggplot(pred, aes(x = draw, fill = party_autonomy)) +\n    stat_halfeye(alpha = .5) +\n    facet_wrap(~ type) +\n    labs(x = \"Media index (predicted)\", \n         y = \"Posterior density\",\n         fill = \"Party autonomy\")\n\n\n\n\n\n8.2.2 Marginal effects and contrasts\nAs noted in the Marginal Effects vignette, there should be one distinct marginal effect for each combination of regressor values. Here, we consider only one combination of regressor values, where region is “Middle East and North Africa”, and civil_liberties is 0.5. Then, we calculate the mean of the posterior distribution of marginal effects:\n\nmfx &lt;- slopes(mod,\n                       newdata = datagrid(civil_liberties = .5,\n                                          region = \"Middle East and North Africa\"))\nmfx\n#&gt; \n#&gt;             Term     Contrast civil_liberties                       region Estimate 2.5 % 97.5 %\n#&gt;  civil_liberties dY/dX                    0.5 Middle East and North Africa    0.816 0.621  1.007\n#&gt;  party_autonomy  TRUE - FALSE             0.5 Middle East and North Africa    0.252 0.166  0.336\n#&gt; \n#&gt; Columns: rowid, term, contrast, estimate, conf.low, conf.high, civil_liberties, region, predicted_lo, predicted_hi, predicted, tmp_idx, media_index, party_autonomy \n#&gt; Type:  response\n\nUse the posterior_draws() to extract draws from the posterior distribution of marginal effects, and plot them:\n\nmfx &lt;- posterior_draws(mfx)\n\nggplot(mfx, aes(x = draw, y = term)) +\n  stat_halfeye() +\n  labs(x = \"Marginal effect\", y = \"\")\n\n\n\n\nPlot marginal effects, conditional on a regressor:\n\nplot_slopes(mod,\n         variables = \"civil_liberties\",\n         condition = \"party_autonomy\")\n\n\n\n\n\n8.2.3 Continuous predictors\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(party_autonomy = FALSE,\n                                       region = \"Middle East and North Africa\",\n                                       civil_liberties = seq(0, 1, by = 0.05))) |&gt;\n        posterior_draws()\n\nggplot(pred, aes(x = civil_liberties, y = draw)) +\n    stat_lineribbon() +\n    scale_fill_brewer(palette = \"Reds\") +\n    labs(x = \"Civil liberties\",\n         y = \"Media index (predicted)\",\n         fill = \"\")\n\n\n\n\nThe slope of this line for different values of civil liberties can be obtained with:\n\nmfx &lt;- slopes(mod,\n    newdata = datagrid(\n        civil_liberties = c(.2, .5, .8),\n        party_autonomy = FALSE,\n        region = \"Middle East and North Africa\"),\n    variables = \"civil_liberties\")\nmfx\n#&gt; \n#&gt;             Term civil_liberties party_autonomy                       region Estimate 2.5 % 97.5 %\n#&gt;  civil_liberties             0.2          FALSE Middle East and North Africa    0.490 0.361  0.639\n#&gt;  civil_liberties             0.5          FALSE Middle East and North Africa    0.807 0.612  0.993\n#&gt;  civil_liberties             0.8          FALSE Middle East and North Africa    0.807 0.674  0.934\n#&gt; \n#&gt; Columns: rowid, term, estimate, conf.low, conf.high, civil_liberties, party_autonomy, region, predicted_lo, predicted_hi, predicted, tmp_idx, media_index \n#&gt; Type:  response\n\nAnd plotted:\n\nmfx &lt;- posterior_draws(mfx)\n\nggplot(mfx, aes(x = draw, fill = factor(civil_liberties))) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Marginal effect of Civil Liberties on Media Index\",\n         y = \"Posterior density\",\n         fill = \"Civil liberties\")\n\n\n\n\nThe slopes() function can use the ellipsis (...) to push any argument forward to the posterior_predict function. This can alter the types of predictions returned. For example, the re_formula=NA argument of the posterior_predict.brmsfit method will compute marginaleffects without including any group-level effects:\n\nmfx &lt;- slopes(\n    mod,\n    newdata = datagrid(\n        civil_liberties = c(.2, .5, .8),\n        party_autonomy = FALSE,\n        region = \"Middle East and North Africa\"),\n    variables = \"civil_liberties\",\n    re_formula = NA) |&gt;\n    posterior_draws()\n\nggplot(mfx, aes(x = draw, fill = factor(civil_liberties))) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Marginal effect of Civil Liberties on Media Index\",\n         y = \"Posterior density\",\n         fill = \"Civil liberties\")\n\n\n\n\n\n8.2.4 Global grand mean\n\npred &lt;- predictions(\n    mod,\n    re_formula = NA,\n    newdata = datagrid(party_autonomy = c(TRUE, FALSE))) |&gt;\n    posterior_draws()\n\nmfx &lt;- slopes(\n    mod,\n    re_formula = NA,\n    variables = \"party_autonomy\") |&gt;\n    posterior_draws()\n\nplot1 &lt;- ggplot(pred, aes(x = draw, fill = party_autonomy)) +\n         stat_halfeye(slab_alpha = .5) +\n         labs(x = \"Media index (Predicted)\",\n              y = \"Posterior density\",\n              fill = \"Party autonomy\")\n\nplot2 &lt;- ggplot(mfx, aes(x = draw)) +\n         stat_halfeye(slab_alpha = .5)  +\n         labs(x = \"Contrast: Party autonomy TRUE - FALSE\",\n              y = \"\",\n              fill = \"Party autonomy\")\n\n## combine plots using the `patchwork` package\nplot1 + plot2\n\n\n\n\n\n8.2.5 Region-specific predictions and contrasts\nPredicted media index by region and level of civil liberties:\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(region = vdem_2015$region,\n                                       party_autonomy = FALSE, \n                                       civil_liberties = seq(0, 1, length.out = 100))) |&gt; \n        posterior_draws()\n\nggplot(pred, aes(x = civil_liberties, y = draw)) +\n    stat_lineribbon() +\n    scale_fill_brewer(palette = \"Reds\") +\n    facet_wrap(~ region) +\n    labs(x = \"Civil liberties\",\n         y = \"Media index (predicted)\",\n         fill = \"\")\n\n\n\n\nPredicted media index by region and level of civil liberties:\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(region = vdem_2015$region,\n                                       civil_liberties = c(.2, .8),\n                                      party_autonomy = FALSE)) |&gt;\n        posterior_draws()\n\nggplot(pred, aes(x = draw, fill = factor(civil_liberties))) +\n    stat_halfeye(slab_alpha = .5) +\n    facet_wrap(~ region) +\n    labs(x = \"Media index (predicted)\",\n         y = \"Posterior density\",\n         fill = \"Civil liberties\")\n\n\n\n\nPredicted media index by region and party autonomy:\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(region = vdem_2015$region,\n                                       party_autonomy = c(TRUE, FALSE),\n                                       civil_liberties = .5)) |&gt;\n        posterior_draws()\n\nggplot(pred, aes(x = draw, y = region , fill = party_autonomy)) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Media index (predicted)\",\n         y = \"\",\n         fill = \"Party autonomy\")\n\n\n\n\nTRUE/FALSE contrasts (marginal effects) of party autonomy by region:\n\nmfx &lt;- slopes(\n    mod,\n    variables = \"party_autonomy\",\n    newdata = datagrid(\n        region = vdem_2015$region,\n        civil_liberties = .5)) |&gt;\n    posterior_draws()\n\nggplot(mfx, aes(x = draw, y = region , fill = party_autonomy)) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Media index (predicted)\",\n         y = \"\",\n         fill = \"Party autonomy\")\n\n\n\n\n\n8.2.6 Hypothetical groups\nWe can also obtain predictions or marginal effects for a hypothetical group instead of one of the observed regions. To achieve this, we create a dataset with NA in the region column. Then we call the marginaleffects or predictions() functions with the allow_new_levels argument. This argument is pushed through via the ellipsis (...) to the posterior_epred function of the brms package:\n\ndat &lt;- data.frame(civil_liberties = .5,\n                  party_autonomy = FALSE,\n                  region = \"New Region\")\n\nmfx &lt;- slopes(\n    mod,\n    variables = \"party_autonomy\",\n    allow_new_levels = TRUE,\n    newdata = dat)\n\ndraws &lt;- posterior_draws(mfx)\n\nggplot(draws, aes(x = draw)) +\n     stat_halfeye() +\n     labs(x = \"Marginal effect of party autonomy in a generic world region\", y = \"\")\n\n\n\n\n\n8.2.7 Averaging, marginalizing, integrating random effects\nConsider a logistic regression model with random effects:\n\ndat &lt;- read.csv(\"https://vincentarelbundock.github.io/Rdatasets/csv/plm/EmplUK.csv\")\ndat$x &lt;- as.numeric(dat$output &gt; median(dat$output))\ndat$y &lt;- as.numeric(dat$emp &gt; median(dat$emp))\nmod &lt;- brm(y ~ x + (1 | firm), data = dat, backend = \"cmdstanr\", family = \"bernoulli\")\n\nWe can compute adjusted predictions for a given value of x and for each firm (random effects) as follows:\n\np &lt;- predictions(mod, newdata = datagrid(x = 0, firm = unique))\nhead(p)\n#&gt; \n#&gt;  x firm Estimate    2.5 % 97.5 %\n#&gt;  0    1  1.0e+00 9.01e-01 1.0000\n#&gt;  0    2  1.0e+00 8.95e-01 1.0000\n#&gt;  0    3  1.0e+00 9.12e-01 1.0000\n#&gt;  0    4  1.0e+00 7.97e-01 1.0000\n#&gt;  0    5  1.0e+00 9.09e-01 1.0000\n#&gt;  0    6  4.9e-08 8.42e-21 0.0019\n#&gt; \n#&gt; Columns: rowid, estimate, conf.low, conf.high, y, x, firm \n#&gt; Type:  response\n\nWe can average/marginalize/integrate across random effects with the avg_predictions() function or the by argument:\n\navg_predictions(mod, newdata = datagrid(x = 0, firm = unique))\n#&gt; \n#&gt;  Estimate 2.5 % 97.5 %\n#&gt;     0.454  0.44  0.468\n#&gt; \n#&gt; Columns: estimate, conf.low, conf.high \n#&gt; Type:  response\n\npredictions(mod, newdata = datagrid(x = 0:1, firm = unique), by = \"x\")\n#&gt; \n#&gt;  x Estimate 2.5 % 97.5 %\n#&gt;  0    0.454 0.440  0.468\n#&gt;  1    0.557 0.546  0.570\n#&gt; \n#&gt; Columns: x, estimate, conf.low, conf.high \n#&gt; Type:  response\n\nWe can also draw from the (assumed gaussian) population distribution of random effects, by asking predictions() to make predictions for new “levels” of the random effects. If we then take an average of predictions using avg_predictions() or the by argument, we will have “integrated out the random effects”, as described in the brmsmargins package vignette. In the code below, we make predictions for 100 firm identifiers which were not in the original dataset. We also ask predictions() to push forward the allow_new_levels and sample_new_levels arguments to the brms::posterior_epred function:\n\npredictions(\n    mod,\n    newdata = datagrid(x = 0:1, firm = -1:-100),\n    allow_new_levels = TRUE,\n    sample_new_levels = \"gaussian\",\n    by = \"x\")\n#&gt; \n#&gt;  x Estimate 2.5 % 97.5 %\n#&gt;  0    0.454 0.339  0.567\n#&gt;  1    0.553 0.433  0.667\n#&gt; \n#&gt; Columns: x, estimate, conf.low, conf.high \n#&gt; Type:  response\n\nWe can “integrate out” random effects in the other slopes() functions too. For instance,\n\navg_comparisons(\n    mod,\n    newdata = datagrid(firm = -1:-100),\n    allow_new_levels = TRUE,\n    sample_new_levels = \"gaussian\")\n#&gt; \n#&gt;  Term Contrast Estimate  2.5 % 97.5 %\n#&gt;     x    1 - 0    0.096 0.0471  0.161\n#&gt; \n#&gt; Columns: term, contrast, estimate, conf.low, conf.high \n#&gt; Type:  response\n\nThis is nearly equivalent the brmsmargins command output (with slight variations due to different random seeds):\n\nlibrary(brmsmargins)\nbm &lt;- brmsmargins(\n  k = 100,\n  object = mod,\n  at = data.frame(x = c(0, 1)),\n  CI = .95,\n  CIType = \"ETI\",\n  contrasts = cbind(\"AME x\" = c(-1, 1)),\n  effects = \"integrateoutRE\")\nbm$ContrastSummary |&gt; data.frame()\n#&gt;            M        Mdn         LL        UL PercentROPE PercentMID   CI CIType ROPE  MID Label\n#&gt; 1 0.09920292 0.09745593 0.04806257 0.1629364          NA         NA 0.95    ETI &lt;NA&gt; &lt;NA&gt; AME x\n\nSee the alternative software vignette for more information on brmsmargins."
+    "text": "8.2 Random effects model\nThis section replicates some of the analyses of a random effects model published in Andrew Heiss’ blog post: “A guide to correctly calculating posterior predictions and average marginal effects with multilevel Bayesian models.” The objective is mainly to illustrate the use of marginaleffects. Please refer to the original post for a detailed discussion of the quantities computed below.\nLoad libraries and download data:\n\nlibrary(brms)\nlibrary(ggdist)\nlibrary(patchwork)\nlibrary(marginaleffects)\n\nvdem_2015 &lt;- read.csv(\"https://github.com/vincentarelbundock/marginaleffects/raw/main/data-raw/vdem_2015.csv\")\n\nhead(vdem_2015)\n#&gt;   country_name country_text_id year                           region media_index party_autonomy_ord polyarchy civil_liberties party_autonomy\n#&gt; 1       Mexico             MEX 2015  Latin America and the Caribbean       0.837                  3     0.631           0.704           TRUE\n#&gt; 2     Suriname             SUR 2015  Latin America and the Caribbean       0.883                  4     0.777           0.887           TRUE\n#&gt; 3       Sweden             SWE 2015 Western Europe and North America       0.956                  4     0.915           0.968           TRUE\n#&gt; 4  Switzerland             CHE 2015 Western Europe and North America       0.939                  4     0.901           0.960           TRUE\n#&gt; 5        Ghana             GHA 2015               Sub-Saharan Africa       0.858                  4     0.724           0.921           TRUE\n#&gt; 6 South Africa             ZAF 2015               Sub-Saharan Africa       0.898                  4     0.752           0.869           TRUE\n\nFit a basic model:\n\nmod &lt;- brm(\n  bf(media_index ~ party_autonomy + civil_liberties + (1 | region),\n     phi ~ (1 | region)),\n  data = vdem_2015,\n  family = Beta(),\n  control = list(adapt_delta = 0.9))\n\n\n8.2.1 Posterior predictions\nTo compute posterior predictions for specific values of the regressors, we use the newdata argument and the datagrid() function. We also use the type argument to compute two types of predictions: accounting for residual (observation-level) residual variance (prediction) or ignoring it (response).\n\nnd = datagrid(model = mod,\n              party_autonomy = c(TRUE, FALSE),\n              civil_liberties = .5,\n              region = \"Middle East and North Africa\")\np1 &lt;- predictions(mod, type = \"response\", newdata = nd) |&gt;\n    posterior_draws() |&gt;\n    transform(type = \"Response\")\np2 &lt;- predictions(mod, type = \"prediction\", newdata = nd) |&gt;\n    posterior_draws() |&gt;\n    transform(type = \"Prediction\")\npred &lt;- rbind(p1, p2)\n\nExtract posterior draws and plot them:\n\nggplot(pred, aes(x = draw, fill = party_autonomy)) +\n    stat_halfeye(alpha = .5) +\n    facet_wrap(~ type) +\n    labs(x = \"Media index (predicted)\", \n         y = \"Posterior density\",\n         fill = \"Party autonomy\")\n\n\n\n\n\n8.2.2 Marginal effects and contrasts\nAs noted in the Marginal Effects vignette, there should be one distinct marginal effect for each combination of regressor values. Here, we consider only one combination of regressor values, where region is “Middle East and North Africa”, and civil_liberties is 0.5. Then, we calculate the mean of the posterior distribution of marginal effects:\n\nmfx &lt;- slopes(mod,\n                       newdata = datagrid(civil_liberties = .5,\n                                          region = \"Middle East and North Africa\"))\nmfx\n#&gt; \n#&gt;             Term     Contrast civil_liberties                       region Estimate 2.5 % 97.5 %\n#&gt;  civil_liberties dY/dX                    0.5 Middle East and North Africa    0.816 0.621  1.007\n#&gt;  party_autonomy  TRUE - FALSE             0.5 Middle East and North Africa    0.252 0.166  0.336\n#&gt; \n#&gt; Columns: rowid, term, contrast, estimate, conf.low, conf.high, civil_liberties, region, predicted_lo, predicted_hi, predicted, tmp_idx, media_index, party_autonomy \n#&gt; Type:  response\n\nUse the posterior_draws() to extract draws from the posterior distribution of marginal effects, and plot them:\n\nmfx &lt;- posterior_draws(mfx)\n\nggplot(mfx, aes(x = draw, y = term)) +\n  stat_halfeye() +\n  labs(x = \"Marginal effect\", y = \"\")\n\n\n\n\nPlot marginal effects, conditional on a regressor:\n\nplot_slopes(mod,\n         variables = \"civil_liberties\",\n         condition = \"party_autonomy\")\n\n\n\n\n\n8.2.3 Continuous predictors\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(party_autonomy = FALSE,\n                                       region = \"Middle East and North Africa\",\n                                       civil_liberties = seq(0, 1, by = 0.05))) |&gt;\n        posterior_draws()\n\nggplot(pred, aes(x = civil_liberties, y = draw)) +\n    stat_lineribbon() +\n    scale_fill_brewer(palette = \"Reds\") +\n    labs(x = \"Civil liberties\",\n         y = \"Media index (predicted)\",\n         fill = \"\")\n\n\n\n\nThe slope of this line for different values of civil liberties can be obtained with:\n\nmfx &lt;- slopes(mod,\n    newdata = datagrid(\n        civil_liberties = c(.2, .5, .8),\n        party_autonomy = FALSE,\n        region = \"Middle East and North Africa\"),\n    variables = \"civil_liberties\")\nmfx\n#&gt; \n#&gt;             Term civil_liberties party_autonomy                       region Estimate 2.5 % 97.5 %\n#&gt;  civil_liberties             0.2          FALSE Middle East and North Africa    0.490 0.361  0.639\n#&gt;  civil_liberties             0.5          FALSE Middle East and North Africa    0.807 0.612  0.993\n#&gt;  civil_liberties             0.8          FALSE Middle East and North Africa    0.807 0.674  0.934\n#&gt; \n#&gt; Columns: rowid, term, estimate, conf.low, conf.high, civil_liberties, party_autonomy, region, predicted_lo, predicted_hi, predicted, tmp_idx, media_index \n#&gt; Type:  response\n\nAnd plotted:\n\nmfx &lt;- posterior_draws(mfx)\n\nggplot(mfx, aes(x = draw, fill = factor(civil_liberties))) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Marginal effect of Civil Liberties on Media Index\",\n         y = \"Posterior density\",\n         fill = \"Civil liberties\")\n\n\n\n\nThe slopes() function can use the ellipsis (...) to push any argument forward to the posterior_predict function. This can alter the types of predictions returned. For example, the re_formula=NA argument of the posterior_predict.brmsfit method will compute marginaleffects without including any group-level effects:\n\nmfx &lt;- slopes(\n    mod,\n    newdata = datagrid(\n        civil_liberties = c(.2, .5, .8),\n        party_autonomy = FALSE,\n        region = \"Middle East and North Africa\"),\n    variables = \"civil_liberties\",\n    re_formula = NA) |&gt;\n    posterior_draws()\n\nggplot(mfx, aes(x = draw, fill = factor(civil_liberties))) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Marginal effect of Civil Liberties on Media Index\",\n         y = \"Posterior density\",\n         fill = \"Civil liberties\")\n\n\n\n\n\n8.2.4 Global grand mean\n\npred &lt;- predictions(\n    mod,\n    re_formula = NA,\n    newdata = datagrid(party_autonomy = c(TRUE, FALSE))) |&gt;\n    posterior_draws()\n\nmfx &lt;- slopes(\n    mod,\n    re_formula = NA,\n    variables = \"party_autonomy\") |&gt;\n    posterior_draws()\n\nplot1 &lt;- ggplot(pred, aes(x = draw, fill = party_autonomy)) +\n         stat_halfeye(slab_alpha = .5) +\n         labs(x = \"Media index (Predicted)\",\n              y = \"Posterior density\",\n              fill = \"Party autonomy\")\n\nplot2 &lt;- ggplot(mfx, aes(x = draw)) +\n         stat_halfeye(slab_alpha = .5)  +\n         labs(x = \"Contrast: Party autonomy TRUE - FALSE\",\n              y = \"\",\n              fill = \"Party autonomy\")\n\n## combine plots using the `patchwork` package\nplot1 + plot2\n\n\n\n\n\n8.2.5 Region-specific predictions and contrasts\nPredicted media index by region and level of civil liberties:\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(region = vdem_2015$region,\n                                       party_autonomy = FALSE, \n                                       civil_liberties = seq(0, 1, length.out = 100))) |&gt; \n        posterior_draws()\n\nggplot(pred, aes(x = civil_liberties, y = draw)) +\n    stat_lineribbon() +\n    scale_fill_brewer(palette = \"Reds\") +\n    facet_wrap(~ region) +\n    labs(x = \"Civil liberties\",\n         y = \"Media index (predicted)\",\n         fill = \"\")\n\n\n\n\nPredicted media index by region and level of civil liberties:\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(region = vdem_2015$region,\n                                       civil_liberties = c(.2, .8),\n                                      party_autonomy = FALSE)) |&gt;\n        posterior_draws()\n\nggplot(pred, aes(x = draw, fill = factor(civil_liberties))) +\n    stat_halfeye(slab_alpha = .5) +\n    facet_wrap(~ region) +\n    labs(x = \"Media index (predicted)\",\n         y = \"Posterior density\",\n         fill = \"Civil liberties\")\n\n\n\n\nPredicted media index by region and party autonomy:\n\npred &lt;- predictions(mod,\n                    newdata = datagrid(region = vdem_2015$region,\n                                       party_autonomy = c(TRUE, FALSE),\n                                       civil_liberties = .5)) |&gt;\n        posterior_draws()\n\nggplot(pred, aes(x = draw, y = region , fill = party_autonomy)) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Media index (predicted)\",\n         y = \"\",\n         fill = \"Party autonomy\")\n\n\n\n\nTRUE/FALSE contrasts (marginal effects) of party autonomy by region:\n\nmfx &lt;- slopes(\n    mod,\n    variables = \"party_autonomy\",\n    newdata = datagrid(\n        region = vdem_2015$region,\n        civil_liberties = .5)) |&gt;\n    posterior_draws()\n\nggplot(mfx, aes(x = draw, y = region , fill = party_autonomy)) +\n    stat_halfeye(slab_alpha = .5) +\n    labs(x = \"Media index (predicted)\",\n         y = \"\",\n         fill = \"Party autonomy\")\n\n\n\n\n\n8.2.6 Hypothetical groups\nWe can also obtain predictions or marginal effects for a hypothetical group instead of one of the observed regions. To achieve this, we create a dataset with NA in the region column. Then we call the marginaleffects or predictions() functions with the allow_new_levels argument. This argument is pushed through via the ellipsis (...) to the posterior_epred function of the brms package:\n\ndat &lt;- data.frame(civil_liberties = .5,\n                  party_autonomy = FALSE,\n                  region = \"New Region\")\n\nmfx &lt;- slopes(\n    mod,\n    variables = \"party_autonomy\",\n    allow_new_levels = TRUE,\n    newdata = dat)\n\ndraws &lt;- posterior_draws(mfx)\n\nggplot(draws, aes(x = draw)) +\n     stat_halfeye() +\n     labs(x = \"Marginal effect of party autonomy in a generic world region\", y = \"\")\n\n\n\n\n\n8.2.7 Averaging, marginalizing, integrating random effects\nConsider a logistic regression model with random effects:\n\ndat &lt;- read.csv(\"https://vincentarelbundock.github.io/Rdatasets/csv/plm/EmplUK.csv\")\ndat$x &lt;- as.numeric(dat$output &gt; median(dat$output))\ndat$y &lt;- as.numeric(dat$emp &gt; median(dat$emp))\nmod &lt;- brm(y ~ x + (1 | firm), data = dat, backend = \"cmdstanr\", family = \"bernoulli\")\n\nWe can compute adjusted predictions for a given value of x and for each firm (random effects) as follows:\n\np &lt;- predictions(mod, newdata = datagrid(x = 0, firm = unique))\nhead(p)\n#&gt; \n#&gt;  x firm Estimate    2.5 % 97.5 %\n#&gt;  0    1  1.0e+00 9.01e-01 1.0000\n#&gt;  0    2  1.0e+00 8.95e-01 1.0000\n#&gt;  0    3  1.0e+00 9.12e-01 1.0000\n#&gt;  0    4  1.0e+00 7.97e-01 1.0000\n#&gt;  0    5  1.0e+00 9.09e-01 1.0000\n#&gt;  0    6  4.9e-08 8.42e-21 0.0019\n#&gt; \n#&gt; Columns: rowid, estimate, conf.low, conf.high, y, x, firm \n#&gt; Type:  response\n\nWe can average/marginalize/integrate across random effects with the avg_predictions() function or the by argument:\n\navg_predictions(mod, newdata = datagrid(x = 0, firm = unique))\n#&gt; \n#&gt;  Estimate 2.5 % 97.5 %\n#&gt;     0.454  0.44  0.468\n#&gt; \n#&gt; Columns: estimate, conf.low, conf.high \n#&gt; Type:  response\n\npredictions(mod, newdata = datagrid(x = 0:1, firm = unique), by = \"x\")\n#&gt; \n#&gt;  x Estimate 2.5 % 97.5 %\n#&gt;  0    0.454 0.440  0.468\n#&gt;  1    0.557 0.546  0.570\n#&gt; \n#&gt; Columns: x, estimate, conf.low, conf.high \n#&gt; Type:  response\n\nWe can also draw from the (assumed gaussian) population distribution of random effects, by asking predictions() to make predictions for new “levels” of the random effects. If we then take an average of predictions using avg_predictions() or the by argument, we will have “integrated out the random effects”, as described in the brmsmargins package vignette. In the code below, we make predictions for 100 firm identifiers which were not in the original dataset. We also ask predictions() to push forward the allow_new_levels and sample_new_levels arguments to the brms::posterior_epred function:\n\npredictions(\n    mod,\n    newdata = datagrid(x = 0:1, firm = -1:-100),\n    allow_new_levels = TRUE,\n    sample_new_levels = \"gaussian\",\n    by = \"x\")\n#&gt; \n#&gt;  x Estimate 2.5 % 97.5 %\n#&gt;  0    0.452 0.339  0.564\n#&gt;  1    0.551 0.431  0.660\n#&gt; \n#&gt; Columns: x, estimate, conf.low, conf.high \n#&gt; Type:  response\n\nWe can “integrate out” random effects in the other slopes() functions too. For instance,\n\navg_comparisons(\n    mod,\n    newdata = datagrid(firm = -1:-100),\n    allow_new_levels = TRUE,\n    sample_new_levels = \"gaussian\")\n#&gt; \n#&gt;  Term Contrast Estimate  2.5 % 97.5 %\n#&gt;     x    1 - 0   0.0968 0.0484  0.161\n#&gt; \n#&gt; Columns: term, contrast, estimate, conf.low, conf.high \n#&gt; Type:  response\n\nThis is nearly equivalent the brmsmargins command output (with slight variations due to different random seeds):\n\nlibrary(brmsmargins)\nbm &lt;- brmsmargins(\n  k = 100,\n  object = mod,\n  at = data.frame(x = c(0, 1)),\n  CI = .95,\n  CIType = \"ETI\",\n  contrasts = cbind(\"AME x\" = c(-1, 1)),\n  effects = \"integrateoutRE\")\nbm$ContrastSummary |&gt; data.frame()\n#&gt;            M        Mdn         LL        UL PercentROPE PercentMID   CI CIType ROPE  MID Label\n#&gt; 1 0.09850989 0.09628107 0.04727819 0.1630572          NA         NA 0.95    ETI &lt;NA&gt; &lt;NA&gt; AME x\n\nSee the alternative software vignette for more information on brmsmargins."
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     "objectID": "articles/brms.html#multinomial-logit",
@@ -501,14 +501,14 @@
     "href": "articles/bootstrap.html#bootstrap",
     "title": "\n9  Bootstrap & Simulation\n",
     "section": "\n9.2 Bootstrap",
-    "text": "9.2 Bootstrap\nmarginaleffects supports three bootstrap frameworks in R: the well-established boot package, the newer rsample package, and the so-called “bayesian bootstrap” in fwb.\n\n9.2.1 boot\n\n\navg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"boot\")\n#&gt; \n#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.266 -0.634  0.459\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.156 -0.337  0.291\n#&gt;  Petal.Width mean(+1) virginica    0.0216      0.174 -0.333  0.364\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, std.error, conf.low, conf.high \n#&gt; Type:  response\n\nAll unknown arguments that we feed to inferences() are pushed forward to boot::boot():\n\nest &lt;- avg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"boot\", sim = \"balanced\", R = 500, conf_type = \"bca\")\nest\n#&gt; \n#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.265 -0.565  0.468\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.159 -0.295  0.312\n#&gt;  Petal.Width mean(+1) virginica    0.0216      0.183 -0.318  0.417\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, std.error, conf.low, conf.high \n#&gt; Type:  response\n\nWe can extract the original boot object from an attribute:\n\nattr(est, \"inferences\")\n#&gt; \n#&gt; BALANCED BOOTSTRAP\n#&gt; \n#&gt; \n#&gt; Call:\n#&gt; bootstrap_boot(model = model, FUN = FUN, newdata = ..1, vcov = ..2, \n#&gt;     variables = ..3, type = ..4, by = ..5, conf_level = ..6, \n#&gt;     cross = ..7, comparison = ..8, transform = ..9, wts = ..10, \n#&gt;     hypothesis = ..11, eps = ..12)\n#&gt; \n#&gt; \n#&gt; Bootstrap Statistics :\n#&gt;        original       bias    std. error\n#&gt; t1* -0.11025325  0.007095463   0.2647118\n#&gt; t2* -0.02006005  0.001977153   0.1588222\n#&gt; t3*  0.02158742 -0.000386270   0.1826585\n\nOr we can extract the individual draws with the posterior_draws() function:\n\nposterior_draws(est) |&gt; head()\n#&gt;   drawid        draw        term contrast    Species    estimate predicted_lo predicted_hi predicted std.error   conf.low conf.high\n#&gt; 1      1  0.34291638 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2647118 -0.5647467 0.4675544\n#&gt; 2      1  0.11507292 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1588222 -0.2951916 0.3121828\n#&gt; 3      1  0.00986429 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1826585 -0.3182129 0.4167165\n#&gt; 4      2 -0.17769879 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2647118 -0.5647467 0.4675544\n#&gt; 5      2 -0.08218494 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1588222 -0.2951916 0.3121828\n#&gt; 6      2 -0.03808062 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1826585 -0.3182129 0.4167165\n\nposterior_draws(est, shape = \"DxP\") |&gt; dim()\n#&gt; [1] 500   3\n\n\n9.2.2 rsample\n\nAs before, we can pass arguments to rsample::bootstraps() through inferences(). For example, for stratified resampling:\n\nest &lt;- avg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"rsample\", R = 100, strata = \"Species\")\nest\n#&gt; \n#&gt;         Term Contrast    Species Estimate  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103 -0.524  0.386\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201 -0.277  0.260\n#&gt;  Petal.Width mean(+1) virginica    0.0216 -0.300  0.362\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, conf.low, conf.high \n#&gt; Type:  response\n\nattr(est, \"inferences\")\n#&gt; # Bootstrap sampling using stratification with apparent sample \n#&gt; # A tibble: 101 × 3\n#&gt;    splits           id           estimates       \n#&gt;    &lt;list&gt;           &lt;chr&gt;        &lt;list&gt;          \n#&gt;  1 &lt;split [150/53]&gt; Bootstrap001 &lt;tibble [3 × 7]&gt;\n#&gt;  2 &lt;split [150/53]&gt; Bootstrap002 &lt;tibble [3 × 7]&gt;\n#&gt;  3 &lt;split [150/57]&gt; Bootstrap003 &lt;tibble [3 × 7]&gt;\n#&gt;  4 &lt;split [150/52]&gt; Bootstrap004 &lt;tibble [3 × 7]&gt;\n#&gt;  5 &lt;split [150/62]&gt; Bootstrap005 &lt;tibble [3 × 7]&gt;\n#&gt;  6 &lt;split [150/47]&gt; Bootstrap006 &lt;tibble [3 × 7]&gt;\n#&gt;  7 &lt;split [150/56]&gt; Bootstrap007 &lt;tibble [3 × 7]&gt;\n#&gt;  8 &lt;split [150/51]&gt; Bootstrap008 &lt;tibble [3 × 7]&gt;\n#&gt;  9 &lt;split [150/54]&gt; Bootstrap009 &lt;tibble [3 × 7]&gt;\n#&gt; 10 &lt;split [150/55]&gt; Bootstrap010 &lt;tibble [3 × 7]&gt;\n#&gt; # ℹ 91 more rows\n\nOr we can extract the individual draws with the posterior_draws() function:\n\nposterior_draws(est) |&gt; head()\n#&gt;   drawid         draw        term contrast    Species    estimate predicted_lo predicted_hi predicted   conf.low conf.high\n#&gt; 1      1 -0.239618402 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.5241231 0.3859502\n#&gt; 2      1 -0.072908264 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.2771160 0.2599758\n#&gt; 3      1  0.004071542 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3000476 0.3619014\n#&gt; 4      2 -0.216492088 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.5241231 0.3859502\n#&gt; 5      2  0.003450908 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.2771160 0.2599758\n#&gt; 6      2  0.105011435 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3000476 0.3619014\n\nposterior_draws(est, shape = \"PxD\") |&gt; dim()\n#&gt; [1]   3 100\n\n\n9.2.3 Fractional Weighted Bootstrap (aka Bayesian Bootstrap)\nThe fwb package implements fractional weighted bootstrap (aka Bayesian bootstrap):\n\n“fwb implements the fractional weighted bootstrap (FWB), also known as the Bayesian bootstrap, following the treatment by Xu et al. (2020). The FWB involves generating sets of weights from a uniform Dirichlet distribution to be used in estimating statistics of interest, which yields a posterior distribution that can be interpreted in the same way the traditional (resampling-based) bootstrap distribution can be.” -Noah Greifer\n\nThe inferences() function makes it easy to apply this inference strategy to marginaleffects objects:\n\navg_comparisons(mod) |&gt; inferences(method = \"fwb\")\n#&gt; \n#&gt;          Term            Contrast Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Length +1                    0.8929     0.0784  0.737  1.046\n#&gt;  Petal.Width  +1                   -0.0362     0.1644 -0.351  0.293\n#&gt;  Species      versicolor - setosa  -1.4629     0.3265 -2.118 -0.851\n#&gt;  Species      virginica - setosa   -1.9842     0.3892 -2.800 -1.257\n#&gt; \n#&gt; Columns: term, contrast, estimate, std.error, conf.low, conf.high \n#&gt; Type:  response"
+    "text": "9.2 Bootstrap\nmarginaleffects supports three bootstrap frameworks in R: the well-established boot package, the newer rsample package, and the so-called “bayesian bootstrap” in fwb.\n\n9.2.1 boot\n\n\navg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"boot\")\n#&gt; \n#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.274 -0.640  0.398\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.159 -0.329  0.319\n#&gt;  Petal.Width mean(+1) virginica    0.0216      0.175 -0.323  0.388\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, std.error, conf.low, conf.high \n#&gt; Type:  response\n\nAll unknown arguments that we feed to inferences() are pushed forward to boot::boot():\n\nest &lt;- avg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"boot\", sim = \"balanced\", R = 500, conf_type = \"bca\")\nest\n#&gt; \n#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.268 -0.646  0.468\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.157 -0.316  0.294\n#&gt;  Petal.Width mean(+1) virginica    0.0216      0.182 -0.357  0.368\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, std.error, conf.low, conf.high \n#&gt; Type:  response\n\nWe can extract the original boot object from an attribute:\n\nattr(est, \"inferences\")\n#&gt; \n#&gt; BALANCED BOOTSTRAP\n#&gt; \n#&gt; \n#&gt; Call:\n#&gt; bootstrap_boot(model = model, FUN = FUN, newdata = ..1, vcov = ..2, \n#&gt;     variables = ..3, type = ..4, by = ..5, conf_level = ..6, \n#&gt;     cross = ..7, comparison = ..8, transform = ..9, wts = ..10, \n#&gt;     hypothesis = ..11, eps = ..12)\n#&gt; \n#&gt; \n#&gt; Bootstrap Statistics :\n#&gt;        original        bias    std. error\n#&gt; t1* -0.11025325  3.620023e-03   0.2682605\n#&gt; t2* -0.02006005  5.498175e-05   0.1569108\n#&gt; t3*  0.02158742 -1.591206e-03   0.1821860\n\nOr we can extract the individual draws with the posterior_draws() function:\n\nposterior_draws(est) |&gt; head()\n#&gt;   drawid        draw        term contrast    Species    estimate predicted_lo predicted_hi predicted std.error   conf.low conf.high\n#&gt; 1      1 -0.20715923 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2682605 -0.6455666 0.4681477\n#&gt; 2      1  0.11246218 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1569108 -0.3156269 0.2944124\n#&gt; 3      1  0.26005004 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1821860 -0.3565338 0.3684294\n#&gt; 4      2 -0.02019376 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 0.2682605 -0.6455666 0.4681477\n#&gt; 5      2 -0.12237572 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 0.1569108 -0.3156269 0.2944124\n#&gt; 6      2 -0.16955910 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 0.1821860 -0.3565338 0.3684294\n\nposterior_draws(est, shape = \"DxP\") |&gt; dim()\n#&gt; [1] 500   3\n\n\n9.2.2 rsample\n\nAs before, we can pass arguments to rsample::bootstraps() through inferences(). For example, for stratified resampling:\n\nest &lt;- avg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"rsample\", R = 100, strata = \"Species\")\nest\n#&gt; \n#&gt;         Term Contrast    Species Estimate  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103 -0.619  0.581\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201 -0.355  0.265\n#&gt;  Petal.Width mean(+1) virginica    0.0216 -0.356  0.399\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, predicted_lo, predicted_hi, predicted, conf.low, conf.high \n#&gt; Type:  response\n\nattr(est, \"inferences\")\n#&gt; # Bootstrap sampling using stratification with apparent sample \n#&gt; # A tibble: 101 × 3\n#&gt;    splits           id           estimates       \n#&gt;    &lt;list&gt;           &lt;chr&gt;        &lt;list&gt;          \n#&gt;  1 &lt;split [150/48]&gt; Bootstrap001 &lt;tibble [3 × 7]&gt;\n#&gt;  2 &lt;split [150/55]&gt; Bootstrap002 &lt;tibble [3 × 7]&gt;\n#&gt;  3 &lt;split [150/50]&gt; Bootstrap003 &lt;tibble [3 × 7]&gt;\n#&gt;  4 &lt;split [150/50]&gt; Bootstrap004 &lt;tibble [3 × 7]&gt;\n#&gt;  5 &lt;split [150/50]&gt; Bootstrap005 &lt;tibble [3 × 7]&gt;\n#&gt;  6 &lt;split [150/53]&gt; Bootstrap006 &lt;tibble [3 × 7]&gt;\n#&gt;  7 &lt;split [150/58]&gt; Bootstrap007 &lt;tibble [3 × 7]&gt;\n#&gt;  8 &lt;split [150/58]&gt; Bootstrap008 &lt;tibble [3 × 7]&gt;\n#&gt;  9 &lt;split [150/55]&gt; Bootstrap009 &lt;tibble [3 × 7]&gt;\n#&gt; 10 &lt;split [150/57]&gt; Bootstrap010 &lt;tibble [3 × 7]&gt;\n#&gt; # ℹ 91 more rows\n\nOr we can extract the individual draws with the posterior_draws() function:\n\nposterior_draws(est) |&gt; head()\n#&gt;   drawid        draw        term contrast    Species    estimate predicted_lo predicted_hi predicted   conf.low conf.high\n#&gt; 1      1 -0.06230329 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.6185060 0.5805186\n#&gt; 2      1 -0.11938081 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.3551267 0.2650783\n#&gt; 3      1 -0.14573683 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3555306 0.3991535\n#&gt; 4      2  0.12473364 Petal.Width mean(+1)     setosa -0.11025325     5.013640     4.901389  4.957514 -0.6185060 0.5805186\n#&gt; 5      2  0.09889677 Petal.Width mean(+1) versicolor -0.02006005     6.330887     6.325011  6.327949 -0.3551267 0.2650783\n#&gt; 6      2  0.08696638 Petal.Width mean(+1)  virginica  0.02158742     6.997499     7.033528  7.015513 -0.3555306 0.3991535\n\nposterior_draws(est, shape = \"PxD\") |&gt; dim()\n#&gt; [1]   3 100\n\n\n9.2.3 Fractional Weighted Bootstrap (aka Bayesian Bootstrap)\nThe fwb package implements fractional weighted bootstrap (aka Bayesian bootstrap):\n\n“fwb implements the fractional weighted bootstrap (FWB), also known as the Bayesian bootstrap, following the treatment by Xu et al. (2020). The FWB involves generating sets of weights from a uniform Dirichlet distribution to be used in estimating statistics of interest, which yields a posterior distribution that can be interpreted in the same way the traditional (resampling-based) bootstrap distribution can be.” -Noah Greifer\n\nThe inferences() function makes it easy to apply this inference strategy to marginaleffects objects:\n\navg_comparisons(mod) |&gt; inferences(method = \"fwb\")\n#&gt; \n#&gt;          Term            Contrast Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Length +1                    0.8929     0.0807  0.736  1.052\n#&gt;  Petal.Width  +1                   -0.0362     0.1551 -0.341  0.279\n#&gt;  Species      versicolor - setosa  -1.4629     0.3221 -2.099 -0.813\n#&gt;  Species      virginica - setosa   -1.9842     0.3781 -2.724 -1.221\n#&gt; \n#&gt; Columns: term, contrast, estimate, std.error, conf.low, conf.high \n#&gt; Type:  response"
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     "title": "\n9  Bootstrap & Simulation\n",
     "section": "\n9.3 Simulation-based inference",
-    "text": "9.3 Simulation-based inference\nThis simulation-based strategy to compute confidence intervals was described in Krinsky & Robb (1986) and popularized by King, Tomz, Wittenberg (2000). We proceed in 3 steps:\n\nDraw R sets of simulated coefficients from a multivariate normal distribution with mean equal to the original model’s estimated coefficients and variance equal to the model’s variance-covariance matrix (classical, “HC3”, or other).\nUse the R sets of coefficients to compute R sets of estimands: predictions, comparisons, or slopes.\nTake quantiles of the resulting distribution of estimands to obtain a confidence interval and the standard deviation of simulated estimates to estimate the standard error.\n\nHere are a few examples:\n\nlibrary(ggplot2)\nlibrary(ggdist)\n\navg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"simulation\")\n#&gt; \n#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.291 -0.741  0.435\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.164 -0.347  0.299\n#&gt;  Petal.Width mean(+1) virginica    0.0216      0.168 -0.306  0.341\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, std.error, conf.low, conf.high, predicted_lo, predicted_hi, predicted, tmp_idx \n#&gt; Type:  response\n\nSince simulation based inference generates R estimates of the quantities of interest, we can treat them similarly to draws from the posterior distribution in bayesian models. For example, we can extract draws using the posterior_draws() function, and plot their distributions using packages likeggplot2 and ggdist:\n\navg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"simulation\") |&gt;\n  posterior_draws(\"rvar\") |&gt;\n  ggplot(aes(y = Species, xdist = rvar)) +\n  stat_slabinterval()"
+    "text": "9.3 Simulation-based inference\nThis simulation-based strategy to compute confidence intervals was described in Krinsky & Robb (1986) and popularized by King, Tomz, Wittenberg (2000). We proceed in 3 steps:\n\nDraw R sets of simulated coefficients from a multivariate normal distribution with mean equal to the original model’s estimated coefficients and variance equal to the model’s variance-covariance matrix (classical, “HC3”, or other).\nUse the R sets of coefficients to compute R sets of estimands: predictions, comparisons, or slopes.\nTake quantiles of the resulting distribution of estimands to obtain a confidence interval and the standard deviation of simulated estimates to estimate the standard error.\n\nHere are a few examples:\n\nlibrary(ggplot2)\nlibrary(ggdist)\n\navg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"simulation\")\n#&gt; \n#&gt;         Term Contrast    Species Estimate Std. Error  2.5 % 97.5 %\n#&gt;  Petal.Width mean(+1) setosa      -0.1103      0.283 -0.662  0.437\n#&gt;  Petal.Width mean(+1) versicolor  -0.0201      0.151 -0.324  0.268\n#&gt;  Petal.Width mean(+1) virginica    0.0216      0.159 -0.290  0.313\n#&gt; \n#&gt; Columns: term, contrast, Species, estimate, std.error, conf.low, conf.high, predicted_lo, predicted_hi, predicted, tmp_idx \n#&gt; Type:  response\n\nSince simulation based inference generates R estimates of the quantities of interest, we can treat them similarly to draws from the posterior distribution in bayesian models. For example, we can extract draws using the posterior_draws() function, and plot their distributions using packages likeggplot2 and ggdist:\n\navg_comparisons(mod, by = \"Species\", variables = \"Petal.Width\") |&gt;\n  inferences(method = \"simulation\") |&gt;\n  posterior_draws(\"rvar\") |&gt;\n  ggplot(aes(y = Species, xdist = rvar)) +\n  stat_slabinterval()"
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     "href": "articles/gformula.html#example-with-real-world-data",
     "title": "\n12  Causal Inference\n",
     "section": "\n12.3 Example with real-world data",
-    "text": "12.3 Example with real-world data\nLet’s illustrate this method by replicating an example from Chapter 13 of Hernán and Robins. The data come from the National Health and Nutrition Examination Survey Data I Epidemiologic Follow-up Study (NHEFS). The outcome is wt82_71, a measure of weight gain. The treatment is qsmk, a binary measure of smoking cessation. There are many confounders.\nStep 1 is to fit a regression model of the outcome on the treatment and control variables:\n\nlibrary(boot)\nlibrary(marginaleffects)\n\nf &lt;- wt82_71 ~ qsmk + sex + race + age + I(age * age) + factor(education) +\n     smokeintensity + I(smokeintensity * smokeintensity) + smokeyrs +\n     I(smokeyrs * smokeyrs) + factor(exercise) + factor(active) + wt71 +\n     I(wt71 * wt71) + I(qsmk * smokeintensity)\n\nurl &lt;- \"https://raw.githubusercontent.com/vincentarelbundock/modelarchive/main/data-raw/nhefs.csv\"\nnhefs &lt;- read.csv(url)\nnhefs &lt;- na.omit(nhefs[, all.vars(f)])\n\nfit &lt;- glm(f, data = nhefs)\n\nSteps 2 and 3 require us to replicate the full dataset by setting the qsmk treatment to counterfactual values. We can do this automatically by calling comparisons().\n\n12.3.1 TLDR\nThese simple commands do everything we need to apply the parametric g-formula:\n\navg_comparisons(fit, variables = list(qsmk = 0:1))\n\n\n Term Contrast Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n qsmk    1 - 0     3.52       0.44 7.99   &lt;0.001 49.4  2.65   4.38\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\nThe rest of the vignette walks through the process in a bit more detail and compares to replication code from Hernán and Robins.\n\n12.3.2 Adjusted Predictions\nWe can compute average predictions in the original data, and average predictions in the two counterfactual datasets like this:\n\n## average predicted outcome in the original data\np &lt;- predictions(fit)\nmean(p$estimate)\n\n[1] 2.6383\n\n## average predicted outcome in the two counterfactual datasets\np &lt;- predictions(fit, newdata = datagrid(qsmk = 0:1, grid_type = \"counterfactual\"))\naggregate(estimate ~ qsmk, data = p, FUN = mean)\n\n  qsmk estimate\n1    0 1.756213\n2    1 5.273587\n\n\nIn the R code that accompanies their book, Hernán and Robins compute the same quantities manually, as follows:\n\n## create a dataset with 3 copies of each subject\nnhefs$interv &lt;- -1 # 1st copy: equal to original one\n\ninterv0 &lt;- nhefs # 2nd copy: treatment set to 0, outcome to missing\ninterv0$interv &lt;- 0\ninterv0$qsmk &lt;- 0\ninterv0$wt82_71 &lt;- NA\n\ninterv1 &lt;- nhefs # 3rd copy: treatment set to 1, outcome to missing\ninterv1$interv &lt;- 1\ninterv1$qsmk &lt;- 1\ninterv1$wt82_71 &lt;- NA\n\nonesample &lt;- rbind(nhefs, interv0, interv1) # combining datasets\n\n## linear model to estimate mean outcome conditional on treatment and confounders\n## parameters are estimated using original observations only (nhefs)\n## parameter estimates are used to predict mean outcome for observations with \n## treatment set to 0 (interv=0) and to 1 (interv=1)\n\nstd &lt;- glm(f, data = onesample)\nonesample$predicted_meanY &lt;- predict(std, onesample)\n\n## estimate mean outcome in each of the groups interv=0, and interv=1\n## this mean outcome is a weighted average of the mean outcomes in each combination \n## of values of treatment and confounders, that is, the standardized outcome\nmean(onesample[which(onesample$interv == -1), ]$predicted_meanY)\n\n[1] 2.6383\n\nmean(onesample[which(onesample$interv == 0), ]$predicted_meanY)\n\n[1] 1.756213\n\nmean(onesample[which(onesample$interv == 1), ]$predicted_meanY)\n\n[1] 5.273587\n\n\nIt may be useful to note that the datagrid() function provided by marginaleffects can create counterfactual datasets automatically. This is equivalent to the onesample dataset:\n\nnd &lt;- datagrid(\n    model = fit,\n    qsmk = c(0, 1),\n    grid_type = \"counterfactual\")\n\n\n12.3.3 Contrast\nNow we want to compute the treatment effect with the parametric g-formula, which is the difference in average predicted outcomes in the two counterfactual datasets. This is equivalent to taking the average contrast with the comparisons() function. There are three important things to note in the command that follows:\n\nThe variables argument is used to indicate that we want to estimate a “contrast” between adjusted predictions when qsmk is equal to 1 or 0.\n\ncomparisons() automatically produces estimates of uncertainty.\n\n\navg_comparisons(std, variables = list(qsmk = 0:1))\n\n\n Term Contrast Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n qsmk    1 - 0     3.52       0.44 7.99   &lt;0.001 49.4  2.65   4.38\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\nUnder the hood, comparisons() did exactly what we described in the g-formula steps above:\nWe can obtain the same result by manually computing the quantities, using the replication code from Hernán and Robins:\n\nmean(onesample[which(onesample$interv == 1), ]$predicted_meanY) -\nmean(onesample[which(onesample$interv == 0), ]$predicted_meanY)\n\n[1] 3.517374\n\n\nAlthough manual computation is simple, it does not provide uncertainty estimates. In contrast, comparisons() has already computed the standard error and confidence interval using the delta method.\nInstead of the delta method, most analysts will rely on bootstrapping. For example, the replication code from Hernán and Robins does this:\n\n## function to calculate difference in means\nstandardization &lt;- function(data, indices) {\n    # create a dataset with 3 copies of each subject\n    d &lt;- data[indices, ] # 1st copy: equal to original one`\n    d$interv &lt;- -1\n    d0 &lt;- d # 2nd copy: treatment set to 0, outcome to missing\n    d0$interv &lt;- 0\n    d0$qsmk &lt;- 0\n    d0$wt82_71 &lt;- NA\n    d1 &lt;- d # 3rd copy: treatment set to 1, outcome to missing\n    d1$interv &lt;- 1\n    d1$qsmk &lt;- 1\n    d1$wt82_71 &lt;- NA\n    d.onesample &lt;- rbind(d, d0, d1) # combining datasets\n\n    # linear model to estimate mean outcome conditional on treatment and confounders\n    # parameters are estimated using original observations only (interv= -1)\n    # parameter estimates are used to predict mean outcome for observations with set\n    # treatment (interv=0 and interv=1)\n    fit &lt;- glm(f, data = d.onesample)\n\n    d.onesample$predicted_meanY &lt;- predict(fit, d.onesample)\n\n    # estimate mean outcome in each of the groups interv=-1, interv=0, and interv=1\n    return(mean(d.onesample$predicted_meanY[d.onesample$interv == 1]) -\n           mean(d.onesample$predicted_meanY[d.onesample$interv == 0]))\n}\n\n## bootstrap\nresults &lt;- boot(data = nhefs, statistic = standardization, R = 1000)\n\n## generating confidence intervals\nse &lt;- sd(results$t[, 1])\nmeant0 &lt;- results$t0\nll &lt;- meant0 - qnorm(0.975) * se\nul &lt;- meant0 + qnorm(0.975) * se\n\nbootstrap &lt;- data.frame(\n    \" \" = \"Treatment - No Treatment\",\n    estimate = meant0,\n    std.error = se,\n    conf.low = ll,\n    conf.high = ul,\n    check.names = FALSE)\nbootstrap\n\n                           estimate std.error conf.low conf.high\n1 Treatment - No Treatment 3.517374 0.4849327 2.566924  4.467825\n\n\nThe results are close to those that we obtained with comparisons(), but the confidence interval differs slightly because of the difference between bootstrapping and the delta method.\n\navg_comparisons(fit, variables = list(qsmk = 0:1))\n\n\n Term Contrast Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n qsmk    1 - 0     3.52       0.44 7.99   &lt;0.001 49.4  2.65   4.38\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response"
+    "text": "12.3 Example with real-world data\nLet’s illustrate this method by replicating an example from Chapter 13 of Hernán and Robins. The data come from the National Health and Nutrition Examination Survey Data I Epidemiologic Follow-up Study (NHEFS). The outcome is wt82_71, a measure of weight gain. The treatment is qsmk, a binary measure of smoking cessation. There are many confounders.\nStep 1 is to fit a regression model of the outcome on the treatment and control variables:\n\nlibrary(boot)\nlibrary(marginaleffects)\n\nf &lt;- wt82_71 ~ qsmk + sex + race + age + I(age * age) + factor(education) +\n     smokeintensity + I(smokeintensity * smokeintensity) + smokeyrs +\n     I(smokeyrs * smokeyrs) + factor(exercise) + factor(active) + wt71 +\n     I(wt71 * wt71) + I(qsmk * smokeintensity)\n\nurl &lt;- \"https://raw.githubusercontent.com/vincentarelbundock/modelarchive/main/data-raw/nhefs.csv\"\nnhefs &lt;- read.csv(url)\nnhefs &lt;- na.omit(nhefs[, all.vars(f)])\n\nfit &lt;- glm(f, data = nhefs)\n\nSteps 2 and 3 require us to replicate the full dataset by setting the qsmk treatment to counterfactual values. We can do this automatically by calling comparisons().\n\n12.3.1 TLDR\nThese simple commands do everything we need to apply the parametric g-formula:\n\navg_comparisons(fit, variables = list(qsmk = 0:1))\n\n\n Term Contrast Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n qsmk    1 - 0     3.52       0.44 7.99   &lt;0.001 49.4  2.65   4.38\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\nThe rest of the vignette walks through the process in a bit more detail and compares to replication code from Hernán and Robins.\n\n12.3.2 Adjusted Predictions\nWe can compute average predictions in the original data, and average predictions in the two counterfactual datasets like this:\n\n## average predicted outcome in the original data\np &lt;- predictions(fit)\nmean(p$estimate)\n\n[1] 2.6383\n\n## average predicted outcome in the two counterfactual datasets\np &lt;- predictions(fit, newdata = datagrid(qsmk = 0:1, grid_type = \"counterfactual\"))\naggregate(estimate ~ qsmk, data = p, FUN = mean)\n\n  qsmk estimate\n1    0 1.756213\n2    1 5.273587\n\n\nIn the R code that accompanies their book, Hernán and Robins compute the same quantities manually, as follows:\n\n## create a dataset with 3 copies of each subject\nnhefs$interv &lt;- -1 # 1st copy: equal to original one\n\ninterv0 &lt;- nhefs # 2nd copy: treatment set to 0, outcome to missing\ninterv0$interv &lt;- 0\ninterv0$qsmk &lt;- 0\ninterv0$wt82_71 &lt;- NA\n\ninterv1 &lt;- nhefs # 3rd copy: treatment set to 1, outcome to missing\ninterv1$interv &lt;- 1\ninterv1$qsmk &lt;- 1\ninterv1$wt82_71 &lt;- NA\n\nonesample &lt;- rbind(nhefs, interv0, interv1) # combining datasets\n\n## linear model to estimate mean outcome conditional on treatment and confounders\n## parameters are estimated using original observations only (nhefs)\n## parameter estimates are used to predict mean outcome for observations with \n## treatment set to 0 (interv=0) and to 1 (interv=1)\n\nstd &lt;- glm(f, data = onesample)\nonesample$predicted_meanY &lt;- predict(std, onesample)\n\n## estimate mean outcome in each of the groups interv=0, and interv=1\n## this mean outcome is a weighted average of the mean outcomes in each combination \n## of values of treatment and confounders, that is, the standardized outcome\nmean(onesample[which(onesample$interv == -1), ]$predicted_meanY)\n\n[1] 2.6383\n\nmean(onesample[which(onesample$interv == 0), ]$predicted_meanY)\n\n[1] 1.756213\n\nmean(onesample[which(onesample$interv == 1), ]$predicted_meanY)\n\n[1] 5.273587\n\n\nIt may be useful to note that the datagrid() function provided by marginaleffects can create counterfactual datasets automatically. This is equivalent to the onesample dataset:\n\nnd &lt;- datagrid(\n    model = fit,\n    qsmk = c(0, 1),\n    grid_type = \"counterfactual\")\n\n\n12.3.3 Contrast\nNow we want to compute the treatment effect with the parametric g-formula, which is the difference in average predicted outcomes in the two counterfactual datasets. This is equivalent to taking the average contrast with the comparisons() function. There are three important things to note in the command that follows:\n\nThe variables argument is used to indicate that we want to estimate a “contrast” between adjusted predictions when qsmk is equal to 1 or 0.\n\ncomparisons() automatically produces estimates of uncertainty.\n\n\navg_comparisons(std, variables = list(qsmk = 0:1))\n\n\n Term Contrast Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n qsmk    1 - 0     3.52       0.44 7.99   &lt;0.001 49.4  2.65   4.38\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response \n\n\nUnder the hood, comparisons() did exactly what we described in the g-formula steps above:\nWe can obtain the same result by manually computing the quantities, using the replication code from Hernán and Robins:\n\nmean(onesample[which(onesample$interv == 1), ]$predicted_meanY) -\nmean(onesample[which(onesample$interv == 0), ]$predicted_meanY)\n\n[1] 3.517374\n\n\nAlthough manual computation is simple, it does not provide uncertainty estimates. In contrast, comparisons() has already computed the standard error and confidence interval using the delta method.\nInstead of the delta method, most analysts will rely on bootstrapping. For example, the replication code from Hernán and Robins does this:\n\n## function to calculate difference in means\nstandardization &lt;- function(data, indices) {\n    # create a dataset with 3 copies of each subject\n    d &lt;- data[indices, ] # 1st copy: equal to original one`\n    d$interv &lt;- -1\n    d0 &lt;- d # 2nd copy: treatment set to 0, outcome to missing\n    d0$interv &lt;- 0\n    d0$qsmk &lt;- 0\n    d0$wt82_71 &lt;- NA\n    d1 &lt;- d # 3rd copy: treatment set to 1, outcome to missing\n    d1$interv &lt;- 1\n    d1$qsmk &lt;- 1\n    d1$wt82_71 &lt;- NA\n    d.onesample &lt;- rbind(d, d0, d1) # combining datasets\n\n    # linear model to estimate mean outcome conditional on treatment and confounders\n    # parameters are estimated using original observations only (interv= -1)\n    # parameter estimates are used to predict mean outcome for observations with set\n    # treatment (interv=0 and interv=1)\n    fit &lt;- glm(f, data = d.onesample)\n\n    d.onesample$predicted_meanY &lt;- predict(fit, d.onesample)\n\n    # estimate mean outcome in each of the groups interv=-1, interv=0, and interv=1\n    return(mean(d.onesample$predicted_meanY[d.onesample$interv == 1]) -\n           mean(d.onesample$predicted_meanY[d.onesample$interv == 0]))\n}\n\n## bootstrap\nresults &lt;- boot(data = nhefs, statistic = standardization, R = 1000)\n\n## generating confidence intervals\nse &lt;- sd(results$t[, 1])\nmeant0 &lt;- results$t0\nll &lt;- meant0 - qnorm(0.975) * se\nul &lt;- meant0 + qnorm(0.975) * se\n\nbootstrap &lt;- data.frame(\n    \" \" = \"Treatment - No Treatment\",\n    estimate = meant0,\n    std.error = se,\n    conf.low = ll,\n    conf.high = ul,\n    check.names = FALSE)\nbootstrap\n\n                           estimate std.error conf.low conf.high\n1 Treatment - No Treatment 3.517374 0.4786991 2.579141  4.455607\n\n\nThe results are close to those that we obtained with comparisons(), but the confidence interval differs slightly because of the difference between bootstrapping and the delta method.\n\navg_comparisons(fit, variables = list(qsmk = 0:1))\n\n\n Term Contrast Estimate Std. Error    z Pr(&gt;|z|)    S 2.5 % 97.5 %\n qsmk    1 - 0     3.52       0.44 7.99   &lt;0.001 49.4  2.65   4.38\n\nColumns: term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high \nType:  response"
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     "title": "\n27  Standard Errors\n",
     "section": "\n27.5 Simulation-based inference",
-    "text": "27.5 Simulation-based inference\nmarginaleffects offers an experimental inferences() function to conduct simulation-based inference following the strategy proposed by Krinsky & Robb (1986):\n\nDraw iter sets of simulated coefficients from a multivariate normal distribution with mean equal to the original model’s estimated coefficients and variance equal to the model’s variance-covariance matrix (classical, “HC3”, or other).\nUse the iter sets of coefficients to compute iter sets of estimands: predictions, comparisons, or slopes.\nTake quantiles of the resulting distribution of estimands to obtain a confidence interval and the standard deviation of simulated estimates to estimate the standard error.\n\nHere are a few examples:\n\nlibrary(marginaleffects)\nlibrary(ggplot2)\nlibrary(ggdist)\n\nmod &lt;- glm(vs ~ hp * wt + factor(gear), data = mtcars, family = binomial)\n\nmod |&gt; predictions() |&gt; inferences(method = \"simulation\")\n#&gt; \n#&gt;  Estimate Std. Error    2.5 % 97.5 %\n#&gt;  7.84e-01      0.188 2.86e-01  0.974\n#&gt;  7.84e-01      0.161 3.68e-01  0.962\n#&gt;  8.98e-01      0.138 4.74e-01  0.990\n#&gt;  8.74e-01      0.228 1.80e-01  0.995\n#&gt;  1.31e-02      0.179 5.19e-05  0.716\n#&gt; --- 22 rows omitted. See ?avg_predictions and ?print.marginaleffects --- \n#&gt;  3.83e-01      0.296 2.50e-02  0.944\n#&gt;  1.21e-06      0.126 8.11e-13  0.346\n#&gt;  6.89e-03      0.145 3.10e-05  0.565\n#&gt;  8.07e-11      0.148 2.22e-16  0.600\n#&gt;  7.95e-01      0.162 3.68e-01  0.969\n#&gt; Columns: rowid, estimate, std.error, conf.low, conf.high, vs, hp, wt, gear \n#&gt; Type:  response\n\nmod |&gt; avg_slopes(vcov = ~gear) |&gt; inferences(method = \"simulation\")\n#&gt; \n#&gt;  Term Contrast  Estimate Std. Error   2.5 %  97.5 %\n#&gt;  gear    4 - 3 -3.92e-02    0.05506 -0.0914 0.12902\n#&gt;  gear    5 - 3 -1.93e-01    0.26716 -0.4888 0.33333\n#&gt;  hp      dY/dX -5.02e-03    0.00439 -0.0115 0.00378\n#&gt;  wt      dY/dX -4.06e-05    0.30024 -0.5342 0.71972\n#&gt; \n#&gt; Columns: term, contrast, estimate, std.error, conf.low, conf.high \n#&gt; Type:  response\n\nSince simulation based inference generates iter estimates of the quantities of interest, we can treat them similarly to draws from the posterior distribution in bayesian models. For example, we can extract draws using the posterior_draws() function, and plot their distributions using packages likeggplot2 and ggdist:\n\nmod |&gt;\n  avg_comparisons(variables = \"gear\") |&gt;\n  inferences(method = \"simulation\") |&gt;\n  posterior_draws(\"rvar\") |&gt;\n  ggplot(aes(y = contrast, xdist = rvar)) +\n  stat_slabinterval()"
+    "text": "27.5 Simulation-based inference\nmarginaleffects offers an experimental inferences() function to conduct simulation-based inference following the strategy proposed by Krinsky & Robb (1986):\n\nDraw iter sets of simulated coefficients from a multivariate normal distribution with mean equal to the original model’s estimated coefficients and variance equal to the model’s variance-covariance matrix (classical, “HC3”, or other).\nUse the iter sets of coefficients to compute iter sets of estimands: predictions, comparisons, or slopes.\nTake quantiles of the resulting distribution of estimands to obtain a confidence interval and the standard deviation of simulated estimates to estimate the standard error.\n\nHere are a few examples:\n\nlibrary(marginaleffects)\nlibrary(ggplot2)\nlibrary(ggdist)\n\nmod &lt;- glm(vs ~ hp * wt + factor(gear), data = mtcars, family = binomial)\n\nmod |&gt; predictions() |&gt; inferences(method = \"simulation\")\n#&gt; \n#&gt;  Estimate Std. Error    2.5 % 97.5 %\n#&gt;  7.84e-01      0.197 2.54e-01  0.976\n#&gt;  7.84e-01      0.166 3.47e-01  0.961\n#&gt;  8.98e-01      0.146 4.64e-01  0.989\n#&gt;  8.74e-01      0.240 1.71e-01  0.996\n#&gt;  1.31e-02      0.180 8.50e-05  0.750\n#&gt; --- 22 rows omitted. See ?avg_predictions and ?print.marginaleffects --- \n#&gt;  3.83e-01      0.293 1.21e-02  0.957\n#&gt;  1.21e-06      0.140 5.46e-12  0.447\n#&gt;  6.89e-03      0.169 2.66e-05  0.700\n#&gt;  8.07e-11      0.161 2.22e-16  0.787\n#&gt;  7.95e-01      0.169 3.50e-01  0.968\n#&gt; Columns: rowid, estimate, std.error, conf.low, conf.high, vs, hp, wt, gear \n#&gt; Type:  response\n\nmod |&gt; avg_slopes(vcov = ~gear) |&gt; inferences(method = \"simulation\")\n#&gt; \n#&gt;  Term Contrast  Estimate Std. Error   2.5 % 97.5 %\n#&gt;  gear    4 - 3 -3.92e-02    0.05652 -0.0900 0.1301\n#&gt;  gear    5 - 3 -1.93e-01    0.27567 -0.4868 0.3329\n#&gt;  hp      dY/dX -5.02e-03    0.00464 -0.0114 0.0053\n#&gt;  wt      dY/dX -4.06e-05    0.32928 -0.6744 0.7151\n#&gt; \n#&gt; Columns: term, contrast, estimate, std.error, conf.low, conf.high \n#&gt; Type:  response\n\nSince simulation based inference generates iter estimates of the quantities of interest, we can treat them similarly to draws from the posterior distribution in bayesian models. For example, we can extract draws using the posterior_draws() function, and plot their distributions using packages likeggplot2 and ggdist:\n\nmod |&gt;\n  avg_comparisons(variables = \"gear\") |&gt;\n  inferences(method = \"simulation\") |&gt;\n  posterior_draws(\"rvar\") |&gt;\n  ggplot(aes(y = contrast, xdist = rvar)) +\n  stat_slabinterval()"
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     "text": "30.2 Calling marginaleffects in functions, loops, environments, or after re-assigning variables\nFunctions from the marginaleffects package can sometimes fail when they are called inside a function, loop, or other environments. To see why, it is important to know that marginaleffects often needs to operate on the original data that was used to fit the model. To extract this original data, we use the get_data() function from the insight package.\nIn most cases, get_data() can extract the data which is stored inside the model object created by the modeling package. However, some modeling packages do not save the original data in the model object (in order to save memory). In those cases, get_data() will parse the call to find the name of the data object, and will search for that data object in the global environment. When users fit models in a different environment (e.g., function calls), get_data() may not be able to retrieve the original data.\nA related problem can arise if users fit a model, but then assign a new value to the variable that used to store the dataset.\nRecommendations:\n\nSupply your dataset explicitly to the newdata argument of slopes() functions.\nAvoid assigning a new value to a variable that you use to store a dataset for model fitting."
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     "title": "31  News",
-    "section": "dev",
-    "text": "dev\n\nhypotheses(): The FUN argument handles group columns gracefully.\nNative support for Amelia for multiple imputation.\n\nDocumentation:\n\nNew section on “Complex aggregations” in the Hypothesis testing vignette.\n\nBug fix:\n\nResults of the predictions() function can be undesirable in version 0.15.0 when (a) type=NULL, (b) model is glm(), and (c) the hypothesis argument is non-numeric. This is because the default type was then set to invlink(link), but the backtransformation using link function was applied to the quantity after applying the (non-linear) hypothesis test (ex: pairwise comparison or contrast). The hypothesis funtion should be applied to predictions on the response scale instead. A warning is now issued and the value of type is automatically switched to “response” when appropriate. Thanks to @strengejacke for report [#903](https://github.com/vincentarelbundock/marginaleffects/issues/903)"
+    "section": "0.15.1",
+    "text": "0.15.1\n\nhypotheses(): The FUN argument handles group columns gracefully.\nNative support for Amelia for multiple imputation.\n\nDocumentation:\n\nNew section on “Complex aggregations” in the Hypothesis testing vignette.\n\nBug fix:\n\nResults of the predictions() function could be inaccurate when (a) running version 0.15.0, (b) type is NULL or invlink(link), (c) model is glm(), and (d) the hypothesis argument is non-numeric. Thanks to @strengejacke for report [#903](https://github.com/vincentarelbundock/marginaleffects/issues/903)"
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     "title": "31  News",
     "section": "0.15.0",
     "text": "0.15.0\nNew:\n\nConformal prediction via inferences()\nhypothesis argument now accepts multiple string formulas.\nThe type argument now accepts an explicit invlink(link) value instead of silently back-transforming. Users are no longer pointed to type_dictionary. Instead, they should call their function with a bad type value, and they will obtain a list of valid types. The default type value is printed in the output. This is useful because the default type value is NULL, so the user often does not explicitly decide.\nAllow install with Rcpp 1.0.0 and greater.\n\nSupport new models:\n\nsurvey::svyolr()\n\nMisc:\n\ninferences(method=\"simulation\") uses the original point estimate rather than the mean of the simulation distribution. Issue #851.\nBetter documentation and error messages for newdata=NULL\nSome performance improvements for predictions() and marginalmeans() (#880, #882, @etiennebacher).\n\nBug fix:\n\nnewdata=\"median\" returned mean of binary variables. Thanks to @jkhanson1970 for report #896."
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     "title": "31  News",
     "section": "0.14.0",
     "text": "0.14.0\nBreaking changes:\n\nRow order of the output changes for some objects. Rows are not sorted alphabetically by term, by, and variables explicitly supplied to datagrid. This can affect hypothesis tests computed using the b1, b2, b3, and other indices.\nNew procedure numderiv argument use a different procedure to select the step size used in the finite difference numeric derivative used to compute standard errors: abs(x) * sqrt(.Machine$double.eps). The numerical results may not be exactly identical to previous versions of marginaleffects, but the step size should be adequate in a broader variety of cases. Note that users can use the numderiv argument for more control on numeric differentiation, as documented.\nbife models are no longer supported pending investigation in weird results in the tests. Looking for volunteers write more thorough tests.\n\nNew:\n\nSupport: logistf package.\nSupport: DCchoice package.\nSupport: stats::nls\nhypotheses() can now accept raw data frame, which gives a lot of flexibility for custom contrasts and functions. See the Hypothesis vignette for an example.\nnumderiv argument allows users to use finite difference (center or forward) or Richardson’s method to compute the numerical derivatives used in the calculation of standard errors.\n\nBug fixes:\n\ninferences() supports the cross argument for comparisons() objects. Thanks to Kirill Solovev for report #856.\nsplines::bs() in formulas could produce incorrect results due to weirdness in stats::model.matrix(). Thanks to @chiungming for report #831.\nmgcv with ocat are now supported. Thanks to Lorenzo Fabbri for Issue #844.\nquantreg problem with rowid merge did not affect estimates but did not return the full original data. Issue #829.\nget_modeldata() extracts weights variable when available.\npredictions() is no longer broken in some inferences() calls. Issue #853\nInaccurate averaging with comparison=differenceavg some models where all predictors are categorical. Thanks to Karl Ove Hufthammer for report #865.\n\nMisc:\n\nMajor refactor to simplify the code base and make maintenance easier."
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     "title": "31  News",
     "section": "0.13.0",
     "text": "0.13.0\nBreaking change:\n\nglmmTMB: Standard errors are no longer supported because they may have been erroneous. Follow Issue #810 on Github for developments: https://github.com/vincentarelbundock/marginaleffects/issues/810\n\nNew:\n\nhypothesis argument accepts wildcards: hypothesis = \"b*=b1\"\ns.value column in all output: Shannon transforms for p values. See Greenland (2019).\nmarginal_means supports mira (mice objects).\ncomparisons(): The variables arguments now accepts arbitrary numeric vectors of length equal to the number of rows in newdata. This allows users to specify fully custom treatment sizes. In the documentation examples, we show how to estimate the difference for a 1 standard deviation shift in a regressor, where the standard deviation is calculated on a group-wise basis.\ncomparisons(): the variables argument now accepts “revpairwise”, “revsequential”, “revreference” for factor and character variables.\ncomparisons(): the comparison argument now accept “lift” and “liftavg”.\n\nPerformance:\n\nComputing elasticities for linear models is now up to 30% faster (#787, @etiennebacher).\n\nBug fixes:\n\nBetter handling of environments when newdata is a function call. Thanks to @jcccf for report #814 and to @capnrefsmmat for the proposed fix using the rlang package.\nDegrees of freedom mismatch for joint hypothesis tests. Thanks to @snhansen for report #789."
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     "title": "31  News",
     "section": "0.12.0",
     "text": "0.12.0\nBreaking change:\n\nRow order of output has changed for many calls, especially those using the by argument. This may break hypothesis tests conducted by indexing b1, b2, etc. This was necessary to fix Issue #776. Thanks to @marcora for the report.\n\nNew:\n\nhypotheses(): Joint hypothesis tests (F and Chi-square) with the joint and joint_test arguments.\nvcov.hypotheses method.\nwts is now available in plot_predictions(), plot_comparisons(), and plot_slopes().\n\nBug:\n\nWrong order of rows in bayesian models with by argument. Thanks to @shirdekel for report #782."
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     "title": "31  News",
     "section": "0.11.2",
     "text": "0.11.2\n\nvcov() and coef() methods for marginaleffects objects.\nStrings in wts are accepted with the by argument.\npredictions() and avg_predictions() no longer use an automatic backtransformation for GLM models unless hypothesis is NULL.\nvcov() can be used to retrieve a full variance-covariance matrix from objects produced by comparisons(), slopes(), predictions(), or marginal_means() objects.\nWhen processing objects obtained using mice multiple imputation, the pooled model using mice::pool is attached to the model attribute of the output. This means that functions like modelsummary::modelsummary() will not erroneously report goodness-of-fit statistics from just a single model and will instead appropriately report the statistics for the pooled model. Thanks to @Tristan-Siegfried for PR #740.\nMore informative error messages on some prediction problems. Thanks to @andymilne for Report #751.\n\nPerformance:\n\ninferences() is now up to 17x faster and much more memory-efficient when method is \"boot\" or \"rsample\" (#770, #771, @etiennebacher).\n\nBugs:\n\nbrms models with nl=TRUE and a single predictor generated an error. Thanks to @Tristan-Siegried for Report #759.\navg_predictions(): Incorrect group-wise averaging when all predictors are categorical, the variables variable is used, and we are averaging with avg_ or the by argument. Thanks to BorgeJorge for report #766.\nBug when datagrid() when called inside a user-written function. Thanks to @NickCH-K for report #769 and to @capnrefsmmat for the diagnostics."
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     "title": "31  News",
     "section": "0.11.1",
     "text": "0.11.1\nBreaking change:\n\nRow orders are now more consistent, but may have changed from previous version. This could affect results from hypothesis with b1, b2, … indexing.\n\nSupport new models:\n\nnlme::lme()\nphylolm::phylolm()\nphylolm::phyloglm()\n\nNew:\n\nVignette on 2x2 experimental designs. Thanks to Demetri Pananos.\ncomparisons() accepts data frames with two numeric columns (“low” and “high”) to specify fully customizable contrasts.\ndatagrid() gets a new by argument to create apply grid-making functions within groups.\nplot_*() gain a newdata argument for use with by.\n\nBug:\n\ncomparisons(comparison = \"lnratioavg\") ignored wts argument. Thanks to Demetri Pananos for report #737.\nordinal::clm(): incorrect standard errors when location and scale parameters are the same. Thanks to MrJerryTAO for report #718.\nIncorrect label for “2sd” comparisons. Thanks to Andy Milne for report #720.\nInvalid factor levels in datagrid() means newdata argument gets ignored. Thanks to Josh Errickson for report #721.\nError in models with only categorical predictors and the by argument. Thanks to Sam Brilleman for report #723.\nElasticities are now supported for ordinal::clm() models. Thanks to MrJerryTAO for report #729.\nglmmTMB models with zero-inflated components are supported. Thanks to @Helsinki-Ronan and @strengejacke for report #734."
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     "title": "31  News",
     "section": "0.11.0",
     "text": "0.11.0\nBreaking changes:\n\ntype column is replaced by type attribute.\npredictions() only works with officially supported model types (same list as comparisons() and slopes()).\n\nRenamed arguments (backward compatibility is preserved):\n\ntransform_pre -&gt; comparison\ntransform_post -&gt; transform\n\nNew:\n\np_adjust argument: Adjust p-values for multiple comparisons.\nequivalence argument available everywhere.\n\nPerformance:\n\nMuch faster results in avg_*() functions for models with only categorical predictors and many rows of data, using deduplication and weights instead of unit-level estimates.\nFaster predictions in lm() and glm() models using RcppEigen.\nBayesian models with many rows. Thanks to Etienne Bacher. #694\nFaster predictions, especially with standard errors and large datasets.\n\nBugs:\n\nMultiple imputation with mira objects was not pooling all datasets. Thanks to @Generalized for report #711.\nSupport for more models with offsets. Thanks to @mariofiorini for report #705.\nError on predictions() with by and wts. Thanks to Noah Greifer for report #695.\nafex: some models generated errors. Thanks to Daniel Lüdecke for report #696.\ngroup column name is always forbidden. Thanks to Daniel Lüdecke for report #697.\nBlank graphs in plot_comparisons() with a list in variables.\ntype=\"link\" produced an error with some categorical brms models. Thanks to @shirdekel for report #703.\nError on predictions(variables = ...) for glmmTMB models. Thanks to Daniel Lüdecke for report #707.\nby with user-specified function in comparison and factor predictor did not aggregate correctly. Thanks to @joaotedde for report #715.\nordinal::clm: Support cum.prob and linear.predictor prediction types. Thanks to @MrJerryTAO for report #717."
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     "title": "31  News",
     "section": "0.10.0",
     "text": "0.10.0\nPerformance:\n\n2-4x faster execution for many calls. Thanks to Etienne Bacher.\n\nNew models supported:\n\nMCMCglmm::MCMCglmm\nRchoice::hetprob\nRchoice::ivpml\nMultiple imputation using mice and any package which can return a list of imputed data frames (e.g., Amelia, missRanger, etc.)\n\nPlot improvements:\n\nNew by argument to display marginal estimates by subgroup.\nNew rug argument to display tick marks in the margins.\nNew points argument in plot_predictions() to display a scatter plot.\nNew gray argument to plot in grayscale using line types and shapes instead of color.\nThe effect argument is renamed to variables in plot_slopes() and plot_comparisons(). This improves consistency with the analogous slopes() and comparisons() functions.\nThe plotting vignette was re-written.\n\nOther:\n\nSupport multiple imputation with mice mira objects. The multiple imputation vignette was rewritten.\nThe variables_grid argument in marginal_means() is renamed newdata. Backward compatibility is maintained.\navg_*() returns an informative error when vcov is “satterthwaite” or “kenward-roger”\n“satterthwaite” and “kenward-roger” are now supported when newdata is not NULL\nInformative error when hypothesis includes a b# larger than the available number of estimates.\navg_predictions(model, variables = \"x\") computes average counterfactual predictions by subgroups of x\ndatagrid() and plot_*() functions are faster in datasets with many extraneous columns.\nIn predictions(type = NULL) with glm() and Gam() we first make predictions on the link scale and then backtransform them. Setting type=\"response\" explicitly makes predictions directly on the response scale without backtransformation.\nStandard errors now supported for more glmmTMB models.\nUse the numDeriv package for numeric differentiation in the calculation of delta method standard error. A global option can now be passed to numDeriv::jacobian:\n\noptions(marginaleffects_numDeriv = list(method = \"simple\", method.args = list(eps = 1e-6)))\noptions(marginaleffects_numDeriv = list(method = \"Richardson\", method.args = list(eps = 1e-6)))\noptions(marginaleffects_numDeriv = NULL)\n\nPrint:\n\nPrint fewer significant digits.\nprint.marginaleffects now prints all columns supplied to newdata\nLess redundant labels when using hypothesis\n\nMany improvements to documentation.\n\nBugfixes:\n\nStandard errors could be inaccurate in models with non-linear components (and interactions) when some of the coefficients were very small. This was related to the step size used for numerical differentiation for the delta method. Issue #684.\navg_predictions(by =) did not work when the dataset included a column named term. Issue #683.\nbrms models with multivariate outcome collapsed categories in comparisons(). Issue #639.\nhypotheses() now works on lists and in calls to lapply(), purrr::map(), etc. Issue #660."
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     "title": "31  News",
     "section": "0.9.0",
     "text": "0.9.0\nBreaking changes:\n\nAll functions return an estimate column instead of the function-specific predicted, comparisons, dydx, etc. This change only affects unit-level estimates, and not average estimates, which already used the estimate column name.\nThe transform_avg argument in tidy() deprecated. Use transform_post instead.\nplot_*(draw=FALSE) now return the actual variable names supplied to the condition argument, rather than the opaque “condition1”, “condition2”, etc.\n\nNew models supported:\n\nblme package.\n\nNew features:\n\nNew functions: avg_predictions(), avg_comparisons(), avg_slopes()\nEquivalence, non-inferiority, and non-superiority tests with the hypotheses() function and equivalence argument.\nNew experimental inferences() function: simulation-based inferences and bootstrap using the boot, rsample, and fwb package.\nNew df argument to set degrees of freedom manually for p and CI.\nPretty print() for all objects.\nby argument\n\nTRUE returns average (marginal) predictions, comparisons, or slopes.\nSupports bayesian models.\n\nhypothesis argument\n\nNumeric value sets the null used in calculating Z and p.\nExample: comparisons(mod, transform_pre = \"ratio\", hypothesis = 1)\n\nAll arguments from the main functions are now available through tidy(), and summary(): conf_level, transform_post, etc.\nBayesian posterior distribution summaries (median, mean, HDI, quantiles) can be customized using global options. See ?comparisons\n\nRenamed functions (backward-compatibility is maintained by keeping the old function names as aliases):\n\nmarginaleffects() -&gt; slopes()\nposteriordraws() -&gt; posterior_draws()\nmarginalmeans() -&gt; marginal_means()\nplot_cap() -&gt; plot_predictions()\nplot_cme() -&gt; plot_slopes()\nplot_cco() -&gt; plot_comparisons()\n\nBug fixes:\n\nIncorrect results: In 0.8.1, plot_*() the threenum and minmax labels did not correspond to the correct numeric values.\nFix corner case for slopes when the dataset includes infinite values.\nmlogit error with factors.\nThe vcov argument now accepts functions for most models.\n\nOther:\n\nRemoved major performance bottleneck for slopes()"
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     "title": "31  News",
     "section": "0.8.1",
     "text": "0.8.1\n\ndeltamethod() can run hypothesis tests on objects produced by the comparisons(), marginaleffects(), predictions(), and marginalmeans() functions. This feature relies on match.call(), which means it may not always work when used programmatically, inside functions and nested environments. It is generally safer and more efficient to use the hypothesis argument.\nplot_cme() and plot_cco() accept lists with user-specified values for the regressors, and can display nice labels for shortcut string-functions like “threenum” or “quartile”.\nposterior_draws: new shape argument to return MCMC draws in various formats, including the new rvar structure from the posterior package.\ntransform_avg function gets printed in summary() output.\ntransform_post and transform_avg support string shortcuts: “exp” and “ln”\nAdded support for mlm models from lm(). Thanks to Noah Greifer.\n\nBug fixes:\n\nhypothesis argument with bayesian models and tidy() used to raise an error.\nMissing values for some regressors in the comparisons() output for brms models."
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     "section": "0.8.0",
     "text": "0.8.0\nBreaking change:\n\nThe interaction argument is deprecated and replaced by the cross argument. This is to reduce ambiguity with respect to the interaction argument in emmeans, which does something completely different, akin to the difference-in-differences illustrated in the Interactions vignette.\n\n71 classes of models supported, including the new:\n\nrms::ols\nrms::lrm\nrms::orm\n\nNew features:\n\nPlots: plot_cme(), plot_cap(), and plot_cco() are now much more flexible in specifying the comparisons to display. The condition argument accepts lists, functions, and shortcuts for common reference values, such as “minmax”, “threenum”, etc.\nvariables argument of the comparisons() function is more flexible:\n\nAccepts functions to specify custom differences in numeric variables (e.g., forward and backward differencing).\nCan specify pairs of factors to compare in the variables argument of the comparisons function.\n\nvariables argument of the predictions() function is more flexible:\n\nAccepts shortcut strings, functions, and vectors of arbitrary length.\n\nIntegrate out random effects in bayesian brms models (see Bayesian analysis vignette)\n\nNew vignettes:\n\nExperiments\nExtending marginal effects\nIntegrating out random effects in bayesian models\n\nBug fixes and minor improvements:\n\nThe default value of conf_level in summary() and tidy() is now NULL, which inherits the conf_level value in the original comparisons/marginaleffects/predictions calls.\nFix typo in function names for missing “lnratioavgwts”\nInteractions with fixest::i() are parsed properly as categorical variables\nFor betareg objects, inference can now be done on all coefficients using deltamethod(). previously only the location coefficients were available.\nFor objects from crch package, a number of bugs have been fixed; standard errors should now be correct for deltamethod(), marginaleffects(), etc.\nFixed a bug in the tidy() function for glmmTMB models without random effects, which caused all t statistics to be identical."
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     "title": "31  News",
     "section": "0.7.1",
     "text": "0.7.1\n\nNew supported model class: gamlss. Thanks to Marcio Augusto Diniz.\nmarginalmeans() accepts a wts argument with values: “equal”, “proportional”, “cells”.\nby argument\n\naccepts data frames for complex groupings.\nin marginalmeans only accepts data frames.\naccepts “group” to group by response level.\nworks with bayesian models.\n\nbyfun argument for the predictions() function to aggregate using different functions.\nhypothesis argument\n\nThe matrix column names are used as labels for hypothesis tests.\nBetter labels with “sequential”, “reference”, “pairwise”.\nnew shortcuts “revpairwise”, “revsequential”, “revreference”\n\nwts argument is respected in by argument and with *avg shortcuts in the transform_pre argument.\ntidy.predictions() and tidy.marginalmeans() get a new transform_avg argument.\nNew vignettes:\n\nUnit-level contrasts in logistic regressions. Thanks to @arthur-albuquerque.\nPython Numpy models in marginaleffects. Thanks to @timpipeseek.\nBootstrap example in standard errors vignette."
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     "title": "31  News",
     "section": "0.7.0",
     "text": "0.7.0\nBreaking changes:\n\nby is deprecated in summary() and tidy(). Use the same by argument in the main functions instead: comparisons(), marginaleffects(), predictions()\nCharacter vectors are no longer supported in the variables argument of the predictions() function. Use newdata=\"fivenum\" or “grid”, “mean”, or “median” instead.\n\nCritical bug fix:\n\nContrasts with interactions were incorrect in version 0.6.0. The error should have been obvious to most analysts in most cases (weird-looking alignment). Thanks to @vmikk.\n\nNew supported packages and models:\n\nsurvival::clogit\nbiglm: The main quantities can be computed, but not the delta method standard errors. See https://github.com/vincentarelbundock/marginaleffects/issues/387\n\nNew vignette:\n\nElasticity\nFrequently Asked Questions\n\nNew features:\n\nElasticity and semi-elasticity using the new slope argument in marginaleffects(): eyex, dyex, eydx\ndatagrid() accepts functions: datagrid(newdata = mtcars, hp = range, mpg = fivenum, wt = sd)\nNew datagridcf() function to create counterfactual datasets. This is a shortcut to the datagrid() function with default to grid_type = \"counterfactual\"\nNew by arguments in predictions(), comparisons(), marginaleffects()\nNew newdata shortcuts: “tukey”, “grid”\nNew string shortcuts for transform_pre in comparisons()\nmarginalmeans() now back transforms confidence intervals when possible.\nvcov argument string shortcuts are now case-insensitive\nThe default contrast in comparisons() for binary predictors is now a difference between 1 and 0, rather than +1 relative to baseline.\ndocumentation improvements"
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     "title": "31  News",
     "section": "0.6.0",
     "text": "0.6.0\nNew supported packages and models:\n\ntidymodels objects of class tidy_model are supported if the fit engine is supported by marginaleffects.\n\nNew function:\n\ndeltamethod(): Hypothesis tests on functions of parameters\nplot_cco(): Plot conditional contrasts\n\nNew arguments:\n\nhypothesis for hypothesis tests and custom contrasts\ntransform_post in predictions()\nwts argument in predictions() only affects average predictions in tidy() or summary().\n\nNew or improved vignettes:\n\nHypothesis Tests and Custom Contrasts using the Delta Method: https://marginaleffects.com/articles/hypothesis.html\nMultiple Imputation: https://marginaleffects.com/articles/multiple_imputation.html\nCausal Inference with the g-Formula: https://marginaleffects.com/articles/gformula.html (Thanks to Rohan Kapre for the idea)\n\nDeprecated or renamed arguments:\n\ncontrast_factor and contrast_numeric arguments are deprecated in comparisons(). Use a named list in the variables argument instead. Backward compatibility is maintained.\nThe transform_post argument in tidy() and summary() is renamed to transform_avg to disambiguate against the argument of the same name in comparisons(). Backward compatibility is preserved.\n\nMisc:\n\ntidy.predictions() computes standard errors using the delta method for average predictions\nSupport gam models with matrix columns.\neps in marginaleffects() is now “adaptive” by default: it equals 0.0001 multiplied the range of the predictor variable\ncomparisons() now supports “log of marginal odds ratio” in the transform_pre argument. Thanks to Noah Greifer.\nNew transform_pre shortcuts: dydx, expdydx\ntidy.predictions() computes standard errors and confidence intervals for linear models or GLM on the link scale."
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     "title": "31  News",
     "section": "0.5.0",
     "text": "0.5.0\nBreaking changes:\n\ntype no longer accepts a character vector. Must be a single string.\nconf.int argument deprecated. Use vcov = FALSE instead.\n\nNew supported packages and models:\n\nmlogit\nmhurdle\ntobit1\nglmmTMB\n\nNew features:\n\ninteraction argument in comparisons() to compute interactions between contrasts (cross-contrasts).\nby argument in tidy() and summary() computes group-average marginal effects and comparisons.\ntransform_pre argument can define custom contrasts between adjusted predictions (e.g., log adjusted risk ratios). Available in comparisons().\ntransform_post argument allows back transformation before returning the final results. Available in comparisons(), marginalmeans(), summary(), tidy().\nThe variables argument of the comparisons() function accepts a named list to specify variable-specific contrast types.\nRobust standard errors with the vcov argument. This requires version 0.17.1 of the insight package.\n\nsandwich package shortcuts: vcov = \"HC3\", \"HC2\", \"NeweyWest\", and more.\nMixed effects models: vcov = \"satterthwaite\" or \"kenward-roger\"\nOne-sided formula to clusters: vcov = ~cluster_variable\nVariance-covariance matrix\nFunction which returns a named squared matrix\n\nmarginalmeans() allows interactions\nBayesian Model Averaging for brms models using type = \"average\". See vignette on the marginaleffects website.\neps argument for step size of numerical derivative\nmarginaleffects and comparisons now report confidence intervals by default.\nNew dependency on the data.table package yields substantial performance improvements.\nMore informative error messages and warnings\nBug fixes and performance improvements\n\nNew pages on the marginaleffects website: https://marginaleffects.com/\n\nAlternative software packages\nRobust standard errors (and more)\nPerformance tips\nTables and plots\nMultinomial Logit and Discrete Choice Models\nGeneralized Additive Models\nMixed effects models (Bayesian and Frequentist)\nTransformations and Custom Contrasts: Adjusted Risk Ratio Example\n\nArgument name changes (backward compatibility is preserved:\n\nEverywhere:\n\nconf.level -&gt; conf_level\n\ndatagrid():\n\nFUN.factor -&gt; FUN_factor (same for related arguments)\ngrid.type -&gt; grid_type"
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     "section": "0.4.1",
     "text": "0.4.1\nNew supported packages and models:\n\nstats::loess\nsampleSelection::selection\nsampleSelection::heckit\n\nMisc:\n\nmgcv::bam models allow exclude argument.\nGam models allow include_smooth argument.\nNew tests\nBug fixes"
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     "section": "0.4.0",
     "text": "0.4.0\nNew function:\n\ncomparisons() computes contrasts\n\nMisc:\n\nSpeed optimizations\npredictions() and plot_cap() include confidence intervals for linear models\nMore robust handling of in-formula functions: factor(), strata(), mo()\nDo not overwrite user’s ggplot2::theme_set() call"
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     "section": "0.3.4",
     "text": "0.3.4\n\nBug fixes"
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     "section": "0.3.3",
     "text": "0.3.3\nNew supported models:\n\nmclogit::mclogit\nrobust::lmRob\nrobustlmm::rlmer\nfixest confidence intervals in predictions\n\nMisc:\n\nSupport modelbased::visualisation_matrix in newdata without having to specify x explicitly.\ntidy.predictions() and summary.predictions() methods.\nDocumentation improvements.\nCRAN test fixes"
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     "section": "0.3.2",
     "text": "0.3.2\nSupport for new models and packages:\n\nbrglm2::bracl\nmclogit::mblogit\nscam::scam\nlmerTest::lmer\n\nMisc:\n\nDrop numDeriv dependency, but make it available via a global option: options(“marginaleffects_numDeriv” = list(method = “Richardson”, method.args = list(eps = 1e-5, d = 0.0001)))\nBugfixes\nDocumentation improvements\nCRAN tests"
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     "title": "31  News",
     "section": "0.3.1",
     "text": "0.3.1\ndocumentation bugfix"
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     "section": "0.3.0",
     "text": "0.3.0\nBreaking changes:\n\npredictions returns predictions for every observation in the original dataset instead of newdata=datagrid().\nmarginalmeans objects have new column names, as do the corresponding tidy and summary outputs.\n\nNew supported packages and models:\n\nbrms::brm\nrstanarm::stanglm\nbrglm2::brmultinom\nMASS::glmmPQL\naod::betabin\n\nMisc:\n\ndatagrid function supersedes typical and counterfactual with the grid.type argument. The typical and counterfactual functions will remain available and exported, but their use is not encouraged.\nposterior_draws function can be applied to a predictions or a marginaleffects object to extract draws from the posterior distribution.\nmarginalmeans standard errors are now computed using the delta method.\npredictions standard errors are now computed using the delta method when they are not available from insight::get_predicted.\nNew vignette on Bayesian models with brms\nNew vignette on Mixed effects models with lme4\nIf the data.table package is installed, marginaleffects will automatically use it to speed things up.\nContrast definition reported in a separate column of marginaleffects output.\nSafer handling of the type argument.\nComprehensive list of supported and tests models on the website.\nMany bug fixes\nMany new tests, including several against emmeans"
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     "title": "31  News",
     "section": "0.2.0",
     "text": "0.2.0\nBreaking change:\n\ndata argument becomes newdata in all functions.\n\nNew supported packages and models:\n\nlme4:glmer.nb\nmgcv::gam\nordinal::clm\nmgcv\n\nmarginalmeans:\n\nNew variables_grid argument\n\npredictions:\n\nSupport mgcv\n\nplot_cap\n\nNew type argument\n\nMisc:\n\nNew validity checks and tests"
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     "section": "0.1.0",
     "text": "0.1.0\nFirst release. Bravo!\nThanks to Marco Avina Mendoza, Resul Umit, and all those who offered comments and suggestions."