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<!DOCTYPE html>
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<head>
<title>chapter_8.knit</title>
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class: center, middle
<span style="font-size: 50px;">**第八章**</span> <br>
<span style="font-size: 50px;">回归模型(一)</span> <br>
<span style="font-size: 30px;">胡传鹏</span> <br>
<span style="font-size: 20px;"> </span> <br>
<span style="font-size: 30px;">2024-04-19</span> <br>
<span style="font-size: 20px;"> Made with Rmarkdown</span> <br>
<style type="text/css">
/* ---- extra.css ---- */
.bigfont {
font-size: 30px;
}
.size5{
font-size: 20px;
}
.tit_font{
font-size: 60px;
}
</style>
---
## 每个小组均有对应的助教同学负责
---
<h1 lang="zh-CN" style="font-size: 60px;"> </h1>
<br>
<br>
## 纯粹的R代码学习 → 使用R语言来实现**统计知识** <br>
<br>
## (1) R: 更灵活的统计分析方法，与统计知识结合更加紧密<br>
<br>
## (2) 心理学/社会科学中常用的统计检验均是回归模型<br>
---
<h1 lang="zh-CN" style="font-size: 60px;">研究问题</h1>
<br>
## 在penguin数据中，恋爱状态(romantic)对被试核心体温(Temperature)的影响
<br>
<br>
## 在penguin数据中，距赤道距离(DEQ)和恋爱状态(romantic)对于被试核心体温(Temperature)的影响
<br>
<br>
---
<h1 lang="en" style="font-size: 60px;">Contents</h1>
<br>
<span style="font-size: 45px;">8.1 *t*-test & linear regression</span></center> <br>
<span style="font-size: 30px;">&emsp;8.1.1 独立样本*t*检验</span></center> <br>
<span style="font-size: 30px;">&emsp;8.1.2 线性回归</span></center> <br>
<span style="font-size: 30px;">&emsp;8.1.3 单样本*t*检验</span></center> <br>
<span style="font-size: 30px;">&emsp;8.1.4 配对样本*t*检验</span></center> <br>
<span style="font-size: 30px;">&emsp;8.1.5 bruceR::TTEST</span></center> <br>
<br>
<span style="font-size: 45px;">8.2 ANOVA & linear regression</span></center> <br>
<span style="font-size: 30px;">&emsp;8.2.1 研究问题</span></center> <br>
<span style="font-size: 30px;">&emsp;8.2.2 代码实操</span></center> <br>
<span style="font-size: 30px;">&emsp;8.2.3 线性回归</span></center> <br>
<span style="font-size: 30px;">&emsp;8.2.4 知识延申</span></center> <br>
---
class: center, middle
<span style="font-size: 60px;">8.1 *t*-test & linear regression</span> <br>
---
class: left, middle
<span style="font-size: 60px;">8.1 研究问题</span> <br>
<!-- -->
(引自[IJzerman et al., 2018](https://doi.org/10.1525/collabra.165))
<br>
<span style="font-size: 35px;">Q: 如何检查恋爱状态(romantic)对核心体温(Temperature)的影响？</span> <br>
<br>
<span style="font-size: 40px;">A:独立样本*t*检验</span> <br>
---
# 8.1 *t*-test
## 8.1.1 独立样本*t*检验(independent *t*-test)
.panelset[
.panel[.panel-name[基础知识]
<br>
**比较两个独立样本群体的均值是否有显著差异。**
<br>
**前提条件**
* 正态性：两个样本数据都应该来自正态分布的总体。样本量足够大时，即使不严格服从正态分布，结果也是稳健的。
* 同方差性：两个样本的方差应该是相等的。
* 独立性：两个样本应该是独立的，即一个样本的观测值不应影响另一个样本的观测值。<br>
<br>
**假设**
* `\(H_0\)`: 两个独立样本群体的均值没有显著差异，即 `\(μ_1\)` = `\(μ_2\)`
* `\(H_1\)`: 两个独立样本群体的均值有显著差异，即 `\(μ_1\)` ≠ `\(μ_2\)`
<br>
`$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$`
.panel[.panel-name[数据清理]
```r
df.penguin <- bruceR::import(here::here('data', 'penguin', 'penguin_rawdata.csv')) %>%
dplyr::mutate(subjID = row_number()) %>%
dplyr::select(subjID,Temperature_t1, Temperature_t2, socialdiversity,
Site, DEQ, romantic, ALEX1:ALEX16) %>% # 选择变量
dplyr::filter(!is.na(Temperature_t1) & !is.na(Temperature_t2) & !is.na(DEQ)) %>% # 处理缺失值
dplyr::mutate(romantic = factor(romantic, levels = c(1,2), labels = c("恋爱", "单身")), # 转化为因子
Temperature = rowMeans(select(., starts_with("Temperature"))), # 计算两次核心温度的均值
ALEX4 = case_when(TRUE ~ 6 - ALEX4),
ALEX12 = case_when(TRUE ~ 6 - ALEX12),
ALEX14 = case_when(TRUE ~ 6 - ALEX14),
ALEX16 = case_when(TRUE ~ 6 - ALEX16),
ALEX = rowSums(select(., starts_with("ALEX")))) # 反向计分后计算总分
```
<div class="datatables html-widget html-fill-item" id="htmlwidget-27888db58a4768197891" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-27888db58a4768197891">{"x":{"filter":"none","vertical":false,"fillContainer":true,"data":[["1","2","3","4","5","6"],[1,2,3,4,5,6],[36.8,34.2,35,36.1,35.72222222,35.3],[36.7,36.6,35.2,35.8,36.44444444,35.9],[8,6,5,7,5,7],["Tsinghua","Oxford","Oxford","Oxford","Chile","Bamberg"],[26.88780212,51.75,51.75,51.75,42.07719421,51.65660095],["单身","恋爱","恋爱","恋爱","恋爱","恋爱"],[2,1,2,3,1,1],[2,1,2,4,1,2],[2,1,4,1,1,1],[4,5,4,4,5,4],[2,1,2,4,1,1],[2,1,2,2,1,3],[2,1,2,3,2,4],[2,1,2,4,1,2],[2,1,2,2,2,4],[1,1,1,1,1,2],[2,1,2,2,1,2],[4,4,3,4,4,2],[2,3,2,2,2,4],[4,4,4,4,4,5],[2,3,2,4,1,1],[4,4,4,4,2,5],[36.75,35.40000000000001,35.1,35.95,36.08333333,35.59999999999999],[31,23,34,40,24,35]],"container":"<table class=\"display fill-container\">\n <thead>\n <tr>\n <th> <\/th>\n <th>subjID<\/th>\n <th>Temperature_t1<\/th>\n <th>Temperature_t2<\/th>\n <th>socialdiversity<\/th>\n <th>Site<\/th>\n <th>DEQ<\/th>\n <th>romantic<\/th>\n <th>ALEX1<\/th>\n <th>ALEX2<\/th>\n <th>ALEX3<\/th>\n <th>ALEX4<\/th>\n <th>ALEX5<\/th>\n <th>ALEX6<\/th>\n <th>ALEX7<\/th>\n <th>ALEX8<\/th>\n <th>ALEX9<\/th>\n <th>ALEX10<\/th>\n <th>ALEX11<\/th>\n <th>ALEX12<\/th>\n <th>ALEX13<\/th>\n <th>ALEX14<\/th>\n <th>ALEX15<\/th>\n <th>ALEX16<\/th>\n <th>Temperature<\/th>\n <th>ALEX<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":4,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"subjID","targets":1},{"name":"Temperature_t1","targets":2},{"name":"Temperature_t2","targets":3},{"name":"socialdiversity","targets":4},{"name":"Site","targets":5},{"name":"DEQ","targets":6},{"name":"romantic","targets":7},{"name":"ALEX1","targets":8},{"name":"ALEX2","targets":9},{"name":"ALEX3","targets":10},{"name":"ALEX4","targets":11},{"name":"ALEX5","targets":12},{"name":"ALEX6","targets":13},{"name":"ALEX7","targets":14},{"name":"ALEX8","targets":15},{"name":"ALEX9","targets":16},{"name":"ALEX10","targets":17},{"name":"ALEX11","targets":18},{"name":"ALEX12","targets":19},{"name":"ALEX13","targets":20},{"name":"ALEX14","targets":21},{"name":"ALEX15","targets":22},{"name":"ALEX16","targets":23},{"name":"Temperature","targets":24},{"name":"ALEX","targets":25}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>
.panel[.panel-name[代码实操]
```r
stats::t.test(data = df.penguin, # 数据框
Temperature ~ romantic, # 因变量~自变量
var.equal = TRUE) %>%
capture.output() # 将输出变整齐
```
```
## [1] ""
## [2] "\tTwo Sample t-test"
## [3] ""
## [4] "data: Temperature by romantic"
## [5] "t = -0.34664, df = 1425, p-value = 0.7289"
## [6] "alternative hypothesis: true difference in means between group 恋爱 and group 单身 is not equal to 0"
## [7] "95 percent confidence interval:"
## [8] " -0.0555949 0.0388971"
## [9] "sample estimates:"
## [10] "mean in group 恋爱 mean in group 单身 "
## [11] " 36.38498 36.39333 "
## [12] ""
```
]]]]
---
# 8.1 *t*-test作为回归模型的特例
## 8.1.2 线性回归(linear regression)
<br>
**回归分析**
* 回归分析用于研究一个或多个自变量（预测变量）与一个因变量（响应变量）之间的关系。<br>
<br>
<br>
--
**线性回归**
* 线性回归的基本思想是通过数据拟合一条直线，使得这条直线尽可能地接近所有的数据点，从而实现对新数据点的预测。<br>
* 线性回归模型可以表示为：
`$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon$$`
* 其中，$y$是因变量，x是自变量， `\(\beta_i\)` 是模型参数，表示截距和斜率，$\epsilon$是误差项，表示模型未能解释的随机误差。
---
# 8.1 *t*-test & linear regression
## 8.1.2 线性回归(linear regression)
<br>
* 独立样本*t*检验是线性模型的特殊形式(自变量为二分变量）
--
<p align="center">
<img src="./picture/chp8/indet-lm.png" width="55%">
</p>
---
# 8.1 *t*-test & linear regression
.pull-left[
```r
# t检验
stats::t.test(
data = df.penguin,
Temperature ~ romantic,
var.equal = TRUE) %>%
capture.output() # 将输出变整齐
```
<p align="center">
<img src="./picture/chp8/compare1.1.png" width="100%">
</p>
]
.pull-right[
```r
# 线性回归
model.inde <- stats::lm(
data = df.penguin,
formula = Temperature ~ 1 + romantic
)
summary(model.inde)
```
<p align="center">
<img src="./picture/chp8/compare1.2.png" width="100%">
</p>
]
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.3 单样本*t*检验(one sample *t*-test)
* 例如：在penguin数据中，全体被试的核心体温(Temperature)是否等于36.6？<br>
<br>
--
<br>
**比较单个样本的平均值(m)与已知的总体平均值(μ)之间是否存在显著差异**<br>
<br>
**前提条件**
* 正态性：样本数据应来自正态分布的总体。样本量足够大时，即使不严格服从正态分布，结果也是稳健的。
* 独立性：样本中的观测值必须是独立的，即一个观测值不应影响另一个观测值。<br>
<br>
**假设**
* `\(H_0\)`: 样本的均值(m)与给定的总体均值或假设的总体均值(μ)之间没有显著差异。
* `\(H_1\)`: 样本的均值(m)与给定的总体均值或假设的总体均值(μ)之间有显著差异。
<br>
`$$t = \frac{\bar{X} - \mu}{s / \sqrt{n}}$$`
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.3 单样本*t*检验(one sample *t*-test)
.pull-left[
<br>
<br>
`$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p + \epsilon$$`
单样本*t*检验中，仅截距不为0。此时公式为：<br>
--
`$$y = \beta_0$$`
`$$H_0: \beta_0 = 0$$`
]
.pull-right[
<p align="center">
<img src="./picture/chp8/singlet-lm.png" width="75%">
</p>
]
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.3 单样本*t*检验(one sample *t*-test)
.pull-left[
```r
stats::t.test(
x = df.penguin$Temperature, # 核心体温均值
mu = 36.6)
```
<br>
<br>
<p align="center">
<img src="./picture/chp8/compare2.1.png" width="100%">
</p>
]
.pull-right[
```r
model.single <- lm(
data = df.penguin,
formula = Temperature - 36.6 ~ 1
)
summary(model.single)
```
<br>
<br>
<p align="center">
<img src="./picture/chp8/compare2.2.png" width="100%">
</p>
]
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.4 配对样本*t*检验(paired *t*-test)
* 例如，在penguin数据中，被试报告的两次核心温度(Temperature_t1, Temperature_t2)是否有显著差异？<br>
<br>
--
**比较两个相关的样本组（例如，同一组受试者在不同条件下的测量）的平均值是否存在显著差异。**<br>
<br>
**前提条件**
* 正态性：样本数据应来自正态分布的总体。样本量足够大时，即使不严格服从正态分布，结果也是稳健的。
* 独立性：配对样本中的观测值必须是独立的，即每一对观测值不应影响其他对的观测值。
* 配对设计：数据必须是以配对形式收集的。<br>
<br>
**假设**
* `\(H_0\)`: 配对样本的总体平均差与零没有显著差异（两个配对样本的均值没有显著差异）。
* `\(H_1\)`: 配对样本的总体平均差与零有显著差异（两个配对样本的均值存在显著差异）。
<br>
`$$t = \frac{\bar{X} - \mu}{s / \sqrt{n}}$$`
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.4 配对样本*t*检验(paired *t*-test)
$$y_1 - y_2 = \beta_0 $$
$$H_0: \beta_0 = 0 $$
--
可以将配对样本*t*检验理解为对差值进行的单样本*t*检验，即单独用一个数字来预测对应的差值（见图的右半部分）。<br>
也可以认为这些组间之差是斜率（见图的左半部分）。<br>
<p align="center">
<img src="./picture/chp8/pairt-lm.png" width="70%">
</p>
---
# 8.1 *t*-test系列均为回归模型的特例
## 8.1.4 配对样本*t*检验(paired *t*-test)
.pull-left[
```r
stats::t.test(
x = df.penguin$Temperature_t1,
y = df.penguin$Temperature_t2,
paired = TRUE
)
```
<br>
<br>
<p align="center">
<img src="./picture/chp8/compare3.1.png" width="100%">
</p>
]
.pull-right[
```r
model.paired <- lm(
Temperature_t1 - Temperature_t2 ~ 1,
data = df.penguin
)
summary(model.paired)
```
<br>
<br>
<p align="center">
<img src="./picture/chp8/compare3.2.png" width="100%">
</p>
]
---
# 8.1 *t*-test
## 8.1.5 bruceR::TTEST
* 如果偏好按传统方式使用*t*检验，推荐`bruceR::TTEST`函数
<p align="center">
<img src="./picture/chp8/TTEST.png" width="70%">
</p>
中文帮助文档：https://zhuanlan.zhihu.com/p/281150493
---
# 8.1 *t*-test
## 8.1.5 bruceR::TTEST
.panelset[
.panel[.panel-name[独立样本t检验]
.pull-left[
```r
stats::t.test(
data = df.penguin,
Temperature ~ romantic,
var.equal = TRUE
)
```
<br>
<p align="center">
<img src="./picture/chp8/compare1.1.png" width="100%">
</p>
]
.pull-right[
```r
bruceR::TTEST(
data = df.penguin, # 数据
y = "Temperature", # 因变量
x = "romantic" # 自变量
)
```
<br>
<p align="center">
<img src="./picture/chp8/leven.png" width="100%">
</p>
]
.panel[.panel-name[单样本t检验]
.pull-left[
```r
stats::t.test(
x = df.penguin$Temperature,
mu = 36.6
)
```
<br>
<p align="center">
<img src="./picture/chp8/compare2.1.png" width="100%">
</p>
]
.pull-right[
```r
bruceR::TTEST(
data = df.penguin, # 数据
y = "Temperature", # 确定变量
test.value = 36.6, # 固定值
test.sided = "=") # 假设的方向
```
<br>
<p align="center">
<img src="./picture/chp8/sample.png" width="100%">
</p>
]
.panel[.panel-name[配对样本t检验]
.pull-left[
```r
stats::t.test(
x = df.penguin$Temperature_t1, #第1次
y = df.penguin$Temperature_t2, #第2次
paired = TRUE)
```
<br>
<p align="center">
<img src="./picture/chp8/compare3.1.png" width="100%">
</p>
]
.pull-right[
```r
bruceR::TTEST(
data = df.penguin, # 数据
y = c("Temperature_t1",
"Temperature_t2"), # 变量为两次核心体温
paired = T) # 配对数据，默认是FALSE
```
<br>
<p align="center">
<img src="./picture/chp8/pair.png" width="100%">
</p>
]
]]]]
---
# 8.1 *t*-test系列均为回归模型的特例
<br>
<br>
| | R自带函数 | 线性模型 | 解释 |
|-------|-------|-------|-------|
| 单样本*t* | t.test(y, mu = 0) | lm(y ~ 1)| 仅有截距的回归模型 |
| 独立样本*t* | t.test($y_1$, `\(y_2\)`) | lm(y ~ 1 + `\(G_2\)`)| 自变量为二分变量的回归模型 |
| 配对样本*t* | t.test($y_1$, `\(y_2\)`, paired=T) | lm($y_1$ - `\(y_2\)` ~ 1)| 仅有截距的回归模型)|
---
class: center, middle
<span style="font-size: 60px;">8.2 ANOVA & linear regression</span> <br>
---
class: left, middle
<span style="font-size: 60px;">8.2.1 研究问题</span> <br>
<br>
<span style="font-size: 35px;">Q: 如何同时检验距赤道距离(DEQ)和恋爱状态(romantic)对于被试体温的影响</span> <br>
<br>
<span style="font-size: 40px;">A:双因素被试间方差分析</span> <br>
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操 & 知识回顾
.pull-left[
<br>
**方法简介：**
当研究者想要比较两个或多个组之间的均值差异时，可使用方差分析(Analysis of Variance，简称ANOVA)。<br>
<br>
它由英国统计学家R.A.Fisher提出，基本思想是将测量数据的总变异（即总方差）按照变异来源分为处理（组间）效应和误差（组内）效应，并作出其数量估计，从而确定实验处理对研究结果影响力的大小。
**假设：**
* `\(H_0\)`: 各因素各个水平下，因变量的均值完全相同
* `\(H_1\)`: 各因素各个水平下，因变量的均值不完全相同
]
.pull-right[
<br>
<br>
<br>
<br>
**前提条件：**
* 可加性：各效应可加，即观测值是由各主效应，交互作用以及误差通过相加得到的<br>
* 随机性：各样本（观测值）是随机样本<br>
* 正态性：各样本来自于正态分布的总体<br>
* 独立性：各样本观测值互相独立<br>
* 方差齐性：各样本来自的总体方差相同<br>
* 因变量应为连续变量<br>
]
---
# 8.2 ANOVA
## 8.2.2 代码实操|数据预处理
.panelset[
.panel[.panel-name[vars]
```r
summary(df.penguin$DEQ)
```
```
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.293 34.433 39.912 39.842 51.317 60.391
```
```r
# 设定分割点
# [0-23.5 热带， 23.5-35 亚热带]， [35-40 暖温带， 40-50 中温带]， [50-66.5 寒温带]
breaks <- c(0, 35, 50, 66.5)
# 设定相应的标签
labels <- c('热带', '温带', '寒温带')
# 创建新的变量
df.penguin$climate <- cut(df.penguin$DEQ,
breaks = breaks,
labels = labels)
summary(df.penguin$climate)
```
```
## 热带 温带 寒温带
## 396 592 439
```
.panel[.panel-name[tidy data]
```r
df <- df.penguin %>%
select(subjID, climate, romantic, Temperature)
```
<div class="datatables html-widget html-fill-item" id="htmlwidget-bec36c95e7b22d5fe1bc" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-bec36c95e7b22d5fe1bc">{"x":{"filter":"none","vertical":false,"fillContainer":true,"data":[["1","2","3","4","5","6"],[1,2,3,4,5,6],["热带","寒温带","寒温带","寒温带","温带","寒温带"],["单身","恋爱","恋爱","恋爱","恋爱","恋爱"],[36.75,35.40000000000001,35.1,35.95,36.08333333,35.59999999999999]],"container":"<table class=\"display fill-container\">\n <thead>\n <tr>\n <th> <\/th>\n <th>subjID<\/th>\n <th>climate<\/th>\n <th>romantic<\/th>\n <th>Temperature<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"columnDefs":[{"className":"dt-right","targets":[1,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"subjID","targets":1},{"name":"climate","targets":2},{"name":"romantic","targets":3},{"name":"Temperature","targets":4}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>
]]]
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操|正态性检验
.panelset[
.panel[.panel-name[KS检验]
```r
# 正态性检验-Kolmogorov-Smirnov检验
# 若p >.05，不能拒绝数据符合正态分布的零假设
ks.test(df$Temperature, 'pnorm')
```
```
##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: df$Temperature
## D = 1, p-value < 0.00000000000000022
## alternative hypothesis: two-sided
```
```r
# 进行数据转换，转换后仍非正态分布
df$Temperature_log <- log(df$Temperature)
ks.test(df$Temperature_log, 'pnorm')
```
```
##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: df$Temperature_log
## D = 0.99981, p-value < 0.00000000000000022
## alternative hypothesis: two-sided
```
.panel[.panel-name[qq图]
```r
# 正态性检验-qq图
qqnorm(df$Temperature)
qqline(df$Temperature, col = "red") # 添加理论正态分布线
```
<!-- -->
.panel[.panel-name[直方图]
```r
ggplot(df, aes(Temperature)) +
geom_histogram(aes(y =..density..), color='black', fill='white', bins=30) +
geom_density(alpha=.5, fill='red')
```
<!-- -->
]]]]
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操|双因素被试间方差分析
.panelset[
.panel[.panel-name[stats::aov()]
```r
aov1 <- stats::aov(Temperature ~ climate * romantic, data = df)
summary(aov1)
```
```
## Df Sum Sq Mean Sq F value Pr(>F)
## climate 2 18.82 9.408 49.392 < 0.0000000000000002 ***
## romantic 1 0.24 0.244 1.280 0.25807
## climate:romantic 2 1.91 0.955 5.014 0.00676 **
## Residuals 1421 270.65 0.190
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
.panel[.panel-name[SPSS]
<!-- -->
]]]
---
class: center, middle
<span style="font-size: 60px;">结果为什么不相同？</span> <br>
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操|平方和(SS)的计算
<br>
在平衡设计中，三种类型的平方和的结果会很清晰，并且方差分析的结果独立于平方和的类型；<br>
而在非平衡设计中，尤其是当各组样本量差距较大时，三种类型的平方和计算结果可能会不同，此时需要根据具体研究设计和问题来选择使用哪一种类型的平方和。<br>
<br>
对于 Y ~ A + B + A * B<br>
<br>
.panelset[
.panel[.panel-name[Type I SS]
<br>
解释变量的顺序会影响到类型I平方和的计算结果，通常用于顺序重要的模型。<br>
效应根据表达式中先出现的效应做调整。A不做调整，B根据A调整，A:B交互项根据A和B调整。<br>
<br>
**stats::aov()函数默认采用的就是Type I SS，它逐步将每一个因子引入模型进行计算。**<br>
.panel[.panel-name[Type II SS]
<br>
忽略了因子之间可能存在的交互作用，适用于所有主效应不涉及交互效应的情况。<br>
假定所有的因子都是同时进入模型的，并且它们都是等价的。<br>
效应根据同水平或低水平的效应做调整。A根据B调整，B依据A调整，A:B交互项同时根据A和B调整。<br>
<br>
**car::Anova()函数默认计算Type II SS，可以通过type = 3调整为Type III SS。**<br>
.panel[.panel-name[Type III SS]
<br>
更全面，假定所有因子（以及它们的交互项）都是重要的，并考虑所有因素。<br>
每个效应根据模型其他各效应做相应调整。A根据B和A:B做调整，A:B交互项根据A和B调整。<br>
<br>
**SPSS默认采用Type III SS。**<br>
**bruceR::MANOVA*()函数也默认采用Type III SS，可以通过ss.type = 2调整为Type II SS。**<br>
]]]]
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操|双因素被试间方差分析
.panelset[
.panel[.panel-name[car::Anova()]
```r
# 结果不一致，原因PPT显示不全，请回到rmd文档查看
aov1 <- car::Anova(stats::aov(Temperature ~ climate * romantic, data = df))
aov1
```
```
## Anova Table (Type II tests)
##
## Response: Temperature
## Sum Sq Df F value Pr(>F)
## climate 19.034 2 49.9680 < 0.00000000000000022 ***
## romantic 0.244 1 1.2801 0.258072
## climate:romantic 1.910 2 5.0136 0.006765 **
## Residuals 270.654 1421
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
```r
# 原因debug
# 查看R的默认对比设置
options("contrasts")
```
```
## $contrasts
## unordered ordered
## "contr.treatment" "contr.poly"
```
```r
# 从输出结果可知，无序默认为contr.treatment()，有序默认为contr.poly()
# factor()函数来创建无序因子，ordered()函数创建有序因子
is.factor(df$climate)
```
```
## [1] TRUE
```
```r
is.ordered(df$climate)
```
```
## [1] FALSE
```
```r
# climate是无序因子
# 创建一个3水平的因子的基准对比
c1 <- contr.treatment(3)
# 创建一个新的对比，这个编码假设分类水平之间的差异被等分，每一个水平与总均值的差异等于1/3
my.coding <- matrix(rep(1/3, 6), ncol=2)
# 将对比调整为每个水平与第一个水平的振幅减去1/3
# 可能的原因：除了关心每个水平对应的效果，同时也关心水平与水平之间的效果
my.simple <- c1-my.coding
my.simple
```
```
## 2 3
## 1 -0.3333333 -0.3333333
## 2 0.6666667 -0.3333333
## 3 -0.3333333 0.6666667
```
```r
# 更改climate的对比
contrasts(df$climate) <- my.simple
# 将数据集df的romantic列的对比设为等距对比，它假设分类水平之间的差异为等距离
contrasts(df$romantic) <- contr.sum(2)/2
# 方差分析
aov1 <- car::Anova(lm(Temperature ~ climate * romantic, data = df),
type = 3)
aov1
```
```
## Anova Table (Type III tests)
##
## Response: Temperature
## Sum Sq Df F value Pr(>F)
## (Intercept) 1768309 1 9284069.6498 < 0.00000000000000022 ***
## climate 19 2 50.8832 < 0.00000000000000022 ***
## romantic 0 1 1.0006 0.317336
## climate:romantic 2 2 5.0136 0.006765 **
## Residuals 271 1421
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
.panel[.panel-name[afex::aov_ez()]
```r
afex::aov_ez(id = "subjID",
dv = "Temperature",
data = df,
between = c("climate", "romantic"),
type = 3)
```
```
## Anova Table (Type 3 tests)
##
## Response: Temperature
## Effect df MSE F ges p.value
## 1 climate 2, 1421 0.19 50.88 *** .067 <.001
## 2 romantic 1, 1421 0.19 1.00 <.001 .317
## 3 climate:romantic 2, 1421 0.19 5.01 ** .007 .007
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1
```
```r
# afex中的其他函数可以得到同样的结果
afex::aov_car(Temperature ~ climate * romantic + Error(subjID), data = df, type = 3)
afex::aov_4(Temperature ~ climate * romantic + (1|subjID), data = df)
```
]]]
---
# 8.2 ANOVA & linear regression
## 8.2.3 线性回归
.pull-left[
<br>
<br>
<br>
这里的ANOVA仍然是线性回归模型的特例，即自变量是离散变量的情况。
<br>
如果使用哑变量（dummy coding）对自变量进行编码后进入回归方程，线性模型中的斜率即是对组间差异的估计。<br>
<br>
**双因素方差分析可以看作是一种特殊的线性回归模型，自变量为两个分类变量。**<br>
]
.pull-right[
<br>
<br>
<!-- -->
]
---
# 8.2 ANOVA & linear regression
## 8.2.3 线性回归
.pull-left[
```r
aov1 <- car::Anova(
aov(Temperature ~ climate * romantic,
data = df),
type = 3
)
aov1
```
<br>
]
.pull-right[
```r
lm1 <- car::Anova(
lm(Temperature ~ climate * romantic,
data = df),
type = 3
)
lm1
```
<br>
]
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操: `bruceR`
* 使用`bruceR`可以更简单地实现心理学中习惯的ANOVA，但要注意数据格式
<p align="center">
<img src="./picture/chp8/ANOVA data.png" width="65%">
</p>
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操: `bruceR::MANOVA`
<p align="center">
<img src="./picture/chp8/MANOVA.png" width="70%">
</p>
---
# 8.2 ANOVA & linear regression
## 8.2.2 代码实操: `bruceR::MANOVA`
```r
res1 <- bruceR::MANOVA(data = df,
dv = "Temperature",
between = c("climate", "romantic"))
```
```
## [1] "Anova Table (Type III tests)"
## [2] ""
## [3] "Response: Temperature"
## [4] " Effect df MSE F ges p.value"
## [5] "1 climate 2, 1421 0.19 50.88 *** .067 <.001"
## [6] "2 romantic 1, 1421 0.19 1.00 <.001 .317"
## [7] "3 climate:romantic 2, 1421 0.19 5.01 ** .007 .007"
## [8] "---"
## [9] "Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1"
```