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PRS_relative_risk_experiment.html
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<h1 class="title toc-ignore">Calculating PRS-based absolute risk</h1>
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<p>This page explores ways of converting PRS Z-scores into estimates of absolute risk by accounting for the prevelance of the outcome and the variance explained by the polygenic score. A useful way of describing risk of a binary outcome to an individual is by stating the percentage of people like them that have the outcome. 23andMe do this by calculating genetic risk for all individuals in their sample, seperating individuals into quantiles of genetic risk, and then calculating the proportion of individuals within each quantile that have the outcome based on observed questionaire data. However, often individual level data with observed phenotypic data is not available, but it prevelance and an estimate of variance explained by the polygenic score is. The variance explained can also be estimated based on sample size, heritabiliy and polygenicity estimates, as is done by the <a href="https://gwasrocs.ca/">GWAS-ROCS Database</a>. Here, we explore ways of converting genetic risk estimates into absolute risk without individual level observed data.</p>
<hr />
<div id="simulation-based-approach" class="section level1">
<h1><span class="header-section-number">1</span> Simulation-based approach</h1>
<div id="create-function" class="section level2">
<h2><span class="header-section-number">1.1</span> Create function</h2>
<p>This function simulates a binary phenotype of a given prevelance, and a polygenic score with a given correlation with the phenotype. The function uses an AUC value to specify the correlation with the phenotype, but other metrics could be used. The function then uses the simulated data to determine how many cases and controls would be within each quantile of genetic risk.</p>
<pre class="r"><code>PRS_abs_risk<-function(n=10000, PRS_auc=0.6, prev=0.3, n_quantile=20, seed=1){
# Create function to convert AUC to R2 on observed scale
h2_obs_AUC <- function(k,auc) {
T0 <- qnorm(1 - k)
z <- dnorm(T0)
i <- z / k
v <- -i * (k / (1-k))
q <- qnorm(auc)
h2l <- 2 * q^2 / ((v - i)^2 + q^2 * i * (i - T0) + v * (v - T0)) # eq 4
p<-k
x= qnorm(1-k)
z= dnorm(x)
i=z/k
C= k*(1-k)*k*(1-k)/(z^2*p*(1-p))
h2_obs = h2l/C
}
# Calculate r2 on observed scale corresonding to desired AUC
r2<-h2_obs_AUC(k=prev, auc=PRS_auc)
# Simulate binary and continuous variable with desired correlation and prevelance
library(SimMultiCorrData)
sim<-rcorrvar(n = n, k_cat = 1, k_cont = 1, method = "Polynomial",
means = 0, vars = 1, skews = 0, skurts = 0, fifths = 0, sixths = 0,
marginal = list(1-prev), support = list(0:2),
rho = matrix(c(1, sqrt(r2), sqrt(r2), 1), 2, 2), errorloop=T, seed=seed)
sim_dat<-data.frame(Ord=sim$ordinal_variables, Cont=sim$continuous_variables)
names(sim_dat)<-c('y','x')
# Calculate simulated AUC
mod <- glm(y ~ x, data=sim_dat, family = "binomial")
prob=predict(mod,type=c("response"))
library(pROC)
prs_roc<-roc(sim_dat$y ~ prob)
# Split data into quantiles
by_quant<-1/n_quantile
perc<-quantile(sim_dat$x, probs = seq(0, 1, by=by_quant))
sim_dat<-data.frame(y=sim_dat$y, x=sim_dat$x, quantile=cut(sim_dat$x, quantile(sim_dat$x, prob = seq(0, 1, length = 1/by_quant+1), type = 5) ))
# Calculate proportion of cases in each quantile
Risk_per_bin<-NULL
for(i in 1:max(as.numeric(sim_dat$quantile), na.rm=T)){
PRS_bin_temp<-sim_dat$y[as.numeric(sim_dat$quantile) == i]
temp<-data.frame(Quantile=i,
PRS_range=levels(sim_dat$quantile)[i],
perc_con=paste0(round(sum(PRS_bin_temp == 0, na.rm=T)/length(PRS_bin_temp)*100, 2),'%'),
perc_case=paste0(round(sum(PRS_bin_temp == 1, na.rm=T)/length(PRS_bin_temp)*100, 2),'%'))
Risk_per_bin<-rbind(Risk_per_bin,temp)
}
print(Risk_per_bin)
print(prs_roc$auc)
output<-list()
output[['Risk']]<-Risk_per_bin
output[['AUC']]<-as.numeric(prs_roc$auc)
output[['Prevelance']]<-mean(sim_dat$y)
output[['Data']]<-sim_dat
output[['Input']]<-data.frame(Option=c('n', 'PRS_auc', 'prev', 'n_quantile','seed'),
Value=c(n, PRS_auc, prev, n_quantile, seed))
return(output)
}</code></pre>
<hr />
</div>
<div id="evaluate-function" class="section level2">
<h2><span class="header-section-number">1.2</span> Evaluate function</h2>
<p>Here will see whether this function can replicate information provided by 23andMe using simulated data alone. We will use the prevelence and AUC reported by 23andMe for earlobe type.</p>
<hr />
<div id="simulated-result" class="section level3">
<h3><span class="header-section-number">1.2.1</span> Simulated result</h3>
<pre class="r"><code>test<-PRS_abs_risk(PRS_auc=0.641, prev=0.7463, n_quantile=20)</code></pre>
<p>Show the AUC and prevelance of the simulated data:</p>
<pre><code>## Simulation AUC = 0.6245574</code></pre>
<pre><code>## Simulation prevelance = 0.7452</code></pre>
<p>Show the proprtion of cases within each quantile of the simulated data:</p>
<table>
<caption>Absolute risk for PRS Z-score vigintiles</caption>
<thead>
<tr class="header">
<th align="right">Quantile</th>
<th align="left">PRS_range</th>
<th align="left">perc_con</th>
<th align="left">perc_case</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="right">1</td>
<td align="left">(-3.67,-1.65]</td>
<td align="left">49.8%</td>
<td align="left">50%</td>
</tr>
<tr class="even">
<td align="right">2</td>
<td align="left">(-1.65,-1.29]</td>
<td align="left">38.12%</td>
<td align="left">61.68%</td>
</tr>
<tr class="odd">
<td align="right">3</td>
<td align="left">(-1.29,-1.04]</td>
<td align="left">36.33%</td>
<td align="left">63.47%</td>
</tr>
<tr class="even">
<td align="right">4</td>
<td align="left">(-1.04,-0.849]</td>
<td align="left">34.53%</td>
<td align="left">65.27%</td>
</tr>
<tr class="odd">
<td align="right">5</td>
<td align="left">(-0.849,-0.679]</td>
<td align="left">29.14%</td>
<td align="left">70.66%</td>
</tr>
<tr class="even">
<td align="right">6</td>
<td align="left">(-0.679,-0.51]</td>
<td align="left">28.34%</td>
<td align="left">71.46%</td>
</tr>
<tr class="odd">
<td align="right">7</td>
<td align="left">(-0.51,-0.391]</td>
<td align="left">26.35%</td>
<td align="left">73.45%</td>
</tr>
<tr class="even">
<td align="right">8</td>
<td align="left">(-0.391,-0.254]</td>
<td align="left">25.35%</td>
<td align="left">74.45%</td>
</tr>
<tr class="odd">
<td align="right">9</td>
<td align="left">(-0.254,-0.127]</td>
<td align="left">24.55%</td>
<td align="left">75.25%</td>
</tr>
<tr class="even">
<td align="right">10</td>
<td align="left">(-0.127,0.00894]</td>
<td align="left">28.94%</td>
<td align="left">70.86%</td>
</tr>
<tr class="odd">
<td align="right">11</td>
<td align="left">(0.00894,0.131]</td>
<td align="left">22.95%</td>
<td align="left">76.85%</td>
</tr>
<tr class="even">
<td align="right">12</td>
<td align="left">(0.131,0.262]</td>
<td align="left">22.75%</td>
<td align="left">77.05%</td>
</tr>
<tr class="odd">
<td align="right">13</td>
<td align="left">(0.262,0.39]</td>
<td align="left">20.36%</td>
<td align="left">79.44%</td>
</tr>
<tr class="even">
<td align="right">14</td>
<td align="left">(0.39,0.525]</td>
<td align="left">19.16%</td>
<td align="left">80.64%</td>
</tr>
<tr class="odd">
<td align="right">15</td>
<td align="left">(0.525,0.674]</td>
<td align="left">18.96%</td>
<td align="left">80.84%</td>
</tr>
<tr class="even">
<td align="right">16</td>
<td align="left">(0.674,0.833]</td>
<td align="left">19.36%</td>
<td align="left">80.44%</td>
</tr>
<tr class="odd">
<td align="right">17</td>
<td align="left">(0.833,1.03]</td>
<td align="left">18.56%</td>
<td align="left">81.24%</td>
</tr>
<tr class="even">
<td align="right">18</td>
<td align="left">(1.03,1.27]</td>
<td align="left">19.36%</td>
<td align="left">80.44%</td>
</tr>
<tr class="odd">
<td align="right">19</td>
<td align="left">(1.27,1.64]</td>
<td align="left">14.37%</td>
<td align="left">85.43%</td>
</tr>
<tr class="even">
<td align="right">20</td>
<td align="left">(1.64,3.62]</td>
<td align="left">11.38%</td>
<td align="left">88.42%</td>
</tr>
</tbody>
</table>
<hr />
</div>
<div id="observed-23andme-result" class="section level3">
<h3><span class="header-section-number">1.2.2</span> Observed 23andMe result</h3>
<div class="figure" style="text-align: center">
<img src="/scratch/users/k1806347/Software/MyGit/GenoPred/Images/PRS_relative_risk_experiment/23andMe_Earlobe.png" alt="23andMe Earlobe table" width="75%" />
<p class="caption">
23andMe Earlobe table
</p>
</div>
<hr />
</div>
</div>
<div id="conclusion" class="section level2">
<h2><span class="header-section-number">1.3</span> Conclusion</h2>
<p>The results is highly concordant! The main limitation here is that the simulation does not perfectly reflect the specified prevelance or AUC, meaning the results are slightly different.</p>
<p>Future work should explore the use of normal distribution theory, rather than simulated data, to estimate the proportion of cases within quantiles.</p>
<hr />
</div>
</div>
<div id="theory-based-approach" class="section level1">
<h1><span class="header-section-number">2</span> Theory-based approach</h1>
<p>Under development….</p>
<hr />
</div>
</div>
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