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<title>FDR / s value on Correlated Null: betahat = betahat, sebetahat = betahat / zscore</title>

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<h1 class="title toc-ignore">FDR / <span class="math inline">\(s\)</span> value on Correlated Null: <code>betahat = betahat</code>, <code>sebetahat = betahat / zscore</code></h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-02-16</em></h4>

</div>


<p><strong>Last updated:</strong> 2017-02-16</p>
<p><strong>Code version:</strong> ca57b48</p>
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Apply two FDR-controlling procedures, Benjamini–Hochberg 1995 (“<a href="https://www.jstor.org/stable/2346101">BH</a>”) and Benjamini-Yekutieli 2001 (“<a href="https://projecteuclid.org/euclid.aos/1013699998">BY</a>”), and two <span class="math inline">\(s\)</span> value models, <code>ash</code> and <code>truncash</code> (with the threshold <span class="math inline">\(T = 1.96\)</span>) to the simulated, correlated null data. The data are obtained from 5 vs 5 GTEx/Liver samples and 10K top expressed genes, and <span class="math inline">\(1000\)</span> independent simulation trials.</p>
<p>Compare the numbers of false discoveries (by definition, all discoveries should be false) obtained by these four methods, using FDR <span class="math inline">\(\leq 0.05\)</span> and <span class="math inline">\(s\)</span>-value <span class="math inline">\(\leq 0.05\)</span> as cutoffs.</p>
</div>
<div id="simulation-p-values-for-bh-and-by-procedures-hatbeta-hatbeta-hat-s-hatbeta-hat-z-for-ash-and-truncash." class="section level2">
<h2>Simulation: <span class="math inline">\(p\)</span> values for <a href="https://www.jstor.org/stable/2346101">BH</a> and <a href="https://projecteuclid.org/euclid.aos/1013699998">BY</a> procedures; <span class="math inline">\(\hat\beta = \hat\beta\)</span>, <span class="math inline">\(\hat s = \hat\beta / \hat z\)</span>, for <code>ash</code> and <code>truncash</code>.</h2>
<p><strong><span class="math inline">\(\hat\beta\)</span> obtained from <a href="nullpipeline.html">the <code>limma::lmFit</code> step of the pipeline</a>, <span class="math inline">\(\hat z\)</span> obtained from <a href="nullpipeline.html">the last step of the pipeline</a>.</strong></p>
<pre class="r"><code>library(ashr)
source(&quot;../code/truncash.R&quot;)</code></pre>
<pre class="r"><code>p = read.table(&quot;../output/p_null_liver_777.txt&quot;)
z = read.table(&quot;../output/z_null_liver_777.txt&quot;)
betahat = read.table(&quot;../output/betahat_null_liver_777.txt&quot;)

m = dim(p)[1]
n = dim(p)[2]
fd.bh = fd.by = fd.ash = fd.truncash = c()

for (i in 1:m) {
  p_BH = p.adjust(p[i, ], method = &quot;BH&quot;)
  fd.bh[i] = sum(p_BH &lt;= 0.05)
  p_BY = p.adjust(p[i, ], method = &quot;BY&quot;)
  fd.by[i] = sum(p_BY &lt;= 0.05)
  betahat_trial = as.numeric(betahat[i, ])
  sebetahat_trial = - betahat_trial / as.numeric(z[i, ])
  fit.ash = ashr::ash(betahat_trial, sebetahat_trial, method = &quot;fdr&quot;, mixcompdist = &quot;normal&quot;)
  fd.ash[i] = sum(ashr::get_svalue(fit.ash) &lt;= 0.05)
  fit.truncash = truncash(betahat_trial, sebetahat_trial, t = qnorm(0.975))
  fd.truncash[i] = sum(get_svalue(fit.truncash) &lt;= 0.05)
}</code></pre>
</div>
<div id="result" class="section level2">
<h2>Result</h2>
<p>Simulated under the global null, FWER <span class="math inline">\(=\)</span> FDR.</p>
<div id="estimated-fwer-or-fdr-by-bh" class="section level3">
<h3>Estimated FWER or FDR by BH</h3>
<pre class="r"><code>fdr.bh = mean(fd.bh &gt;= 1)
fdr.bh</code></pre>
<pre><code>[1] 0.046</code></pre>
<p>Estimated FWER or FDR by BY</p>
<pre class="r"><code>fdr.by = mean(fd.by &gt;= 1)
fdr.by</code></pre>
<pre><code>[1] 0.006</code></pre>
<p>Estimated FWER or FDR by <code>ash</code></p>
<pre class="r"><code>fdr.ash = mean(fd.ash &gt;= 1)
fdr.ash</code></pre>
<pre><code>[1] 0.096</code></pre>
<p>Estimated FWER or FDR by <code>truncash</code></p>
<pre class="r"><code>fdr.truncash = mean(fd.truncash &gt;= 1)
fdr.truncash</code></pre>
<pre><code>[1] 0.062</code></pre>
</div>
<div id="happenstance-of-false-discoveries-by-four-approaches" class="section level3">
<h3>Happenstance of false discoveries by four approaches</h3>
<pre class="r"><code>maxcount = max(c(fd.bh, fd.by, fd.ash, fd.truncash))
xlim = c(0, maxcount)
maxfreq = max(c(max(table(fd.bh)), max(table(fd.by)), max(table(fd.ash)), max(table(fd.truncash))))
ylim = c(0, maxfreq)
plot(table(fd.bh), xlab = &quot;Number of False Discoveries / 10K&quot;, ylab = &quot;Frequency&quot;, main = &quot;Benjamini - Hochberg 1995&quot;, xlim = xlim, ylim = ylim)</code></pre>
<p><img src="figure/FDR_null_betahat.Rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(table(fd.by), xlab = &quot;Number of False Discoveries / 10K&quot;, ylab = &quot;Frequency&quot;, main = &quot;Benjamini - Yekutieli 2001&quot;, xlim = xlim, ylim = ylim)</code></pre>
<p><img src="figure/FDR_null_betahat.Rmd/unnamed-chunk-7-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(table(fd.ash), xlab = &quot;Number of False Discoveries / 10K&quot;, ylab = &quot;Frequency&quot;, main = &quot;ash&quot;, xlim = xlim, ylim = ylim)</code></pre>
<p><img src="figure/FDR_null_betahat.Rmd/unnamed-chunk-7-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(table(fd.truncash), xlab = &quot;Number of False Discoveries / 10K&quot;, ylab = &quot;Frequency&quot;, main = &quot;truncash&quot;, xlim = xlim, ylim = ylim)</code></pre>
<p><img src="figure/FDR_null_betahat.Rmd/unnamed-chunk-7-4.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="comparison-of-the-numbers-of-false-discoveries-by-four-approaches" class="section level3">
<h3>Comparison of the numbers of false discoveries by four approaches</h3>
<pre class="r"><code>m = length(fd.bh)
fd.ind = (1:m)[!((fd.bh == 0) &amp; (fd.by == 0) &amp; (fd.ash == 0) &amp; (fd.truncash == 0))]
plot(1:length(fd.ind), fd.bh[fd.ind], pch = 4, ylim = xlim, xlab = &quot;Trials with False Discoveries&quot;, ylab = &quot;Number of False Discoveries / 10K&quot;)
points(1:length(fd.ind), fd.by[fd.ind], pch = 4, col = 2)
points(1:length(fd.ind), fd.ash[fd.ind], pch = 4, col = 3)
points(1:length(fd.ind), fd.truncash[fd.ind], pch = 4, col = 4)
legend(&quot;topright&quot;, c(&quot;BH&quot;, &quot;BY&quot;, &quot;ash&quot;, &quot;truncash&quot;), col = 1:4, pch = 4)</code></pre>
<p><img src="figure/FDR_null_betahat.Rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="session-information" class="section level2">
<h2>Session Information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: macOS Sierra 10.12.3

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] SQUAREM_2016.10-1 ashr_2.1.2       

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.9       knitr_1.15.1      magrittr_1.5     
 [4] workflowr_0.3.0   MASS_7.3-45       doParallel_1.0.10
 [7] pscl_1.4.9        lattice_0.20-34   foreach_1.4.3    
[10] stringr_1.1.0     tools_3.3.2       parallel_3.3.2   
[13] grid_3.3.2        git2r_0.18.0      htmltools_0.3.5  
[16] iterators_1.0.8   yaml_2.1.14       rprojroot_1.2    
[19] digest_0.6.9      codetools_0.2-15  evaluate_0.10    
[22] rmarkdown_1.3     stringi_1.1.2     backports_1.0.5  
[25] truncnorm_1.0-7  </code></pre>
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