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<title>FWER Procedures on Simulated Correlated Null</title>

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<h1 class="title toc-ignore">FWER Procedures on Simulated Correlated Null</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2016-11-29</em></h4>

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<p><strong>Last updated:</strong> 2018-05-12</p>
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<p><strong>Last updated:</strong> 2018-05-12</p>
<p><strong>Code version:</strong> 140be7f</p>
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>In order to understand the behavior of <span class="math inline">\(p\)</span>-values of top expressed, correlated genes under the global null, simulated from GTEx data, we apply two FWER-controlling multiple comparison procedures, Holm’s “step-down” (<a href="http://www.jstor.org/stable/4615733?seq=1#page_scan_tab_contents">Holm 1979</a>) and Hochberg’s “step-up.” (<a href="https://academic.oup.com/biomet/article-abstract/75/4/800/423177/A-sharper-Bonferroni-procedure-for-multiple-tests">Hochberg 1988</a>)</p>
</div>
<div id="holms-step-down-procedure-start-from-the-smallest-p-value" class="section level2">
<h2>Holm’s step-down procedure: start from the smallest <span class="math inline">\(p\)</span>-value</h2>
<p><em>It can be shown that Holm’s procedure conservatively controls the FWER in the strong sense, under arbitrary correlation among <span class="math inline">\(p\)</span>-values.</em></p>
<p>First, order the <span class="math inline">\(p\)</span>-values</p>
<p><span class="math display">\[
p_{(1)} \leq p_{(2)} \leq \cdots \leq p_{(n)}
\]</span></p>
<p>and let <span class="math inline">\(H_{(1)}, H_{(2)}, \ldots, H_{(n)}\)</span> be the corresponding hypotheses. Then examine the <span class="math inline">\(p\)</span>-values in order.</p>
<p>Step 1: If <span class="math inline">\(p_{(1)} \leq \alpha/n\)</span> reject <span class="math inline">\(H_{(1)}\)</span> and go to Step 2. Otherwise, accept <span class="math inline">\(H_{(1)}, H_{(2)}, \ldots, H_{(n)}\)</span> and stop.</p>
<p>……</p>
<p>Step <span class="math inline">\(i\)</span>: If <span class="math inline">\(p_{(i)} \leq \alpha / (n − i + 1)\)</span> reject <span class="math inline">\(H_{(i)}\)</span> and go to step <span class="math inline">\(i + 1\)</span>. Otherwise, accept <span class="math inline">\(H_{(i)}, H_{(i + 1)}, \ldots, H_{(n)}\)</span> and stop.</p>
<p>……</p>
<p>Step <span class="math inline">\(n\)</span>: If <span class="math inline">\(p_{(n)} \leq \alpha\)</span>, reject <span class="math inline">\(H_{(n)}\)</span>. Otherwise, accept <span class="math inline">\(H_{(n)}\)</span>.</p>
<p>Hence the procedure starts with the most extreme (smallest) <span class="math inline">\(p\)</span>-value and stops the first time <span class="math inline">\(p_{(i)}\)</span> exceeds the critical value <span class="math inline">\(\alpha_i = \alpha/(n − i + 1)\)</span>.</p>
<p><strong>It can be shown that Holm’s procedure conservatively controls the FWER in the strong sense, under arbitrary correlation among <span class="math inline">\(p\)</span>-values.</strong></p>
</div>
<div id="hochbergs-step-up-procedure-start-from-the-largest-p-value" class="section level2">
<h2>Hochberg’s step-up procedure: start from the largest <span class="math inline">\(p\)</span>-value</h2>
<p><em>It can be shown that Hochberg’s procedure conservatively controls the FWER in the strong sense, when <span class="math inline">\(p\)</span>-values are independent.</em></p>
<p>First, order the <span class="math inline">\(p\)</span>-values</p>
<p><span class="math display">\[
p_{(1)} \leq p_{(2)} \leq \cdots \leq p_{(n)}
\]</span></p>
<p>and let <span class="math inline">\(H_{(1)}, H_{(2)}, \ldots, H_{(n)}\)</span> be the corresponding hypotheses. Then examine the <span class="math inline">\(p\)</span>-values in order.</p>
<p>Step 1: If <span class="math inline">\(p_{(n)} \leq \alpha\)</span> reject <span class="math inline">\(H_{(1)}, \ldots, H_{(n)}\)</span> and stop. Otherwise, accept <span class="math inline">\(H_{(n)}\)</span> and go to step 2.</p>
<p>……</p>
<p>Step <span class="math inline">\(i\)</span>: If <span class="math inline">\(p_{(n - i + 1)} \leq \alpha / i\)</span> reject <span class="math inline">\(H_{(1)}, \ldots, H_{(n - i + 1)}\)</span> and stop. Otherwise, accept <span class="math inline">\(H_{(n - i + 1)}\)</span> and go to step <span class="math inline">\(i + 1\)</span>.</p>
<p>……</p>
<p>Step <span class="math inline">\(n\)</span>: If <span class="math inline">\(p_{(1)} \leq \alpha / n\)</span>, reject <span class="math inline">\(H_{(1)}\)</span>. Otherwise, accept <span class="math inline">\(H_{(1)}\)</span>.</p>
<p>Hence the procedure starts with the least extreme (largest) <span class="math inline">\(p\)</span>-value and stops the first time <span class="math inline">\(p_{(i)}\)</span> falls below the critical value <span class="math inline">\(\alpha_i = \alpha/(n − i + 1)\)</span>.</p>
<p><strong>It can be shown that Hochberg’s procedure conservatively controls the FWER in the strong sense, when <span class="math inline">\(p\)</span>-values are independent.</strong></p>
<p><strong><a href="https://projecteuclid.org/euclid.aos/1028144846">Sarkar 1998</a> also pointed out that Hochberg’s procedure can control the FWER strongly under certain dependency among the test statistics, such as a multivariate normal with a common marginal distribution and positive correlations.</strong></p>
<p><strong>Holm’s procedure is based on Bonferroni correction, whereas Hochberg’s on Sime’s inequality. Both use exactly the same thresholds, comparing <span class="math inline">\(p_{(i)}\)</span> with <span class="math inline">\(\alpha/(n − i + 1)\)</span>, yet Holm’s starts from the smallest <span class="math inline">\(p\)</span>-value, and Hochberg’s from the largest. Hochberg’s is thus strictly more powerful than Holm’s.</strong></p>
</div>
<div id="result" class="section level2">
<h2>Result</h2>
<p>Now we apply the two procedures to the simulated, correlated null data.</p>
<pre class="r"><code>p1 = read.table(&quot;../output/p_null_liver.txt&quot;)
p2 = read.table(&quot;../output/p_null_liver_777.txt&quot;)
p = rbind(p1, p2)
m = nrow(p)
holm = hochberg = matrix(nrow = m, ncol = ncol(p))

for(i in 1:m){
  holm[i, ] = p.adjust(p[i, ], method = &quot;holm&quot;) # p-values adjusted by Holm (1979)
  hochberg[i, ] = p.adjust(p[i, ], method = &quot;hochberg&quot;) # p_values adjusted by Hochberg (1988)
}</code></pre>
<pre class="r"><code>## calculate empirical FWER at 100 nominal FWER&#39;s

alpha = seq(0, 0.15, length = 100)
fwer_holm = fwer_hochberg = c()
for (i in 1:length(alpha)) {
  fwer_holm[i] = mean(apply(holm, 1, function(x) {min(x) &lt;= alpha[i]}))
  fwer_hochberg[i] = mean(apply(hochberg, 1, function(x) {min(x) &lt;= alpha[i]}))
}

fwer_holm_se = sqrt(fwer_holm * (1 - fwer_holm) / m)
fwer_hochberg_se = sqrt(fwer_hochberg * (1 - fwer_hochberg) / m)</code></pre>
<p>Here at each nominal FWER from <span class="math inline">\(0\)</span> to <span class="math inline">\(0.15\)</span>, we plot the empirical FWER, calculated from <span class="math inline">\(m = 2000\)</span> independent simulation trials. Dotted lines indicate one standard error computed from binomial model <span class="math inline">\(= \sqrt{\hat{\text{FWER}}(1 - \hat{\text{FWER}}) / m}\)</span>.</p>
<pre class="r"><code>plot(alpha, fwer_holm, pch = 1, xlab = &quot;nominal FWER&quot;, ylab = &quot;empirical FWER&quot;, xlim = c(0, max(alpha)), ylim = c(0, max(alpha)), cex = 0.75)
points(alpha, fwer_hochberg, col = &quot;blue&quot;, pch = 19, cex = 0.25)
lines(alpha, fwer_holm - fwer_holm_se, lty = 3)
lines(alpha, fwer_holm + fwer_holm_se, lty = 3)
lines(alpha, fwer_hochberg + fwer_hochberg_se, lty = 3, col = &quot;blue&quot;)
lines(alpha, fwer_hochberg - fwer_hochberg_se, lty = 3, col = &quot;blue&quot;)

abline(0, 1, lty = 3, col = &quot;red&quot;)
legend(&quot;topleft&quot;, c(&quot;Holm&quot;, &quot;Hochberg&quot;), col = c(1, &quot;blue&quot;), pch = c(1, 19))</code></pre>
<p><img src="figure/StepDown.Rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-3-1.png:</em></summary>
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Author
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<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/StepDown.Rmd/unnamed-chunk-3-1.png" target="_blank">0f36d99</a>
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<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/2696cb2bfc4ec049384ec72d180c5c79e2e08066/docs/figure/StepDown.Rmd/unnamed-chunk-3-1.png" target="_blank">2696cb2</a>
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<td style="text-align:left;">
LSun
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<td style="text-align:left;">
2017-02-10
</td>
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</table>
</details>
<p>The results from Holm’s step-down and Hochberg’s step-up are almost the same for this simulated data set. They both give almost the same discoveries, although in theory Hochberg’s should be strictly more powerful than Holm’s. <em>The agreement of both procedures may indicate that test statistics are indeed inflated for moderate observations but not extreme observations.</em></p>
</div>
<div id="session-information" class="section level2">
<h2>Session Information</h2>
</div>
<div id="session-information-1" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.4

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

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     

loaded via a namespace (and not attached):
 [1] workflowr_1.0.1   Rcpp_0.12.16      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.21.0      magrittr_1.5      evaluate_0.10.1  
[10] stringi_1.1.6     whisker_0.3-2     R.oo_1.21.0      
[13] R.utils_2.6.0     rmarkdown_1.9     tools_3.4.3      
[16] stringr_1.3.0     yaml_2.1.18       compiler_3.4.3   
[19] htmltools_0.3.6   knitr_1.20       </code></pre>
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