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<title>Ordered p values vs critical values</title>
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<h1 class="title toc-ignore">Ordered <span class="math inline">\(p\)</span> values vs critical values</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-02-22</em></h4>
</div>
<p><strong>Last updated:</strong> 2017-02-22</p>
<p><strong>Code version:</strong> 242b122</p>
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Various FWER and FDR controlling procedures are working well for the simulated correlated global null data, even though some of them are only guaranteed to work under independence or certain types of correlation. Since all of these methods are essentially comparing ordered <span class="math inline">\(p\)</span> values with some critical values, we plot these ordered <span class="math inline">\(p\)</span> values and corresponding critical values for <span class="math inline">\(10\)</span> random data sets to take a closer look.</p>
</div>
<div id="result" class="section level2">
<h2>Result</h2>
<pre class="r"><code>p = read.table("../output/p_null_liver_777.txt")</code></pre>
<p>Here we plot <span class="math inline">\(10\)</span> data sets. For each data set the left plots the ordered <span class="math inline">\(p\)</span> values on the whole, and the right plot takes a look at the most extreme ones, as well as the critical values in Benjamini-Hochberg’s FDR (green) and Hochberg’s FWER (blue).</p>
<p>We can see that with the number of <span class="math inline">\(p\)</span>-values very large, even though the ordered <span class="math inline">\(p\)</span> values may be generally lower than their expected values under independence, the most extreme ones are seldom falling below the critical values used by both procedures, simply because these values are too small. Thus, even though we have unignorable correlation, both procedures could still be able to control FDR / FWER satisfactorily.</p>
<pre class="r"><code>set.seed(777)
m = dim(p)[1]
n = dim(p)[2]
alpha = 0.05
sel = sample(m, 10)
for (i in sel) {
par(mfrow = c(1, 2))
plot((1:n), sort(p[i, ]), ylim = c(0, 1), cex = 0.1, xlab = "order", ylab = "ordered p value")
abline(0, 1 / n, col = "red")
plot((1:n), (1:n) / n * alpha, type = "n", ylim = c(0, 0.01), xlab = "order", ylab = "ordered p value")
lines((1:n), (1:n) / n * alpha, col = "blue")
lines((1:n), alpha / (n - (1:n) + 1), col = "green")
abline(0, 1 / n, col = "red")
points((1:n), sort(p[i, ]), cex = 0.5)
}</code></pre>
<p><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/cutoff_null.Rmd/unnamed-chunk-2-10.png" width="672" style="display: block; margin: auto;" /></p>
</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
loaded via a namespace (and not attached):
[1] backports_1.0.5 magrittr_1.5 rprojroot_1.2 tools_3.3.2
[5] htmltools_0.3.5 yaml_2.1.14 Rcpp_0.12.9 stringi_1.1.2
[9] rmarkdown_1.3 knitr_1.15.1 git2r_0.18.0 stringr_1.1.0
[13] digest_0.6.11 workflowr_0.3.0 evaluate_0.10 </code></pre>
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