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<h1 class="title toc-ignore">Histogram of correlated <span class="math inline">\(z\)</span> scores, Part 3</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-03-10</em></h4>
</div>
<p><strong>Last updated:</strong> 2017-03-10</p>
<p><strong>Code version:</strong> 894c395</p>
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>From previous simulation on <a href="FDR_Null.html">FDR on correlated null</a> we get a sense that Benjamini-Hochberg is surprisingly robust under correlation. At the same time, there are cases when BH erroneously produces a large number of false discoveries. Therefore, we take a close look at those data sets BH fails on.</p>
</div>
<div id="simulation" class="section level2">
<h2>Simulation</h2>
<pre class="r"><code>z = read.table("../output/z_null_liver_777.txt")
p = read.table("../output/p_null_liver_777.txt")
n = dim(z)</code></pre>
<pre class="r"><code>pihat0 = read.table("../output/pihat0_z_null_liver_777.txt")
pihat0 = as.numeric(t(pihat0))</code></pre>
<pre class="r"><code>fd.bh = c()
for (i in 1 : n[1]) {
p_BH = p.adjust(p[i, ], method = "BH")
fd.bh[i] = sum(p_BH <= 0.05)
}</code></pre>
</div>
<div id="result" class="section level2">
<h2>Result</h2>
<p>We take a look that all data sets where BH erroneously produces at least one false discovery (green), compared with <span class="math inline">\(\hat\pi_0\)</span> produced by <code>ash</code> on these same data sets.</p>
<pre class="r"><code>fdn = sum(fd.bh >= 1)
I = order(fd.bh, decreasing = TRUE)[1:fdn]
x = seq(-10, 10, 0.01)
y = dnorm(x)
k = 1
m = n[2]
for (j in I) {
cat("N0.", k, ": Data Set", j, "; Number of False Discoveries:", fd.bh[j], "; pihat0 =", pihat0[j])
hist(as.numeric(z[j, ]), xlab = "z scores", freq = FALSE, ylim = c(0, 0.45), nclass = 100, main = "10000 z scores, 100 bins")
lines(x, y, col = "red")
qqnorm(as.numeric(z[j, ]), main = "Normal Q-Q plot for z scores")
abline(0, 1)
plot(sort(as.numeric(p[j, ])), xlab = "Order", ylab = "Ordered p value", main = "All p values")
abline(0, 1 / m, col = "blue")
abline(0, 0.05 / m, col = "red")
legend("top", lty = 1, col = c("blue", "red"), c("Uniform", "BH, FDR = 0.05"))
pj = sort(as.numeric(p[j, ]))
plot(pj[1:max(100, fd.bh[j])], xlab = "Order", ylab = "Ordered p value", main = "Zoom-in to 100 smallest p values", ylim = c(0, pj[max(100, fd.bh[j])]))
abline(0, 1 / m, col = "blue")
points(pj[1:fd.bh[j]], col = "green", pch = 19)
abline(0, 0.05 / m, col = "red")
legend("top", lty = 1, col = c("blue", "red"), c("Uniform", "BH, FDR = 0.05"))
plot(pj[pj <= 0.01], xlab = "Order", ylab = "Ordered p value", main = "Zoom-in to p values <= 0.01", ylim = c(0, 0.01))
abline(0, 1 / m, col = "blue")
points(pj[1:fd.bh[j]], col = "green", pch = 19)
abline(0, 0.05 / m, col = "red")
legend("top", lty = 1, col = c("blue", "red"), c("Uniform", "BH, FDR = 0.05"))
plot(pj[pj <= 0.05], xlab = "Order", ylab = "Ordered p value", main = "Zoom-in to p values <= 0.05", ylim = c(0, 0.05))
abline(0, 1 / m, col = "blue")
points(pj[1:fd.bh[j]], col = "green", pch = 19)
abline(0, 0.05 / m, col = "red")
legend("top", lty = 1, col = c("blue", "red"), c("Uniform", "BH, FDR = 0.05"))
k = k + 1
}</code></pre>
<pre><code>N0. 1 : Data Set 355 ; Number of False Discoveries: 639 ; pihat0 = 0.04750946</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-6.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 2 : Data Set 327 ; Number of False Discoveries: 489 ; pihat0 = 0.1522796</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-12.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 3 : Data Set 23 ; Number of False Discoveries: 408 ; pihat0 = 0.03139009</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-17.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-18.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 4 : Data Set 122 ; Number of False Discoveries: 331 ; pihat0 = 0.04301549</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-20.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-21.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-22.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-23.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-24.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 5 : Data Set 783 ; Number of False Discoveries: 170 ; pihat0 = 0.06208376</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-25.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-26.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-27.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-28.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-29.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-30.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 6 : Data Set 749 ; Number of False Discoveries: 114 ; pihat0 = 0.02583276</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-31.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-32.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-33.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-34.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-35.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-36.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 7 : Data Set 724 ; Number of False Discoveries: 79 ; pihat0 = 0.01606004</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-37.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-38.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-39.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-40.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-41.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-42.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 8 : Data Set 56 ; Number of False Discoveries: 35 ; pihat0 = 0.04829662</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-43.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-44.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-45.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-46.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-47.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-48.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 9 : Data Set 840 ; Number of False Discoveries: 28 ; pihat0 = 0.02316832</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-49.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-50.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-51.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-52.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-53.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-54.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 10 : Data Set 858 ; Number of False Discoveries: 16 ; pihat0 = 0.05110069</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-55.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-56.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-57.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-58.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-59.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-60.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 11 : Data Set 771 ; Number of False Discoveries: 12 ; pihat0 = 0.04316169</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-61.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-62.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-63.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-64.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-65.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-66.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 12 : Data Set 389 ; Number of False Discoveries: 11 ; pihat0 = 0.03156173</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-67.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-68.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-69.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-70.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-71.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-72.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 13 : Data Set 485 ; Number of False Discoveries: 9 ; pihat0 = 0.04794909</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-73.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-74.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-75.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-76.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-77.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-78.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 14 : Data Set 77 ; Number of False Discoveries: 7 ; pihat0 = 0.05151585</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-79.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-80.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-81.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-82.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-83.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-84.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 15 : Data Set 503 ; Number of False Discoveries: 7 ; pihat0 = 0.1545398</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-85.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-86.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-87.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-88.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-89.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-90.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 16 : Data Set 984 ; Number of False Discoveries: 7 ; pihat0 = 0.04567195</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-91.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-92.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-93.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-94.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-95.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-96.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 17 : Data Set 360 ; Number of False Discoveries: 6 ; pihat0 = 0.0308234</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-97.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-98.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-99.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-100.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-101.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-102.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 18 : Data Set 522 ; Number of False Discoveries: 4 ; pihat0 = 0.02083846</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-103.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-104.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-105.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-106.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-107.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-108.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 19 : Data Set 51 ; Number of False Discoveries: 3 ; pihat0 = 0.2435437</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-109.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-110.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-111.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-112.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-113.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-114.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 20 : Data Set 316 ; Number of False Discoveries: 3 ; pihat0 = 0.2961358</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-115.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-116.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-117.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-118.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-119.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-120.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 21 : Data Set 663 ; Number of False Discoveries: 3 ; pihat0 = 0.2818026</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-121.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-122.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-123.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-124.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-125.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-126.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 22 : Data Set 274 ; Number of False Discoveries: 2 ; pihat0 = 0.08580661</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-127.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-128.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-129.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-130.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-131.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-132.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 23 : Data Set 901 ; Number of False Discoveries: 2 ; pihat0 = 0.9997959</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-133.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-134.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-135.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-136.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-137.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-138.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 24 : Data Set 912 ; Number of False Discoveries: 2 ; pihat0 = 0.106048</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-139.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-140.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-141.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-142.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-143.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-144.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 25 : Data Set 22 ; Number of False Discoveries: 1 ; pihat0 = 0.03271534</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-145.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-146.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-147.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-148.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-149.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-150.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 26 : Data Set 31 ; Number of False Discoveries: 1 ; pihat0 = 0.1874129</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-151.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-152.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-153.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-154.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-155.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-156.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 27 : Data Set 187 ; Number of False Discoveries: 1 ; pihat0 = 0.7071115</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-157.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-158.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-159.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-160.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-161.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-162.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 28 : Data Set 248 ; Number of False Discoveries: 1 ; pihat0 = 0.5095841</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-163.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-164.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-165.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-166.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-167.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-168.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 29 : Data Set 269 ; Number of False Discoveries: 1 ; pihat0 = 0.02787007</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-169.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-170.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-171.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-172.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-173.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-174.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 30 : Data Set 285 ; Number of False Discoveries: 1 ; pihat0 = 0.4624418</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-175.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-176.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-177.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-178.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-179.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-180.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 31 : Data Set 403 ; Number of False Discoveries: 1 ; pihat0 = 0.0531628</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-181.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-182.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-183.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-184.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-185.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-186.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 32 : Data Set 483 ; Number of False Discoveries: 1 ; pihat0 = 0.9998824</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-187.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-188.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-189.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-190.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-191.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-192.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 33 : Data Set 501 ; Number of False Discoveries: 1 ; pihat0 = 0.1090034</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-193.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-194.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-195.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-196.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-197.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-198.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 34 : Data Set 530 ; Number of False Discoveries: 1 ; pihat0 = 0.4743284</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-199.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-200.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-201.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-202.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-203.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-204.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 35 : Data Set 574 ; Number of False Discoveries: 1 ; pihat0 = 0.5260185</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-205.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-206.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-207.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-208.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-209.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-210.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 36 : Data Set 575 ; Number of False Discoveries: 1 ; pihat0 = 0.0946117</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-211.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-212.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-213.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-214.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-215.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-216.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 37 : Data Set 643 ; Number of False Discoveries: 1 ; pihat0 = 0.3321723</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-217.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-218.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-219.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-220.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-221.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-222.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 38 : Data Set 769 ; Number of False Discoveries: 1 ; pihat0 = 0.1187511</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-223.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-224.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-225.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-226.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-227.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-228.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 39 : Data Set 778 ; Number of False Discoveries: 1 ; pihat0 = 0.07619716</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-229.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-230.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-231.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-232.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-233.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-234.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 40 : Data Set 817 ; Number of False Discoveries: 1 ; pihat0 = 0.999989</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-235.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-236.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-237.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-238.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-239.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-240.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 41 : Data Set 837 ; Number of False Discoveries: 1 ; pihat0 = 0.8625374</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-241.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-242.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-243.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-244.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-245.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-246.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 42 : Data Set 897 ; Number of False Discoveries: 1 ; pihat0 = 0.8465903</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-247.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-248.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-249.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-250.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-251.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-252.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 43 : Data Set 923 ; Number of False Discoveries: 1 ; pihat0 = 0.03239883</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-253.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-254.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-255.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-256.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-257.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-258.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 44 : Data Set 955 ; Number of False Discoveries: 1 ; pihat0 = 0.999845</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-259.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-260.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-261.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-262.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-263.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-264.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 45 : Data Set 971 ; Number of False Discoveries: 1 ; pihat0 = 0.2387909</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-265.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-266.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-267.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-268.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-269.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-270.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>N0. 46 : Data Set 997 ; Number of False Discoveries: 1 ; pihat0 = 0.09560788</code></pre>
<p><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-271.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-272.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-273.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-274.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-275.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z_3.Rmd/unnamed-chunk-4-276.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_0.6.9 workflowr_0.3.0 evaluate_0.10 </code></pre>
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