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<h1 class="title toc-ignore">Marginal Distribution of <span class="math inline">\(z\)</span> Scores: Alternative</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-05-08</em></h4>
</div>
<!-- The file analysis/chunks.R contains chunks that define default settings
shared across the workflowr files. -->
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<p><strong>Last updated:</strong> 2017-05-09</p>
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package git2r is installed -->
<p><strong>Code version:</strong> 615c317</p>
<!-- Add your analysis here -->
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p><a href="marginal_z.html">We’ve seen</a> that when generated from the global null, that is, when the cases and controls have no difference, the <span class="math inline">\(z\)</span> scores’ behavior are what one would expect if simulated from correlated marginally <span class="math inline">\(N\left(0, 1\right)\)</span> random variables.</p>
<p>But what if the <span class="math inline">\(z\)</span> scores are not simulated from the global null? Are these <span class="math inline">\(z\)</span> scores going to behave significantly different from correlated marginally <span class="math inline">\(N\left(0, 1\right)\)</span> random samples? Let’s take a look at the real data containing true effects.</p>
<pre class="r"><code>library(limma)
library(edgeR)
library(qvalue)
library(ashr)
#extract top g genes from G by n matrix X of expression
top_genes_index = function (g, X)
{return(order(rowSums(X), decreasing = TRUE)[1 : g])
}
lcpm = function (r) {
R = colSums(r)
t(log2(((t(r) + 0.5) / (R + 1)) * 10^6))
}
# transform counts to z scores
# these z scores are marginally N(0, 1) under null
counts_to_z = function (counts, condition) {
design = model.matrix(~condition)
dgecounts = calcNormFactors(DGEList(counts = counts, group = condition))
v = voom(dgecounts, design, plot = FALSE)
lim = lmFit(v)
r.ebayes = eBayes(lim)
p = r.ebayes$p.value[, 2]
t = r.ebayes$t[, 2]
z = sign(t) * qnorm(1 - p/2)
return (z)
}</code></pre>
</div>
<div id="generating-non-null-z-scores-from-real-data" class="section level2">
<h2>Generating non-null <span class="math inline">\(z\)</span> scores from real data</h2>
<p>For convenience and without loss of generality, we are using the liver tissue as an anchor in the simulation, and always choose top expressed genes in livers.</p>
<pre class="r"><code>r.liver = read.csv("../data/liver.csv")
r.liver = r.liver[, -(1 : 2)] # remove gene name and description
Y = lcpm(r.liver)
G = 1e4
subset = top_genes_index(G, Y)
r.liver = r.liver[subset, ]</code></pre>
<div id="liver-vs-heart" class="section level3">
<h3>Liver vs Heart</h3>
<pre class="r"><code>tissue = "heart"
r = read.csv(paste0("../data/", tissue, ".csv"))
r = r[, -(1 : 2)] # remove gene name and description
## choose top expressed genes in liver
r = r[subset, ]</code></pre>
<pre class="r"><code>set.seed(777)
m = 1e3
n = 5
z.list = list()
condition = c(rep(0, n), rep(1, n))
for (i in 1 : m) {
counts = cbind(r.liver[, sample(1 : ncol(r.liver), n)],
r[, sample(1 : ncol(r), n)])
z.list[[i]] = counts_to_z(counts, condition)
}</code></pre>
<pre class="r"><code>z.mat = matrix(unlist(z.list), nrow = m, byrow = TRUE)
saveRDS(z.mat, "../output/z_5liver_5heart_777.rds")</code></pre>
</div>
<div id="liver-vs-muscle" class="section level3">
<h3>Liver vs Muscle</h3>
<pre class="r"><code>tissue = "muscle"
r = read.csv(paste0("../data/", tissue, ".csv"))
r = r[, -(1 : 2)] # remove gene name and description
## choose top expressed genes in liver
r = r[subset, ]</code></pre>
<pre class="r"><code>set.seed(777)
m = 1e3
n = 5
z.list = list()
condition = c(rep(0, n), rep(1, n))
for (i in 1 : m) {
counts = cbind(r.liver[, sample(1 : ncol(r.liver), n)],
r[, sample(1 : ncol(r), n)])
z.list[[i]] = counts_to_z(counts, condition)
}</code></pre>
<pre class="r"><code>z.mat = matrix(unlist(z.list), nrow = m, byrow = TRUE)
saveRDS(z.mat, "../output/z_5liver_5muscle_777.rds")</code></pre>
</div>
</div>
<div id="marginal-distributions-of-correlated-non-null-z-scores" class="section level2">
<h2>Marginal distributions of correlated non-null <span class="math inline">\(z\)</span> scores</h2>
<p>Now following the simulations to explore <a href="marginal_z.html">the marginal distributions of the correlated null <span class="math inline">\(z\)</span> scores</a>, we are taking a look at the marginal distributions of the correlated non-null <span class="math inline">\(z\)</span> scores. The main focus is still the number of tail observations.</p>
</div>
<div id="liver-vs-heart-1" class="section level2">
<h2>Liver vs Heart</h2>
<pre class="r"><code>z.heart = readRDS("../output/z_5liver_5heart_777.rds")
n = ncol(z.heart)
m = nrow(z.heart)</code></pre>
<div id="row-wise" class="section level3">
<h3>Row-wise</h3>
<p><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-17.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-18.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-20.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20heart-21.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="column-wise" class="section level3">
<h3>Column-wise</h3>
<p><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20heart-17.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="liver-vs-muscle-1" class="section level2">
<h2>Liver vs Muscle</h2>
<pre class="r"><code>z.muscle = readRDS("../output/z_5liver_5muscle_777.rds")
n = ncol(z.muscle)
m = nrow(z.muscle)</code></pre>
<div id="row-wise-1" class="section level3">
<h3>Row-wise</h3>
<p><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-17.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-18.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-20.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20row%20wise%20tail%20liver%20v%20muscle-21.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="column-wise-1" class="section level3">
<h3>Column-wise</h3>
<p><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z_alternative.rmd/plot%20column%20wise%20tail%20liver%20v%20muscle-17.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="conclusion" class="section level2">
<h2>Conclusion</h2>
<p>As expected, the empirical distribution and the indicated marginal distribution of the correlated non-null <span class="math inline">\(z\)</span> scores are starkly different from <a href="marginal_z.html">those of the correlated null ones</a>.</p>
<p>Particular interesting are the column-wise plots. The distribution of the number of tail observations is not only not close to normal, not close to what would be expected under correlated marginally <span class="math inline">\(N\left(0, 1\right)\)</span>, but also not even unimodal, not even peaked at <span class="math inline">\(m\alpha\)</span> when <span class="math inline">\(\alpha \leq0.5\)</span> or <span class="math inline">\(m\left(1-\alpha\right)\)</span> when <span class="math inline">\(\alpha\geq0.5\)</span>. It suggests that there are indeed plenty of true signals, which make the marginal distribution of the <span class="math inline">\(z\)</span> scores for a certain gene often not centered at <span class="math inline">\(0\)</span>.</p>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<!-- Insert the session information into the document -->
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.3.3 (2017-03-06)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: macOS Sierra 10.12.4
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] ashr_2.1-10 qvalue_2.4.2 edgeR_3.14.0 limma_3.28.5
loaded via a namespace (and not attached):
[1] Rcpp_0.12.10 knitr_1.15.1 magrittr_1.5
[4] MASS_7.3-45 splines_3.3.3 pscl_1.4.9
[7] doParallel_1.0.10 munsell_0.4.3 lattice_0.20-34
[10] SQUAREM_2016.10-1 colorspace_1.2-6 foreach_1.4.3
[13] stringr_1.2.0 plyr_1.8.3 tools_3.3.3
[16] parallel_3.3.3 grid_3.3.3 gtable_0.2.0
[19] git2r_0.18.0 iterators_1.0.8 htmltools_0.3.5
[22] yaml_2.1.14 rprojroot_1.2 digest_0.6.11
[25] reshape2_1.4.1 ggplot2_2.1.0 codetools_0.2-15
[28] evaluate_0.10 rmarkdown_1.3 stringi_1.1.2
[31] scales_0.4.0 backports_1.0.5 truncnorm_1.0-7 </code></pre>
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