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<title>The Distribution of W_j</title>

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<h1 class="title toc-ignore">The Distribution of <span class="math inline">\(W_j\)</span></h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2018-04-13</em></h4>

</div>


<!-- The file analysis/chunks.R contains chunks that define default settings
shared across the workflowr files. -->
<!-- Update knitr chunk options -->
<!-- Insert the date the file was last updated -->
<p><strong>Last updated:</strong> 2018-05-06</p>
<!-- Insert the code version (Git commit SHA1) if Git repository exists and R
 package git2r is installed -->
<p><strong>Code version:</strong> 0b0a394</p>
<!-- Add your analysis here -->
<pre class="r"><code>source(&quot;../code/gdash_lik.R&quot;)
source(&quot;../code/gdfit.R&quot;)
source(&quot;../code/count_to_summary.R&quot;)
library(limma)
library(edgeR)
library(ashr)
library(plyr)
library(ggplot2)
library(reshape2)</code></pre>
<div id="introduction" class="section level2">
<h2>Introduction</h2>
</div>
<div id="simulated-data" class="section level2">
<h2>Simulated Data</h2>
<pre class="r"><code>set.seed(777)
d &lt;- 10
n &lt;- 1e4
B &lt;- matrix(rnorm(n * d), n, d)
Sigma &lt;- B %*% t(B) + diag(n)
sigma &lt;- diag(Sigma)
Rho &lt;- cov2cor(Sigma)
rhobar &lt;- c()
for (l in 1 : 10) {
  rhobar[l] &lt;- (sum(Rho^l) - n) / (n * (n - 1))
}</code></pre>
<pre class="r"><code>par(mar = c(5.1, 4.1, 1, 2.1))
hist(Rho[lower.tri(Rho)], xlab = expression(rho[ij]), main = &quot;&quot;)</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>set.seed(20)
z &lt;- rnorm(d)
Z &lt;- B %*% z + rnorm(n)
Z &lt;- Z / sqrt(sigma)
cat(&quot;sd(Z) =&quot;, sd(Z))</code></pre>
<pre><code>sd(Z) = 1.262205</code></pre>
<pre class="r"><code>hist(Z, breaks = 20, prob = TRUE, ylim = c(0, dnorm(0)))
lines(seq(-5, 5, by = 0.1), dnorm(seq(-5, 5, by = 0.1)), col = &quot;blue&quot;)</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>p &lt;- pnorm(-abs(Z)) * 2

par(mfcol = c(2, 2))
par(mar = c(5.1, 4.1, 3, 2.1))
hist(p, breaks = 100, main = &quot;Correlated&quot;, xlab = &quot;p-value&quot;)

par(mar = c(5.1, 4.1, 1, 2.1))
plot(-log(p), ylim = range(-log(p), -log(pnorm(-sqrt(2 * log(n))) * 2), -log(0.05 / n)))
abline(h = -log(pnorm(-sqrt(2 * log(n))) * 2), col = &quot;maroon&quot;)
abline(h = -log(0.05 / n), col = &quot;red&quot;)
abline(h = -log(0.001), col = &quot;green&quot;)
abline(h = -log(0.05), col = &quot;blue&quot;)

Z &lt;- rnorm(n)
p &lt;- pnorm(-abs(Z)) * 2
par(mar = c(5.1, 4.1, 3, 2.1))
hist(p, breaks = 100, main = &quot;Independent&quot;, xlab = &quot;p-value&quot;)

par(mar = c(5.1, 4.1, 1, 2.1))
plot(-log(p), ylim = range(-log(p), -log(pnorm(-sqrt(2 * log(n))) * 2), -log(0.05 / n)))
abline(h = -log(pnorm(-sqrt(2 * log(n))) * 2), col = &quot;maroon&quot;)
abline(h = -log(0.05 / n), col = &quot;red&quot;)
abline(h = -log(0.001), col = &quot;green&quot;)
abline(h = -log(0.05), col = &quot;blue&quot;)</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>set.seed(777)
nsim &lt;- 1e4
Z.list &lt;- W &lt;- list()
for (i in 1 : nsim) {
z &lt;- rnorm(d)
Z &lt;- B %*% z + rnorm(n)
Z &lt;- Z / sqrt(sigma)
Z.list[[i]] &lt;- Z
Z.GD &lt;- gdfit.mom(Z, 100)
W[[i]] &lt;- Z.GD$w
}
Z.sim &lt;- Z.list
W.sim &lt;- W</code></pre>
</div>
<div id="real-data-from-gtex" class="section level2">
<h2>Real Data from GTEx</h2>
<pre class="r"><code>r &lt;- readRDS(&quot;../data/liver.rds&quot;)</code></pre>
<pre class="r"><code>top_genes_index = function (g, X) {
  return(order(rowSums(X), decreasing = TRUE)[1 : g])
}
lcpm = function (r) {
  R = colSums(r)
  t(log2(((t(r) + 0.5) / (R + 1)) * 10^6))
}</code></pre>
<pre class="r"><code>nsamp &lt;- 5
ngene &lt;- 1e4</code></pre>
<pre class="r"><code>Y = lcpm(r)
subset = top_genes_index(ngene, Y)
r = r[subset,]</code></pre>
<pre class="r"><code>set.seed(7)
nsim &lt;- 1e4
Z.list &lt;- W &lt;- list()
for (i in 1 : nsim) {
  ## generate data
  counts &lt;- r[, sample(ncol(r), 2 * nsamp)]
  design &lt;- model.matrix(~c(rep(0, nsamp), rep(1, nsamp)))
  summary &lt;- count_to_summary(counts, design)
  Z &lt;- summary$z
  Z.list[[i]] &lt;- Z
  Z.GD &lt;- gdfit.mom(Z, 100)
  W[[i]] &lt;- Z.GD$w
}
Z.gtex &lt;- Z.list
W.gtex &lt;- W</code></pre>
<pre class="r"><code>quantile.vec1 &lt;- exp(seq(-21, -5, by = 0.01))
quantile.vec2 &lt;- seq(0.007, 0.993, by = 0.001)
quantile.vec3 &lt;- exp(seq(-5, -21, by = -0.01))
emp.cdf.Z1 &lt;- sapply(quantile.vec1, function(x) {sapply(Z.gtex, function(y) mean(y &lt;= qnorm(x)))})
emp.cdf.Z2 &lt;- sapply(quantile.vec2, function(x) {sapply(Z.gtex, function(y) mean(y &lt;= qnorm(x)))})
emp.cdf.Z3 &lt;- sapply(quantile.vec3, function(x) {sapply(Z.gtex, function(y) mean(y &lt;= -qnorm(x)))})
emp.cdf.Z4 &lt;- sapply(quantile.vec3, function(x) {sapply(Z.gtex, function(y) mean(y &gt; -qnorm(x)))})</code></pre>
<pre class="r"><code>ecdf.avg1 &lt;- colMeans(emp.cdf.Z1)
ecdf.avg2 &lt;- colMeans(emp.cdf.Z2)
ecdf.avg3 &lt;- colMeans(emp.cdf.Z3)
ecdf.avg4 &lt;- colMeans(emp.cdf.Z4)
ecdf.avg &lt;- c(ecdf.avg1, ecdf.avg2, ecdf.avg3)
ecdf.tail.avg.conf.int1 &lt;- apply(emp.cdf.Z1, 2, function(x) {t.test(x)$conf.int})
ecdf.tail.avg.conf.int4 &lt;- apply(emp.cdf.Z4, 2, function(x) {t.test(x)$conf.int})</code></pre>
<pre class="r"><code>plot(c(qnorm(quantile.vec1), qnorm(quantile.vec2), -qnorm(quantile.vec3)), ecdf.avg, type = &quot;l&quot;, col = &quot;red&quot;, xlab = &quot;z&quot;, ylab = &quot;Cumulative Distribution Function (CDF)&quot;)
lines(c(qnorm(quantile.vec1), qnorm(quantile.vec2), -qnorm(quantile.vec3)), c(quantile.vec1, quantile.vec2, pnorm(-qnorm(quantile.vec3))), lty = 2)
legend(&quot;bottomright&quot;, lty = c(1, 2), col = c(1, 2), legend = c(expression(bar(&quot;F&quot;)(z)), expression(Phi(z))))</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-13-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(qnorm(quantile.vec1), log(ecdf.avg1), type = &quot;l&quot;,
     ylim = range(log(quantile.vec1), log(ecdf.avg1)),
     xlab = &quot;z&quot;, ylab = &quot;log (CDF)&quot;)
lines(qnorm(quantile.vec1), log(quantile.vec1), lty = 2, col = &quot;red&quot;)
lines(qnorm(quantile.vec1), log(pnorm(qnorm(quantile.vec1), 0, 1.1)), lty = 2, col = &quot;green&quot;)
lines(qnorm(quantile.vec1), log(pnorm(qnorm(quantile.vec1), 0, 1.05)), lty = 2, col = &quot;blue&quot;)
polygon(x = c(qnorm(quantile.vec1), rev(qnorm(quantile.vec1))),
        y = c(log(ecdf.tail.avg.conf.int1[1, ]), rev(log(ecdf.tail.avg.conf.int1[2, ]))),
        border = NA,
        col = grDevices::adjustcolor(&quot;grey75&quot;, alpha.f = 0.5))</code></pre>
<pre><code>Warning in log(ecdf.tail.avg.conf.int1[1, ]): NaNs produced</code></pre>
<pre class="r"><code>legend(&quot;bottomright&quot;, lty = c(1, 2, 2, 2), col = c(1, 2, 4, 3), legend = c(
  expression(log(bar(&quot;F&quot;)(z))),
  expression(log(Phi(z))),
  expression(log(Phi(z / 1.05))),
  expression(log(Phi(z / 1.1)))
))</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-13-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>plot(-qnorm(quantile.vec3), log(ecdf.avg4), type = &quot;l&quot;,
     ylim = range(log(quantile.vec3), log(ecdf.avg4)),
     xlab = &quot;z&quot;, ylab = &quot;log (1 - CDF)&quot;)
lines(-qnorm(quantile.vec3), log(quantile.vec3), lty = 2, col = &quot;red&quot;)
lines(-qnorm(quantile.vec3), log(pnorm(qnorm(quantile.vec3), 0, 1.1)), lty = 2, col = &quot;green&quot;)
lines(-qnorm(quantile.vec3), log(pnorm(qnorm(quantile.vec3), 0, 1.05)), lty = 2, col = &quot;blue&quot;)
polygon(x = c(-qnorm(quantile.vec3), rev(-qnorm(quantile.vec3))),
        y = c(log(ecdf.tail.avg.conf.int4[1, ]), rev(log(ecdf.tail.avg.conf.int4[2, ]))),
        border = NA,
        col = grDevices::adjustcolor(&quot;grey75&quot;, alpha.f = 0.5))
legend(&quot;bottomleft&quot;, lty = c(1, 2, 2, 2), col = c(1, 2, 4, 3), legend = c(
  expression(log(1 - bar(&quot;F&quot;)(z))),
  expression(log(1 - Phi(z))),
  expression(log(1 - Phi(z / 1.05))),
  expression(log(1 - Phi(z / 1.1)))
))</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-13-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>set.seed(777)
nsamp &lt;- 50
nsim &lt;- 1e3
z &lt;- sebetahat &lt;- list()
for (i in 1 : nsim) {
  ## generate data
  counts &lt;- r[, sample(ncol(r), 2 * nsamp)]
  design &lt;- model.matrix(~c(rep(0, nsamp), rep(1, nsamp)))
  summary &lt;- count_to_summary(counts, design)
  z[[i]] &lt;- summary$z
  sebetahat[[i]] &lt;- summary$sebetahat
}</code></pre>
<pre class="r"><code>sd.vec &lt;- sapply(z, sd)
median.vec &lt;- sapply(z, median)
fd.vec &lt;- sapply(z, function(x) {
  p &lt;- pnorm(-abs(x)) * 2
  sum(p &lt;= 0.005)
})
sel &lt;- c(834, 211, 397, 748)
par(mfrow = c(2, 2))
for (i in seq(sel)) {
  fit &lt;- gdfit(z[[sel[i]]], 10)
  plot.gdfit(z[[sel[i]]], fit$w, fit$L, legend = FALSE)
}</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-15-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>set.seed(6)
par(mfrow = c(2, 3))
par(mar = c(4.5, 4.5, 2, 2))
hist(pnorm(-abs(z[[834]])) * 2, prob = TRUE, xlab = &quot;&quot;, breaks = 100, main = &quot;(a): Histogram of two-sided p-values&quot;)
lines(c(0, 1), c(1, 1), col = &quot;red&quot;)
hist(z[[834]], prob = TRUE, breaks = 100, xlab = &quot;&quot;, xlim = c(-4.5, -2), main = &quot;(b): Left tail of correlated z-scores&quot;)
lines(seq(-6, 6, by = 0.01), dnorm(seq(-6, 6, by = 0.01), 0, sd(z[[834]])), col = &quot;blue&quot;)
lines(seq(-6, 6, by = 0.01), dnorm(seq(-6, 6, by = 0.01)), col = &quot;red&quot;)
hist(z[[834]], prob = TRUE, breaks = 100, xlab = &quot;&quot;, xlim = c(2, 4.5), main = &quot;(c): Right tail of correlated z-scores&quot;)
lines(seq(-6, 6, by = 0.01), dnorm(seq(-6, 6, by = 0.01)), col = &quot;red&quot;)
p &lt;- pnorm(-abs(z[[834]])) * 2
plot(sample(-log(pnorm(-abs(z[[834]])) * 2)), ylim = c(0, 20), ylab = &quot;-log(p)&quot;, main = expression(paste(&quot;(d): Correlated &quot;, N(0, 1))))
abline(h = -log(0.005), col = &quot;red&quot;)
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = &quot;blue&quot;)
abline(h = -log(0.05 / 1e4), col = &quot;green&quot;)
plot(-log(pnorm(-abs(rnorm(1e4))) * 2), ylim = c(0, 20), ylab = &quot;-log(p)&quot;, main = expression(paste(&quot;(e): Independent &quot;, N(0, 1))))
abline(h = -log(0.005), col = &quot;red&quot;)
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = &quot;blue&quot;)
abline(h = -log(0.05 / 1e4), col = &quot;green&quot;)
plot(-log(pnorm(-abs(rnorm(1e4, 0, 1.6))) * 2), ylim = c(0, 20), ylab = &quot;-log(p)&quot;, main = expression(paste(&quot;(f): Independent &quot;, N(0, 1.6^2))))
abline(h = -log(0.005), col = &quot;red&quot;)
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = &quot;blue&quot;)
abline(h = -log(0.05 / 1e4), col = &quot;green&quot;)</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-16-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>p.bh &lt;- p.adjust(p, method = &quot;BH&quot;)
sum(p.bh &lt;= 0.05)</code></pre>
<pre><code>[1] 575</code></pre>
<pre class="r"><code>plot(sort(log(p)), cex = 0.25, pch = 19, ylim = c(-19, 0), xlab = &quot;Order&quot;, ylab = &quot;log(p)&quot;)

set.seed(6)
z.indep &lt;- rnorm(1e4)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = &quot;blue&quot;)
z.indep &lt;- rnorm(1e4, 0, 1.6)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = &quot;green&quot;)

plot(sort(log(p)), cex = 0.25, pch = 19, ylim = c(-19, -2.5), xlim = c(1, 850), xlab = &quot;Order&quot;, ylab = &quot;log(p)&quot;)

set.seed(6)
z.indep &lt;- rnorm(1e4)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = &quot;blue&quot;)
z.indep &lt;- rnorm(1e4, 0, 1.6)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = &quot;green&quot;)
abline(h = log(0.005), col = &quot;red&quot;, lty = 2)
abline(h = log(pnorm(-sqrt(2 * log(1e4))) * 2), col = &quot;red&quot;, lty = 2)
abline(h = log(0.05 / 1e4), col = &quot;red&quot;, lty = 2)</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-16-2.png" width="672" style="display: block; margin: auto;" /></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 3.4.3 (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     

other attached packages:
 [1] reshape2_1.4.3    ggplot2_2.2.1     plyr_1.8.4       
 [4] edgeR_3.20.2      limma_3.34.4      ashr_2.2-2       
 [7] Rmosek_8.0.69     PolynomF_1.0-1    CVXR_0.95        
[10] REBayes_1.2       Matrix_1.2-12     SQUAREM_2017.10-1
[13] EQL_1.0-0         ttutils_1.0-1    

loaded via a namespace (and not attached):
 [1] gmp_0.5-13.1      Rcpp_0.12.16      pillar_1.0.1     
 [4] compiler_3.4.3    git2r_0.21.0      R.methodsS3_1.7.1
 [7] R.utils_2.6.0     iterators_1.0.9   tools_3.4.3      
[10] digest_0.6.15     bit_1.1-12        tibble_1.4.1     
[13] gtable_0.2.0      evaluate_0.10.1   lattice_0.20-35  
[16] rlang_0.1.6       foreach_1.4.4     yaml_2.1.18      
[19] parallel_3.4.3    Rmpfr_0.6-1       ECOSolveR_0.4    
[22] stringr_1.3.0     knitr_1.20        locfit_1.5-9.1   
[25] rprojroot_1.3-2   bit64_0.9-7       grid_3.4.3       
[28] R6_2.2.2          rmarkdown_1.9     magrittr_1.5     
[31] scales_0.5.0      MASS_7.3-47       backports_1.1.2  
[34] codetools_0.2-15  htmltools_0.3.6   scs_1.1-1        
[37] colorspace_1.3-2  stringi_1.1.6     lazyeval_0.2.1   
[40] munsell_0.4.3     doParallel_1.0.11 pscl_1.5.2       
[43] truncnorm_1.0-7   R.oo_1.21.0      </code></pre>
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