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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>
<p><strong>Last updated:</strong> 2018-05-29</p>
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<p><summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/LSun/truncash/tree/c1f43e06aec6d1c40d338d132152d8f6c48f9213" target="_blank">c1f43e0</a> </summary></p>
Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated. <br><br> Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use <code>wflow_publish</code> or <code>wflow_git_commit</code>). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:
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2018-05-12
</td>
<td style="text-align:left;">
Update to 1.0
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/cc0ab8379469bc3726f1508cd81e4ecd6fef1b1a/analysis/gd_w.rmd" target="_blank">cc0ab83</a>
</td>
<td style="text-align:left;">
Lei Sun
</td>
<td style="text-align:left;">
2018-05-11
</td>
<td style="text-align:left;">
update
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/807f92420647de4cf0371bdc02df4e8d9b18b191/docs/gd_w.html" target="_blank">807f924</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-11
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/548070eac8fc50f838c644fcfd86d3629055b333/analysis/gd_w.rmd" target="_blank">548070e</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-11
</td>
<td style="text-align:left;">
wflow_publish(“analysis/gd_w.rmd”)
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/84c7d5b765665fc6a0ac8db7d494dc833d002c48/docs/gd_w.html" target="_blank">84c7d5b</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-11
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/5c9aeacd3380e2b33e94ef6df5b785fcc1b27336/analysis/gd_w.rmd" target="_blank">5c9aeac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-11
</td>
<td style="text-align:left;">
wflow_publish(“analysis/gd_w.rmd”)
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/ed9c0d9f3e9849aac735d0edaa0365bd70b53d8b/docs/gd_w.html" target="_blank">ed9c0d9</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-11
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0ea9f778fe7f0894048bf5e397e871aa8bf577ba/analysis/gd_w.rmd" target="_blank">0ea9f77</a>
</td>
<td style="text-align:left;">
Lei Sun
</td>
<td style="text-align:left;">
2018-05-10
</td>
<td style="text-align:left;">
W
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/adeab809ceaaa2df50649ca4769e150d9e5ba13b/docs/gd_w.html" target="_blank">adeab80</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-06
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0b0a394f10deafc6607b96bff23d15a2e7f3b789/analysis/gd_w.rmd" target="_blank">0b0a394</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-06
</td>
<td style="text-align:left;">
wflow_publish(c(“analysis/BH_robustness.rmd”, “analysis/gd_w.rmd”))
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/720d17933c2b646477da82f5d1d8b4ea720bfc15/docs/gd_w.html" target="_blank">720d179</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/b82e2bc13e3a982ca2d6c31e9893377a920d667b/analysis/gd_w.rmd" target="_blank">b82e2bc</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
<td style="text-align:left;">
wflow_publish(c(“analysis/gd_w.rmd”, “analysis/index.Rmd”))
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/c7c59841b71459ea11b0c4ca53d870725a458672/analysis/gd_w.rmd" target="_blank">c7c5984</a>
</td>
<td style="text-align:left;">
Lei Sun
</td>
<td style="text-align:left;">
2018-04-15
</td>
<td style="text-align:left;">
ecdfz
</td>
</tr>
<tr>
<td style="text-align:left;">
html
</td>
<td style="text-align:left;">
<a href="https://cdn.rawgit.com/LSun/truncash/1c2f32e2432b150d762c89a63fb6c3ec43c86b64/docs/gd_w.html" target="_blank">1c2f32e</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-15
</td>
<td style="text-align:left;">
Build site.
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/3b5dcc47b80ecd0ec28888f123f9fdb9d9e00e0d/analysis/gd_w.rmd" target="_blank">3b5dcc4</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-15
</td>
<td style="text-align:left;">
wflow_publish(“analysis/gd_w.rmd”)
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/1b5d40fd2fde02446cf261ce68a851a80f3e4865/analysis/gd_w.rmd" target="_blank">1b5d40f</a>
</td>
<td style="text-align:left;">
Lei Sun
</td>
<td style="text-align:left;">
2018-04-13
</td>
<td style="text-align:left;">
gd w
</td>
</tr>
<tr>
<td style="text-align:left;">
rmd
</td>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/c766b808c28b56f5113e1013ea4dcabed557ec32/analysis/gd_w.rmd" target="_blank">c766b80</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-13
</td>
<td style="text-align:left;">
add lfsr
</td>
</tr>
</tbody>
</table>
</ul>
</details>
<hr />
<pre class="r"><code>source("../code/gdash_lik.R")</code></pre>
<pre><code>Warning: package 'Matrix' was built under R version 3.4.4</code></pre>
<pre class="r"><code>source("../code/gdfit.R")
source("../code/count_to_summary.R")
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 <- 10
n <- 1e4
B <- matrix(rnorm(n * d), n, d)
Sigma <- B %*% t(B) + diag(n)
sigma <- diag(Sigma)
Rho <- cov2cor(Sigma)
rhobar <- c()
for (l in 1 : 10) {
rhobar[l] <- (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 = "")</code></pre>
<p><img src="figure/gd_w.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>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/720d17933c2b646477da82f5d1d8b4ea720bfc15/docs/figure/gd_w.rmd/unnamed-chunk-3-1.png" target="_blank">720d179</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>set.seed(20)
z <- rnorm(d)
Z <- B %*% z + rnorm(n)
Z <- Z / sqrt(sigma)
cat("sd(Z) =", 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 = "blue")</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-4-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/720d17933c2b646477da82f5d1d8b4ea720bfc15/docs/figure/gd_w.rmd/unnamed-chunk-4-1.png" target="_blank">720d179</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>p <- pnorm(-abs(Z)) * 2
par(mfcol = c(2, 2))
par(mar = c(5.1, 4.1, 3, 2.1))
hist(p, breaks = 100, main = "Correlated", xlab = "p-value")
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 = "maroon")
abline(h = -log(0.05 / n), col = "red")
abline(h = -log(0.001), col = "green")
abline(h = -log(0.05), col = "blue")
Z <- rnorm(n)
p <- pnorm(-abs(Z)) * 2
par(mar = c(5.1, 4.1, 3, 2.1))
hist(p, breaks = 100, main = "Independent", xlab = "p-value")
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 = "maroon")
abline(h = -log(0.05 / n), col = "red")
abline(h = -log(0.001), col = "green")
abline(h = -log(0.05), col = "blue")</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-4-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/720d17933c2b646477da82f5d1d8b4ea720bfc15/docs/figure/gd_w.rmd/unnamed-chunk-4-2.png" target="_blank">720d179</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>set.seed(777)
nsim <- 1e4
Z.list <- W <- list()
for (i in 1 : nsim) {
z <- rnorm(d)
Z <- B %*% z + rnorm(n)
Z <- Z / sqrt(sigma)
Z.list[[i]] <- Z
Z.GD <- gdfit.mom(Z, 100)
W[[i]] <- Z.GD$w
}
Z.sim <- Z.list
W.sim <- W</code></pre>
</div>
<div id="real-data-from-gtex" class="section level2">
<h2>Real Data from GTEx</h2>
<pre class="r"><code>r <- readRDS("../data/liver.rds")</code></pre>
<pre class="r"><code>top_genes_index = function (g, X) {
return(order(rowSums(X), decreasing = TRUE)[1 : g])
}
lcpm = function (r) {
R = colSums(r)
t(log2(((t(r) + 0.5) / (R + 1)) * 10^6))
}</code></pre>
<pre class="r"><code>nsamp <- 5
ngene <- 1e4</code></pre>
<pre class="r"><code>Y = lcpm(r)
subset = top_genes_index(ngene, Y)
r = r[subset,]</code></pre>
<pre class="r"><code>set.seed(7)
nsim <- 1e4
Z.list <- W <- list()
for (i in 1 : nsim) {
## generate data
counts <- r[, sample(ncol(r), 2 * nsamp)]
design <- model.matrix(~c(rep(0, nsamp), rep(1, nsamp)))
summary <- count_to_summary(counts, design)
Z <- summary$z
Z.list[[i]] <- Z
Z.GD <- gdfit.mom(Z, 100)
W[[i]] <- Z.GD$w
}
Z.gtex <- Z.list
W.gtex <- W</code></pre>
<pre class="r"><code>quantile.vec1 <- exp(seq(-21, -5, by = 0.01))
quantile.vec2 <- seq(0.007, 0.993, by = 0.001)
quantile.vec3 <- exp(seq(-5, -21, by = -0.01))
emp.cdf.Z1 <- sapply(quantile.vec1, function(x) {sapply(Z.gtex, function(y) mean(y <= qnorm(x)))})
emp.cdf.Z2 <- sapply(quantile.vec2, function(x) {sapply(Z.gtex, function(y) mean(y <= qnorm(x)))})
emp.cdf.Z3 <- sapply(quantile.vec3, function(x) {sapply(Z.gtex, function(y) mean(y <= -qnorm(x)))})
emp.cdf.Z4 <- sapply(quantile.vec3, function(x) {sapply(Z.gtex, function(y) mean(y > -qnorm(x)))})</code></pre>
<pre class="r"><code>ecdf.avg1 <- colMeans(emp.cdf.Z1)
ecdf.avg2 <- colMeans(emp.cdf.Z2)
ecdf.avg3 <- colMeans(emp.cdf.Z3)
ecdf.avg4 <- colMeans(emp.cdf.Z4)
ecdf.avg <- c(ecdf.avg1, ecdf.avg2, ecdf.avg3)
ecdf.tail.avg.conf.int1 <- apply(emp.cdf.Z1, 2, function(x) {t.test(x)$conf.int})
ecdf.tail.avg.conf.int4 <- 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 = "l", col = "red", xlab = "z", ylab = "Cumulative Distribution Function (CDF)")
lines(c(qnorm(quantile.vec1), qnorm(quantile.vec2), -qnorm(quantile.vec3)), c(quantile.vec1, quantile.vec2, pnorm(-qnorm(quantile.vec3))), lty = 2)
legend("bottomright", lty = c(1, 2), col = c(1, 2), legend = c(expression(bar("F")(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>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-13-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/720d17933c2b646477da82f5d1d8b4ea720bfc15/docs/figure/gd_w.rmd/unnamed-chunk-13-1.png" target="_blank">720d179</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>plot(qnorm(quantile.vec1), log(ecdf.avg1), type = "l",
ylim = range(log(quantile.vec1), log(ecdf.avg1)),
xlab = "z", ylab = "log (CDF)")
lines(qnorm(quantile.vec1), log(quantile.vec1), lty = 2, col = "red")
lines(qnorm(quantile.vec1), log(pnorm(qnorm(quantile.vec1), 0, 1.1)), lty = 2, col = "green")
lines(qnorm(quantile.vec1), log(pnorm(qnorm(quantile.vec1), 0, 1.05)), lty = 2, col = "blue")
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("grey75", 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("bottomright", lty = c(1, 2, 2, 2), col = c(1, 2, 4, 3), legend = c(
expression(log(bar("F")(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>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-13-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/720d17933c2b646477da82f5d1d8b4ea720bfc15/docs/figure/gd_w.rmd/unnamed-chunk-13-2.png" target="_blank">720d179</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>plot(-qnorm(quantile.vec3), log(ecdf.avg4), type = "l",
ylim = range(log(quantile.vec3), log(ecdf.avg4)),
xlab = "z", ylab = "log (1 - CDF)")
lines(-qnorm(quantile.vec3), log(quantile.vec3), lty = 2, col = "red")
lines(-qnorm(quantile.vec3), log(pnorm(qnorm(quantile.vec3), 0, 1.1)), lty = 2, col = "green")
lines(-qnorm(quantile.vec3), log(pnorm(qnorm(quantile.vec3), 0, 1.05)), lty = 2, col = "blue")
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("grey75", alpha.f = 0.5))
legend("bottomleft", lty = c(1, 2, 2, 2), col = c(1, 2, 4, 3), legend = c(
expression(log(1 - bar("F")(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>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-13-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/720d17933c2b646477da82f5d1d8b4ea720bfc15/docs/figure/gd_w.rmd/unnamed-chunk-13-3.png" target="_blank">720d179</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-18
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>set.seed(777)
nsamp <- 50
nsim <- 1e3
z <- sebetahat <- list()
for (i in 1 : nsim) {
## generate data
counts <- r[, sample(ncol(r), 2 * nsamp)]
design <- model.matrix(~c(rep(0, nsamp), rep(1, nsamp)))
summary <- count_to_summary(counts, design)
z[[i]] <- summary$z
sebetahat[[i]] <- summary$sebetahat
}</code></pre>
<pre class="r"><code>sd.vec <- sapply(z, sd)
median.vec <- sapply(z, median)
fd.vec <- sapply(z, function(x) {
p <- pnorm(-abs(x)) * 2
sum(p <= 0.005)
})
sel <- c(834, 211, 397, 748)
par(mfrow = c(2, 2))
for (i in seq(sel)) {
fit <- gdfit(z[[sel[i]]], 10)
plot.gdfit(z[[sel[i]]], fit$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>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-15-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/adeab809ceaaa2df50649ca4769e150d9e5ba13b/docs/figure/gd_w.rmd/unnamed-chunk-15-1.png" target="_blank">adeab80</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-06
</td>
</tr>
</tbody>
</table>
</details>
<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 = "", breaks = 100, main = "(a): Histogram of two-sided p-values")
lines(c(0, 1), c(1, 1), col = "red")
hist(z[[834]], prob = TRUE, breaks = 100, xlab = "", xlim = c(-4.5, -2), main = "(b): Left tail of correlated z-scores")
lines(seq(-6, 6, by = 0.01), dnorm(seq(-6, 6, by = 0.01), 0, sd(z[[834]])), col = "blue")
lines(seq(-6, 6, by = 0.01), dnorm(seq(-6, 6, by = 0.01)), col = "red")
hist(z[[834]], prob = TRUE, breaks = 100, xlab = "", xlim = c(2, 4.5), main = "(c): Right tail of correlated z-scores")
lines(seq(-6, 6, by = 0.01), dnorm(seq(-6, 6, by = 0.01)), col = "red")
p <- pnorm(-abs(z[[834]])) * 2
plot(sample(-log(pnorm(-abs(z[[834]])) * 2)), ylim = c(0, 20), ylab = "-log(p)", main = expression(paste("(d): Correlated ", N(0, 1))))
abline(h = -log(0.005), col = "red")
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = "blue")
abline(h = -log(0.05 / 1e4), col = "green")
plot(-log(pnorm(-abs(rnorm(1e4))) * 2), ylim = c(0, 20), ylab = "-log(p)", main = expression(paste("(e): Independent ", N(0, 1))))
abline(h = -log(0.005), col = "red")
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = "blue")
abline(h = -log(0.05 / 1e4), col = "green")
plot(-log(pnorm(-abs(rnorm(1e4, 0, 1.6))) * 2), ylim = c(0, 20), ylab = "-log(p)", main = expression(paste("(f): Independent ", N(0, 1.6^2))))
abline(h = -log(0.005), col = "red")
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = "blue")
abline(h = -log(0.05 / 1e4), col = "green")</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-16-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-16-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/adeab809ceaaa2df50649ca4769e150d9e5ba13b/docs/figure/gd_w.rmd/unnamed-chunk-16-1.png" target="_blank">adeab80</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-06
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>p.bh <- p.adjust(p, method = "BH")
sum(p.bh <= 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 = "Order", ylab = "log(p)")
set.seed(6)
z.indep <- rnorm(1e4)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = "blue")
z.indep <- rnorm(1e4, 0, 1.6)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = "green")
plot(sort(log(p)), cex = 0.25, pch = 19, ylim = c(-19, -2.5), xlim = c(1, 850), xlab = "Order", ylab = "log(p)")
set.seed(6)
z.indep <- rnorm(1e4)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = "blue")
z.indep <- rnorm(1e4, 0, 1.6)
points(sort(log(pnorm(-abs(z.indep)) * 2)), cex = 0.25, pch = 19, col = "green")
abline(h = log(0.005), col = "red", lty = 2)
abline(h = log(pnorm(-sqrt(2 * log(1e4))) * 2), col = "red", lty = 2)
abline(h = log(0.05 / 1e4), col = "red", lty = 2)</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-16-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-16-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/adeab809ceaaa2df50649ca4769e150d9e5ba13b/docs/figure/gd_w.rmd/unnamed-chunk-16-2.png" target="_blank">adeab80</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-05-06
</td>
</tr>
</tbody>
</table>
</details>
<pre class="r"><code>W.sim.gd <- sapply(Z.sim, function (z) {fit.z <- gdfit(z, L = 10); return(
list(w = fit.z$w, status = fit.z$status))})</code></pre>
<pre class="r"><code>W.gtex.gd <- sapply(Z.gtex, function (z) {fit.z <- gdfit(z, L = 10); return(
list(w = fit.z$w, status = fit.z$status))})</code></pre>
</div>
<div id="correlated-null" class="section level2">
<h2>Correlated null</h2>
<pre><code>quartz_off_screen
2 </code></pre>
</div>
<div id="shoulder-inflation" class="section level2">
<h2>Shoulder inflation</h2>
<pre class="r"><code>z <- z.mat.sel[3, ]
p <- pnorm(-abs(z)) * 2
## the image is 7.5 * 3
setEPS()
postscript("../output/fig/cor_z_cdf.eps", width = 7.5, height = 3)
par(mfrow = c(1, 3))
par(oma = c(4, 2.5, 0, 0)) # make room (i.e. the 4's) for the overall x and y axis titles
par(mar = c(2, 2, 2.5, 1)) # make the plots be closer together
plot(ecdf(z), xlab = "", ylab = "", lwd = 2, main = expression("(a): CDF of All"), cex.main = 1.5)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01)), col = "blue", lwd = 2)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01), 0, 1.6), col = "green", lwd = 2)
rect(xleft = c(-5, 2.5),
xright = c(-2.5, 5),
ytop = c(0.05, 1),
ybottom = c(0, 0.95), border = "red", lty = c(1, 5))
plot(ecdf(z), xlab = "", ylab = "", main = expression("(b): Left Tail"), lwd = 2, xlim = c(-5, -2.5), ylim = c(0, 0.05), cex.main = 1.5, bty = "n")
box(col = "red")
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01)), col = "blue", lwd = 2)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01), 0, 1.6), col = "green", lwd = 2)
plot(ecdf(z), xlab = "", ylab = "", main = expression("(c): Right tail"), lwd = 2, xlim = c(2.5, 5), ylim = c(0.95, 1), cex.main = 1.5, bty = "n")
box(col = "red", lty = 5)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01)), col = "blue", lwd = 2)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01), 0, 1.6), col = "green", lwd = 2)
mtext('CDF', side = 2, outer = TRUE, line = 1)
legend("bottomleft", inset = c(-1.275, -0.35), legend = c("N(0, 1)", expression(N(0, 1.6^2))), lty = 1, lwd = 2, xpd = NA, col = c("blue", "green"), ncol = 2, cex = 1.25)
dev.off()</code></pre>
<pre><code>quartz_off_screen
2 </code></pre>
<pre class="r"><code># 7.5 * 3
setEPS()
postscript("../output/fig/cor_z_pval.eps", width = 7.5, height = 3.3)
par(mfrow = c(1, 3))
par(oma = c(4, 2.5, 0, 0)) # make room (i.e. the 4's) for the overall x and y axis titles
par(mar = c(4.5, 2, 2.5, 1)) # make the plots be closer together
set.seed(5)
p.norm.1 <- pnorm(-abs(rnorm(1e4))) * 2
set.seed(25)
p.norm.1.6 <- pnorm(-abs(rnorm(1e4, 0, 1.6))) * 2
y.max <- -log(min(p.norm.1, p, p.norm.1.6))
y.max <- 20
plot(sample(-log(p)), ylim = c(0, y.max), ylab = "-log(p)", main = expression(paste("(d): Correlated ", N(0, 1))), cex.main = 1.5, cex.lab = 1.5)
abline(h = -log(0.005), col = "red", lwd = 2)
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = "orange", lwd = 2)
abline(h = -log(0.05 / 1e4), col = "yellow", lwd = 2)
plot(-log(p.norm.1), ylim = c(0, y.max), ylab = "-log(p)", main = expression(paste("(e): Independent ", N(0, 1))), col = "blue", cex.main = 1.5, cex.lab = 1.5)
abline(h = -log(0.005), col = "red", lwd = 2)
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = "orange", lwd = 2)
abline(h = -log(0.05 / 1e4), col = "yellow", lwd = 2)
plot(-log(p.norm.1.6), ylim = c(0, y.max), ylab = "-log(p)", main = expression(paste("(f): Independent ", N(0, 1.6^2))), col = "green", cex.main = 1.5, cex.lab = 1.5)
abline(h = -log(0.005), col = "red", lwd = 2)
abline(h = -log(pnorm(-sqrt(2 * log(1e4))) * 2), col = "orange", lwd = 2)
abline(h = -log(0.05 / 1e4), col = "yellow", lwd = 2)
mtext('-log(p)', side = 2, outer = TRUE, line = 1)
legend("bottomleft", inset = c(-2.01, -0.51), legend = c("0.005", "Universal Threshold", "Bonferroni"), lty = 1, lwd = 2, xpd = NA, col = c("red", "orange", "yellow"), ncol = 3, cex = 1.25)
dev.off()</code></pre>
<pre><code>quartz_off_screen
2 </code></pre>
<pre class="r"><code>## under 0.005
sum(p <= 0.005)</code></pre>
<pre><code>[1] 809</code></pre>
<pre class="r"><code>p.bh <- p.adjust(p, method = "BH")
## BHq at FDR 0.05
sum(p.bh <= 0.05)</code></pre>
<pre><code>[1] 506</code></pre>
<pre class="r"><code>fit.q <- qvalue::qvalue(p)
## pi0 by qvalue
1 - fit.q$pi0</code></pre>
<pre><code>[1] 0.566462</code></pre>
<pre class="r"><code>## qvalue at FDR 0.05
sum(fit.q$qvalues <= 0.05)</code></pre>
<pre><code>[1] 2162</code></pre>
<pre class="r"><code>## pi0 by ashr
fit.a <- ashr::ash(z, 1, method = "fdr")
1 - ashr::get_pi0(fit.a)</code></pre>
<pre><code>[1] 0.98599</code></pre>
<pre class="r"><code>## ashr at FDR 0.05
sum(ashr::get_qvalue(fit.a) <= 0.05)</code></pre>
<pre><code>[1] 10000</code></pre>
</div>
<div id="deconvolution" class="section level2">
<h2>Deconvolution</h2>
<pre class="r"><code>KFE <- function(y, T = 300, lambda = 1/3){
# Kernel Fourier Estimator: Stefanski and Carroll (Statistics, 1990)
ks <- function(s,x) exp(s^2/2) * cos(s * x)
K <- function(t, y, lambda = 1/3){
k <- y
for(i in 1:length(y)){
k[i] <- integrate(ks, 0, 1/lambda, x = (y[i] - t))$value/pi
}
mean(k)
}
eps <- 1e-04
if(length(T) == 1) T <- seq(min(y)-eps, max(y)+eps, length = T)
g <- T
for(j in 1:length(T))
g[j] <- K(T[j], y, lambda = lambda)
list(x = T, y = g)
}
biweight <- function(x0, x, bw){
t <- (x - x0)/bw
(1-t^2)^2*((t> -1 & t<1)-0) *15/16
}
CDF.KW <- function(h, interp = FALSE, eps = 0.001, bw = 0.7){
#Wasserstein distance: ||G-H||_W
if(interp == "biweight"){
yk = h$x
for (j in 1:length(yk))
yk[j] = sum(biweight(h$x[j], h$x, bw = bw)*h$y/sum(h$y))
H <- cumsum(yk)
H <- H/H[length(H)]
}
else {
H <- cumsum(h$y)
H <- H/H[length(H)]
}
return(H)
}
library(deconvolveR)</code></pre>
<pre class="r"><code>G <- function (t) {
# 0.6 * pnorm(t, 0, 0.1) + 0.3 * pnorm(t, 0, 1) + 0.1 * pnorm(t, 0, 3)
0.6 * pnorm(t, 0, 0) + 0.3 * pnorm(t, 0, 1) + 0.1 * pnorm(t, 0, 3)
}
set.seed(777)
theta <- sample(c(
rnorm(6e3, 0, 0),
rnorm(3e3, 0, 1),
rnorm(1e3, 0, 3)
))</code></pre>
<pre class="r"><code>set.seed(777)
r <- readRDS("../data/liver.rds")
nsamp <- 5
ngene <- 1e4
Y = lcpm(r)
subset = top_genes_index(ngene, Y)
r = r[subset,]
counts <- r[, sample(ncol(r), 2 * nsamp)]
design <- model.matrix(~c(rep(0, nsamp), rep(1, nsamp)))
summary <- count_to_summary(counts, design)
s <- summary$sebetahat
s <- s / sqrt(mean(s^2))</code></pre>
<pre class="r"><code>x.plot <- seq(-6, 6, by = 0.01)
g.plot <- G(x.plot)
for (i in 3 : 4) {
z <- z.mat.sel[i, ]
X <- theta + s * z
Z <- theta + z
## ASH
fit.ash <- ashr::ash(X, s, method = "fdr", mixcompdist = "normal")
ash.plot <- as.numeric(ashr::mixcdf(ashr::get_fitted_g(fit.ash), x.plot))
ASH.data <- cbind.data.frame(
Method = "ASH",
x = x.plot,
cdfhat = ash.plot
)
## CASH
gd.priority = ifelse(i == 3, TRUE, FALSE)
fit.cash <- gdash(X, s)
cash.plot <- as.numeric(ashr::mixcdf(ashr::get_fitted_g(fit.cash), x.plot))
CASH.data <- cbind.data.frame(
Method = "CASH",
x = x.plot,
cdfhat = cash.plot
)
plot(x.plot, g.plot, type = "l")
lines(x.plot, cash.plot, col = "blue")
## Efron's BD (2016)
fit.bd <- deconvolveR::deconv(tau = x.plot, X = Z, family = "Normal", deltaAt = 0)
BD.data <- cbind.data.frame(
Method = "Efron",
x = fit.bd$stats[, 1],
cdfhat = fit.bd$stats[, 4]
)
## Kiefer-Wolfowitz's NPMLE (1956)
fit.kw <- REBayes::GLmix(x = X, sigma = s)
kw.plot <- CDF.KW(fit.kw)
KW.data <- cbind.data.frame(
Method = "KW",
x = fit.kw$x,
cdfhat = kw.plot
)
kws.plot <- CDF.KW(fit.kw, interp = "biweight")
KWs.data <- cbind.data.frame(
Method = "KWs",
x = fit.kw$x,
cdfhat = kws.plot
)
## kernal deconvolution by Stefanski and Carroll 1990
fit.sc <- KFE(Z)
sc.plot <- CDF.KW(fit.sc)
SC.data <- cbind.data.frame(
Method = "SC-FK",
x = fit.sc$x,
cdfhat = sc.plot
)
if (i == 3) {
deconv.inf.ggdata <- cbind.data.frame(
Noise = "Inflated Noise",
rbind.data.frame(
SC.data,
KW.data,
BD.data,
ASH.data,
CASH.data
)
)
} else {
deconv.def.ggdata <- cbind.data.frame(
Noise = "Deflated Noise",
rbind.data.frame(
SC.data,
KW.data,
BD.data,
ASH.data,
CASH.data
)
)
}
}</code></pre>
<pre><code>Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
consider reducing rtol
Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
consider reducing rtol
Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
consider reducing rtol
Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
consider reducing rtol
Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
consider reducing rtol
Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
consider reducing rtol</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-23-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Warning in stats::nlm(f = loglik, p = aStart, gradtol = 1e-10, ...): NA/Inf
replaced by maximum positive value</code></pre>
<pre><code>Warning in stats::nlm(f = loglik, p = aStart, gradtol = 1e-10, ...): NA/Inf
replaced by maximum positive value</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-23-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>deconv.ggdata <- rbind.data.frame(
deconv.inf.ggdata,
deconv.def.ggdata
)</code></pre>
<pre class="r"><code>method.name <- c("Fourier", "KW", "Efron", "ASH", "CASH")
method.col <- scales::hue_pal()(5)
method.linetype <- c("dashed", "dotdash", "twodash", "longdash", "dashed")
## plotting
ggplot(data = deconv.ggdata, aes(x = x, y = cdfhat, col = Method, linetype = Method)) +
geom_line(lwd = 1.1) +
facet_wrap(~Noise, nrow = 1) +
stat_function(fun = G, col = "black", lty = 1) +
xlim(-5, 5) +
scale_linetype_manual(values = method.linetype, labels = method.name) +
scale_color_manual(values = method.col, labels = method.name) +
labs(x = expression(theta), y = "Estimated CDF of g", title = expression(g == 0.6~delta[0] + 0.3~N(0, 1^2) + 0.1~N(0, 3^2))) +
theme(plot.title = element_text(size = 15, hjust = 0.5),
axis.title.x = element_text(size = 15),
axis.text.x = element_text(size = 10),
axis.title.y = element_text(size = 15),
axis.text.y = element_text(size = 10),
strip.text = element_text(size = 15),
legend.position = "bottom",
legend.background = element_rect(color = "grey"),
legend.text = element_text(size = 12))</code></pre>
<pre><code>Warning: Removed 908 rows containing missing values (geom_path).</code></pre>
<p><img src="figure/gd_w.rmd/unnamed-chunk-24-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>ggsave("../output/fig/deconv.eps", height = 4, width = 7)</code></pre>
<pre><code>Warning: Removed 908 rows containing missing values (geom_path).</code></pre>
</div>
<div id="session-information" 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
other attached packages:
[1] deconvolveR_1.1 reshape2_1.4.3 ggplot2_2.2.1
[4] plyr_1.8.4 edgeR_3.20.9 limma_3.34.9
[7] ashr_2.2-7 Rmosek_8.0.69 PolynomF_1.0-2
[10] CVXR_0.95 REBayes_1.3 Matrix_1.2-14
[13] SQUAREM_2017.10-1 EQL_1.0-0 ttutils_1.0-1
loaded via a namespace (and not attached):
[1] qvalue_2.10.0 locfit_1.5-9.1 splines_3.4.3
[4] lattice_0.20-35 colorspace_1.3-2 htmltools_0.3.6
[7] yaml_2.1.19 gmp_0.5-13.1 rlang_0.2.0
[10] R.oo_1.22.0 pillar_1.2.2 Rmpfr_0.7-0
[13] R.utils_2.6.0 bit64_0.9-7 scs_1.1-1
[16] foreach_1.4.4 stringr_1.3.1 munsell_0.4.3
[19] gtable_0.2.0 workflowr_1.0.1 R.methodsS3_1.7.1
[22] codetools_0.2-15 evaluate_0.10.1 labeling_0.3
[25] knitr_1.20 doParallel_1.0.11 pscl_1.5.2
[28] parallel_3.4.3 Rcpp_0.12.16 backports_1.1.2
[31] scales_0.5.0 truncnorm_1.0-8 bit_1.1-13
[34] digest_0.6.15 stringi_1.2.2 grid_3.4.3
[37] rprojroot_1.3-2 ECOSolveR_0.4 tools_3.4.3
[40] magrittr_1.5 lazyeval_0.2.1 tibble_1.4.2
[43] whisker_0.3-2 MASS_7.3-50 assertthat_0.2.0
[46] rmarkdown_1.9 iterators_1.0.9 R6_2.2.2
[49] git2r_0.21.0 compiler_3.4.3 </code></pre>
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