<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta charset="utf-8" />
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="pandoc" />
<meta name="author" content="Lei Sun" />
<meta name="date" content="2018-04-16" />
<title>Re-Analyze Efron’s Leukemia Data</title>
<script src="site_libs/jquery-1.11.3/jquery.min.js"></script>
<meta name="viewport" content="width=device-width, initial-scale=1" />
<link href="site_libs/bootstrap-3.3.5/css/cosmo.min.css" rel="stylesheet" />
<script src="site_libs/bootstrap-3.3.5/js/bootstrap.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/html5shiv.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/respond.min.js"></script>
<script src="site_libs/jqueryui-1.11.4/jquery-ui.min.js"></script>
<link href="site_libs/tocify-1.9.1/jquery.tocify.css" rel="stylesheet" />
<script src="site_libs/tocify-1.9.1/jquery.tocify.js"></script>
<script src="site_libs/navigation-1.1/tabsets.js"></script>
<script src="site_libs/navigation-1.1/codefolding.js"></script>
<link href="site_libs/highlightjs-9.12.0/textmate.css" rel="stylesheet" />
<script src="site_libs/highlightjs-9.12.0/highlight.js"></script>
<link href="site_libs/font-awesome-4.5.0/css/font-awesome.min.css" rel="stylesheet" />
<style type="text/css">code{white-space: pre;}</style>
<style type="text/css">
pre:not([class]) {
background-color: white;
}
</style>
<script type="text/javascript">
if (window.hljs) {
hljs.configure({languages: []});
hljs.initHighlightingOnLoad();
if (document.readyState && document.readyState === "complete") {
window.setTimeout(function() { hljs.initHighlighting(); }, 0);
}
}
</script>
<style type="text/css">
h1 {
font-size: 34px;
}
h1.title {
font-size: 38px;
}
h2 {
font-size: 30px;
}
h3 {
font-size: 24px;
}
h4 {
font-size: 18px;
}
h5 {
font-size: 16px;
}
h6 {
font-size: 12px;
}
.table th:not([align]) {
text-align: left;
}
</style>
</head>
<body>
<style type = "text/css">
.main-container {
max-width: 940px;
margin-left: auto;
margin-right: auto;
}
code {
color: inherit;
background-color: rgba(0, 0, 0, 0.04);
}
img {
max-width:100%;
height: auto;
}
.tabbed-pane {
padding-top: 12px;
}
button.code-folding-btn:focus {
outline: none;
}
</style>
<style type="text/css">
/* padding for bootstrap navbar */
body {
padding-top: 51px;
padding-bottom: 40px;
}
/* offset scroll position for anchor links (for fixed navbar) */
.section h1 {
padding-top: 56px;
margin-top: -56px;
}
.section h2 {
padding-top: 56px;
margin-top: -56px;
}
.section h3 {
padding-top: 56px;
margin-top: -56px;
}
.section h4 {
padding-top: 56px;
margin-top: -56px;
}
.section h5 {
padding-top: 56px;
margin-top: -56px;
}
.section h6 {
padding-top: 56px;
margin-top: -56px;
}
</style>
<script>
// manage active state of menu based on current page
$(document).ready(function () {
// active menu anchor
href = window.location.pathname
href = href.substr(href.lastIndexOf('/') + 1)
if (href === "")
href = "index.html";
var menuAnchor = $('a[href="' + href + '"]');
// mark it active
menuAnchor.parent().addClass('active');
// if it's got a parent navbar menu mark it active as well
menuAnchor.closest('li.dropdown').addClass('active');
});
</script>
<div class="container-fluid main-container">
<!-- tabsets -->
<script>
$(document).ready(function () {
window.buildTabsets("TOC");
});
</script>
<!-- code folding -->
<style type="text/css">
.code-folding-btn { margin-bottom: 4px; }
</style>
<script>
$(document).ready(function () {
window.initializeCodeFolding("hide" === "show");
});
</script>
<script>
$(document).ready(function () {
// move toc-ignore selectors from section div to header
$('div.section.toc-ignore')
.removeClass('toc-ignore')
.children('h1,h2,h3,h4,h5').addClass('toc-ignore');
// establish options
var options = {
selectors: "h1,h2,h3",
theme: "bootstrap3",
context: '.toc-content',
hashGenerator: function (text) {
return text.replace(/[.\\/?&!#<>]/g, '').replace(/\s/g, '_').toLowerCase();
},
ignoreSelector: ".toc-ignore",
scrollTo: 0
};
options.showAndHide = true;
options.smoothScroll = true;
// tocify
var toc = $("#TOC").tocify(options).data("toc-tocify");
});
</script>
<style type="text/css">
#TOC {
margin: 25px 0px 20px 0px;
}
@media (max-width: 768px) {
#TOC {
position: relative;
width: 100%;
}
}
.toc-content {
padding-left: 30px;
padding-right: 40px;
}
div.main-container {
max-width: 1200px;
}
div.tocify {
width: 20%;
max-width: 260px;
max-height: 85%;
}
@media (min-width: 768px) and (max-width: 991px) {
div.tocify {
width: 25%;
}
}
@media (max-width: 767px) {
div.tocify {
width: 100%;
max-width: none;
}
}
.tocify ul, .tocify li {
line-height: 20px;
}
.tocify-subheader .tocify-item {
font-size: 0.90em;
padding-left: 25px;
text-indent: 0;
}
.tocify .list-group-item {
border-radius: 0px;
}
</style>
<!-- setup 3col/9col grid for toc_float and main content -->
<div class="row-fluid">
<div class="col-xs-12 col-sm-4 col-md-3">
<div id="TOC" class="tocify">
</div>
</div>
<div class="toc-content col-xs-12 col-sm-8 col-md-9">
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">truncash</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li>
<a href="index.html">Home</a>
</li>
<li>
<a href="about.html">About</a>
</li>
<li>
<a href="license.html">License</a>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li>
<a href="https://github.com/LSun/truncash">
<span class="fa fa-github"></span>
</a>
</li>
</ul>
</div><!--/.nav-collapse -->
</div><!--/.container -->
</div><!--/.navbar -->
<div class="fluid-row" id="header">
<div class="btn-group pull-right">
<button type="button" class="btn btn-default btn-xs dropdown-toggle" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"><span>Code</span> <span class="caret"></span></button>
<ul class="dropdown-menu" style="min-width: 50px;">
<li><a id="rmd-show-all-code" href="#">Show All Code</a></li>
<li><a id="rmd-hide-all-code" href="#">Hide All Code</a></li>
</ul>
</div>
<h1 class="title toc-ignore">Re-Analyze Efron’s Leukemia Data</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2018-04-16</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> 2018-05-09</p>
<!-- Insert the code version (Git commit SHA1) if Git repository exists and R
package git2r is installed -->
<p><strong>Code version:</strong> 4070fcd</p>
<!-- Add your analysis here -->
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<pre class="r"><code>source("../code/gdash_lik.R")
source("../code/gdfit.R")
library(limma)
library(locfdr)
library(scales)</code></pre>
<pre class="r"><code>load("../data/LSI/leukdata.RData")
load("../data/LSI/leukz.RData")
design <- design <- c(rep(0, 47), rep(1, 25))</code></pre>
<pre class="r"><code>temp <- locfdr(leukz)</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-3-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="reproduce-efrons-empirical-null-analysis" class="section level2">
<h2>Reproduce Efron’s empirical null analysis</h2>
<pre class="r"><code>t <- apply(leukdata, 1, function (x) {
t.test(x[design == 1], x[design == 0], var.equal = TRUE)$statistic
})
z <- -qnorm(pt(-abs(t), 70)) * sign(t)</code></pre>
<pre class="r"><code>temp <- locfdr(z)</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="use-smyths-pipeline-to-obtain-better-summary-statistics" class="section level2">
<h2>Use Smyth’s pipeline to obtain better summary statistics</h2>
<pre class="r"><code>lim = limma::lmFit(leukdata, model.matrix(~design))
r.ebayes = limma::eBayes(lim)
p = r.ebayes$p.value[, 2]
t = r.ebayes$t[, 2]
z = -sign(t) * qnorm(p/2)</code></pre>
<pre class="r"><code>fit.locfdr <- locfdr(z)</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-7-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>fit.qvalue <- qvalue::qvalue(p)</code></pre>
</div>
<div id="cash-fitting" class="section level2">
<h2>CASH fitting</h2>
<ul>
<li>betahat = sebetahat * z</li>
</ul>
<pre class="r"><code>betahat = lim$coefficients[, 2]
sebetahat = betahat / t
betahat = sebetahat * z</code></pre>
<pre class="r"><code>fit.cash <- gdash(betahat, sebetahat, gd.ord = 10)
fit.ash <- ashr::ash(betahat, sebetahat, mixcompdist = "normal", method = "fdr")</code></pre>
<pre class="r"><code>x.plot <- seq(-10, 10, length = 1000)
gd.ord <- 10
hermite = Hermite(gd.ord)
gd0.std = dnorm(x.plot)
matrix_lik_plot = cbind(gd0.std)
for (i in 1 : gd.ord) {
gd.std = (-1)^i * hermite[[i]](x.plot) * gd0.std / sqrt(factorial(i))
matrix_lik_plot = cbind(matrix_lik_plot, gd.std)
}
y.plot = matrix_lik_plot %*% fit.cash$w * fit.cash$fitted_g$pi[1]
method.col <- c("#377eb8", "#984ea3", "#4daf4a", "#ff7f00", "#e41a1c")
par(mfrow = c(1, 2))
hist(z, prob = TRUE, main = "", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
lines(x.plot, y.plot, col = method.col[5], lwd = 2)
lines(x.plot, dnorm(x.plot), col =
"orange"
# method.col[2]
, lty = 2, lwd = 2)
lines(x.plot, dnorm(x.plot, fit.locfdr$fp0[3, 1], fit.locfdr$fp0[3, 2]) * fit.locfdr$fp0[3, 3], col = method.col[3], lty = 2, lwd = 2)
text(-3, 0.22, "N(0,1)", col = "orange")
text(-5, 0.15, bquote(atop("Efron's empirical null", .(round(fit.locfdr$fp0[3, 3], 2)) %*% N(.(round(fit.locfdr$fp0[3, 1], 2)), .(round(fit.locfdr$fp0[3, 2], 2))^2))), col = method.col[3])
text(4.5, 0.10, "CASH null", col = method.col[5])
arrows(-1.8, 0.22, -1.1, 0.215, length = 0.1, angle = 20, col = "orange")
arrows(-2, 0.15, -1.5, 0.145, length = 0.1, angle = 20, col = method.col[3])
arrows(2.9, 0.10, 2, 0.095, length = 0.1, angle = 20, col = method.col[5])
plot(z, fit.cash$lfdr, pch = 18, cex = 0.5, ylim = c(0, 1), col = method.col[5], ylab = "Local FDR", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
points(z, fit.locfdr$fdr, pch = 16, cex = 0.5, col = method.col[3])
points(z, ashr::get_lfdr(fit.ash), pch = 17, cex = 0.5, col = method.col[4])
points(z, fit.qvalue$lfdr, pch = 15, cex = 0.5, col = method.col[2])
abline(h = 0.2, lty = 2)
legend("topright", bty = "n", pch = 15 : 18, col = method.col[2 : 5], c("qvalue", "locfdr", "ASH", "CASH"))</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-11-1.png" width="672" style="display: block; margin: auto;" /></p>
<ul>
<li>sebetahat = betahat / z</li>
</ul>
<pre class="r"><code>betahat = lim$coefficients[, 2]
sebetahat = betahat / z</code></pre>
<pre class="r"><code>fit.cash2 <- gdash(betahat, sebetahat, gd.ord = 10)</code></pre>
<pre class="r"><code>y.plot2 = matrix_lik_plot %*% fit.cash2$w * fit.cash2$fitted_g$pi[1]
par(mfrow = c(1, 2))
hist(z, prob = TRUE, main = "", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
lines(x.plot, y.plot, col = "red", lwd = 2)
lines(x.plot, y.plot2, col = "blue", lwd = 2)
lines(x.plot, dnorm(x.plot, fit.locfdr$fp0[3, 1], fit.locfdr$fp0[3, 2]) * fit.locfdr$fp0[3, 3], col = 'green', lty = 2, lwd = 2)
plot(z, fit.cash$lfdr, pch = 18, cex = 0.5, ylim = c(0, 1), col = "red", ylab = "Local FDR", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
points(z, fit.cash2$lfdr, pch = 17, cex = 0.5, col = "blue")
points(z, fit.locfdr$fdr, pch = 16, cex = 0.5, col = "green")
abline(h = 0.2, lty = 2)
legend("topright", bty = "n", pch = c(16, 18, 17), col = c("green", "red", "blue"), c("qvalue", "CASH: 1", "CASH: 2"))</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-14-1.png" width="672" style="display: block; margin: auto;" /></p>
<ul>
<li>sebetahat = betahat / z</li>
</ul>
<pre class="r"><code>betahat = lim$coefficients[, 2]
sebetahat = betahat / z</code></pre>
<pre class="r"><code>fit.cash <- gdash(betahat, sebetahat, gd.ord = 10)
fit.ash <- ashr::ash(betahat, sebetahat, mixcompdist = "normal", method = "fdr")</code></pre>
<pre class="r"><code>x.plot <- seq(-10, 10, length = 1000)
gd.ord <- 10
hermite = Hermite(gd.ord)
gd0.std = dnorm(x.plot)
matrix_lik_plot = cbind(gd0.std)
for (i in 1 : gd.ord) {
gd.std = (-1)^i * hermite[[i]](x.plot) * gd0.std / sqrt(factorial(i))
matrix_lik_plot = cbind(matrix_lik_plot, gd.std)
}
y.plot = matrix_lik_plot %*% fit.cash$w * fit.cash$fitted_g$pi[1]
method.col <- c("#377eb8", "#984ea3", "#4daf4a", "#ff7f00", "#e41a1c")
setEPS()
postscript("../output/fig/leukemia.eps", height = 5, width = 12)
par(mfrow = c(1, 2))
hist(z, prob = TRUE, main = "", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
lines(x.plot, y.plot, col = method.col[5], lwd = 2)
lines(x.plot, dnorm(x.plot), col =
"orange"
# method.col[2]
, lty = 2, lwd = 2)
lines(x.plot, dnorm(x.plot, fit.locfdr$fp0[3, 1], fit.locfdr$fp0[3, 2]) * fit.locfdr$fp0[3, 3], col = method.col[3], lty = 2, lwd = 2)
text(-3, 0.22, "N(0,1)", col = "orange")
text(-5, 0.15, bquote(atop("Efron's empirical null", .(round(fit.locfdr$fp0[3, 3], 2)) %*% N(.(round(fit.locfdr$fp0[3, 1], 2)), .(round(fit.locfdr$fp0[3, 2], 2))^2))), col = method.col[3])
text(4.5, 0.10, "CASH null", col = method.col[5])
arrows(-1.8, 0.22, -1.1, 0.215, length = 0.1, angle = 20, col = "orange")
arrows(-2, 0.15, -1.5, 0.145, length = 0.1, angle = 20, col = method.col[3])
arrows(2.9, 0.10, 2, 0.095, length = 0.1, angle = 20, col = method.col[5])
plot(z, fit.cash$lfdr, pch = 18, cex = 0.5, ylim = c(0, 1), col = method.col[5], ylab = "Local FDR", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
points(z, fit.locfdr$fdr, pch = 16, cex = 0.5, col = method.col[3])
points(z, ashr::get_lfdr(fit.ash), pch = 17, cex = 0.5, col = method.col[4])
points(z, fit.qvalue$lfdr, pch = 15, cex = 0.5, col = method.col[2])
abline(h = 0.2, lty = 2)
legend("topright", bty = "n", pch = 15 : 18, col = method.col[2 : 5], c("qvalue", "locfdr", "ASH", "CASH"))
dev.off()</code></pre>
<pre><code>quartz_off_screen
2 </code></pre>
</div>
<div id="hiv" class="section level2">
<h2>HIV</h2>
<pre class="r"><code>load("../data/LSI/hivdata.RData")
load("../data/LSI/hivz.RData")
design <- rep(0 : 1, each = 4)</code></pre>
<div id="reproduce-efrons-empirical-null-analysis-1" class="section level3">
<h3>Reproduce Efron’s empirical null analysis</h3>
<pre class="r"><code>temp <- locfdr(hivz)</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-19-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="use-smyths-pipeline-to-obtain-better-summary-statistics-1" class="section level3">
<h3>Use Smyth’s pipeline to obtain better summary statistics</h3>
<pre class="r"><code>loghivdata <- log2(hivdata)
loghivdata <- limma::normalizeBetweenArrays(loghivdata)
lim = limma::lmFit(loghivdata, model.matrix(~design))
r.ebayes = limma::eBayes(lim)
p = r.ebayes$p.value[, 2]
t = r.ebayes$t[, 2]
z = -sign(t) * qnorm(p/2)</code></pre>
<pre class="r"><code>fit.locfdr <- locfdr(z)</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-21-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre class="r"><code>fit.qvalue <- qvalue::qvalue(p)</code></pre>
</div>
<div id="cash-fitting-1" class="section level3">
<h3>CASH fitting</h3>
<ul>
<li>sebetahat = betahat / z</li>
</ul>
<pre class="r"><code>betahat = lim$coefficients[, 2]
sebetahat = betahat / z</code></pre>
<pre class="r"><code>fit.cash <- gdash(betahat, sebetahat, gd.ord = 10)
fit.ash <- ashr::ash(betahat, sebetahat, mixcompdist = "normal", method = "fdr")</code></pre>
<pre class="r"><code>x.plot <- seq(-7, 5, length = 1000)
gd.ord <- 10
hermite = Hermite(gd.ord)
gd0.std = dnorm(x.plot)
matrix_lik_plot = cbind(gd0.std)
for (i in 1 : gd.ord) {
gd.std = (-1)^i * hermite[[i]](x.plot) * gd0.std / sqrt(factorial(i))
matrix_lik_plot = cbind(matrix_lik_plot, gd.std)
}
y.plot = matrix_lik_plot %*% fit.cash$w * fit.cash$fitted_g$pi[1]
method.col <- scales::hue_pal()(5)
par(mfrow = c(1, 2))
hist(z, prob = TRUE, main = "", xlab = expression(paste(z, "-scores")), cex.lab = 1.25, breaks = 50)
lines(x.plot, y.plot, col = method.col[5], lwd = 2)
lines(x.plot, dnorm(x.plot), col =
"orange"
# method.col[2]
, lty = 2, lwd = 2)
lines(x.plot, dnorm(x.plot, fit.locfdr$fp0[3, 1], fit.locfdr$fp0[3, 2]) * fit.locfdr$fp0[3, 3], col = method.col[3], lty = 2, lwd = 2)
text(-3, 0.22, "N(0,1)", col = "orange")
text(-5, 0.15, bquote(atop("Efron's empirical null", .(round(fit.locfdr$fp0[3, 3], 2)) %*% N(.(round(fit.locfdr$fp0[3, 1], 2)), .(round(fit.locfdr$fp0[3, 2], 2))^2))), col = method.col[3])
text(4.5, 0.10, "CASH null", col = method.col[5])
arrows(-1.8, 0.22, -1.1, 0.215, length = 0.1, angle = 20, col = "orange")
arrows(-2, 0.15, -1.5, 0.145, length = 0.1, angle = 20, col = method.col[3])
arrows(2.9, 0.10, 2, 0.095, length = 0.1, angle = 20, col = method.col[5])
plot(z, fit.cash$lfdr, pch = 18, cex = 0.5, ylim = c(0, 1), col = method.col[5], ylab = "Local FDR", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
points(z, fit.locfdr$fdr, pch = 16, cex = 0.5, col = method.col[3])
points(z, ashr::get_lfdr(fit.ash), pch = 17, cex = 0.5, col = method.col[4])
points(z, fit.qvalue$lfdr, pch = 15, cex = 0.5, col = method.col[2])
abline(h = 0.2, lty = 2)
legend("topleft", bty = "n", pch = 15 : 18, col = method.col[2 : 5], c("qvalue", "locfdr", "ASH", "CASH"))</code></pre>
<p><img src="figure/efron_leukemia.rmd/unnamed-chunk-25-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</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] scales_0.5.0 locfdr_1.1-8 limma_3.34.4
[4] ashr_2.2-2 Rmosek_8.0.69 PolynomF_1.0-1
[7] CVXR_0.95 REBayes_1.2 Matrix_1.2-12
[10] SQUAREM_2017.10-1 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] plyr_1.8.4 compiler_3.4.3 git2r_0.21.0
[7] R.methodsS3_1.7.1 R.utils_2.6.0 iterators_1.0.9
[10] tools_3.4.3 digest_0.6.15 bit_1.1-12
[13] tibble_1.4.1 gtable_0.2.0 evaluate_0.10.1
[16] lattice_0.20-35 rlang_0.1.6 foreach_1.4.4
[19] yaml_2.1.18 parallel_3.4.3 Rmpfr_0.6-1
[22] ECOSolveR_0.4 stringr_1.3.0 knitr_1.20
[25] rprojroot_1.3-2 bit64_0.9-7 grid_3.4.3
[28] qvalue_2.10.0 R6_2.2.2 rmarkdown_1.9
[31] reshape2_1.4.3 ggplot2_2.2.1 magrittr_1.5
[34] splines_3.4.3 MASS_7.3-47 backports_1.1.2
[37] codetools_0.2-15 htmltools_0.3.6 scs_1.1-1
[40] assertthat_0.2.0 colorspace_1.3-2 stringi_1.1.6
[43] lazyeval_0.2.1 munsell_0.4.3 doParallel_1.0.11
[46] pscl_1.5.2 truncnorm_1.0-7 R.oo_1.21.0 </code></pre>
</div>
<hr>
<p>
This <a href="http://rmarkdown.rstudio.com">R Markdown</a> site was created with <a href="https://github.com/jdblischak/workflowr">workflowr</a>
</p>
<hr>
<!-- To enable disqus, uncomment the section below and provide your disqus_shortname -->
<!-- disqus
<div id="disqus_thread"></div>
<script type="text/javascript">
/* * * CONFIGURATION VARIABLES: EDIT BEFORE PASTING INTO YOUR WEBPAGE * * */
var disqus_shortname = 'rmarkdown'; // required: replace example with your forum shortname
/* * * DON'T EDIT BELOW THIS LINE * * */
(function() {
var dsq = document.createElement('script'); dsq.type = 'text/javascript'; dsq.async = true;
dsq.src = '//' + disqus_shortname + '.disqus.com/embed.js';
(document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(dsq);
})();
</script>
<noscript>Please enable JavaScript to view the <a href="http://disqus.com/?ref_noscript">comments powered by Disqus.</a></noscript>
<a href="http://disqus.com" class="dsq-brlink">comments powered by <span class="logo-disqus">Disqus</span></a>
-->
</div>
</div>
</div>
<script>
// add bootstrap table styles to pandoc tables
function bootstrapStylePandocTables() {
$('tr.header').parent('thead').parent('table').addClass('table table-condensed');
}
$(document).ready(function () {
bootstrapStylePandocTables();
});
</script>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement("script");
script.type = "text/javascript";
script.src = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML";
document.getElementsByTagName("head")[0].appendChild(script);
})();
</script>
</body>
</html>