<!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="2017-06-14" /> <title>Posterior Inference with Gaussian Derivative Likelihood: Model and Result</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> <link href="site_libs/highlightjs-1.1/textmate.css" rel="stylesheet" /> <script src="site_libs/highlightjs-1.1/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 && 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 --> <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"> <h1 class="title toc-ignore">Posterior Inference with Gaussian Derivative Likelihood: Model and Result</h1> <h4 class="author"><em>Lei Sun</em></h4> <h4 class="date"><em>2017-06-14</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> 2017-06-17</p> <!-- Insert the code version (Git commit SHA1) if Git repository exists and R package git2r is installed --> <p><strong>Code version:</strong> 73b4bb7</p> <!-- Add your analysis here --> <div id="gd-ash-model" class="section level2"> <h2><code>GD-ASH</code> Model</h2> <p>Recall the typical <code>GD-ASH</code> model is</p> <p><span class="math display">\[ \begin{array}{l} \beta_j \sim \sum\pi_k N\left(0, \sigma_k^2\right) \ ;\\ \hat\beta_j = \beta_j + \hat s_j z_j \ ;\\ z_j \sim N\left(0, 1\right), \text{ correlated} \ . \end{array} \]</span> Then we are fitting the empirical distribution of <span class="math inline">\(z\)</span> with Gaussian derivatives</p> <p><span class="math display">\[ f(z) = \sum w_l\frac{1}{\sqrt{l!}}\varphi^{(l)}(z) \ . \]</span> Therefore, in essence, we are changing the likelihood of <span class="math inline">\(\hat\beta_j | \hat s_j, \beta_j\)</span> from correlated <span class="math inline">\(N\left(\beta_j, \hat s_j^2\right)\)</span> to independent <span class="math inline">\(\frac{1}{\hat s_j}f\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right)\)</span>, which using Gaussian derivatives is</p> <p><span class="math display">\[ \frac{1}{\hat s_j}\sum w_l \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) \ . \]</span> Note that when <span class="math inline">\(f = \varphi\)</span> it becomes the independent <span class="math inline">\(N\left(\beta_j, \hat s_j^2\right)\)</span> case.</p> </div> <div id="gd-lik-model" class="section level2"> <h2><code>GD-Lik</code> Model</h2> <p>Therefore, if we use Gaussian derivatives instead of Gaussian as the likelihood, the posterior distribution of <span class="math inline">\(\beta_j | \hat s_j, \hat\beta_j\)</span> should be</p> <p><span class="math display">\[ \begin{array}{rcl} f\left(\beta_j \mid \hat s_j, \hat\beta_j\right) &=& \frac{ \displaystyle g\left(\beta_j\right) f\left(\hat\beta_j \mid \hat s_j, \beta_j \right) }{ \displaystyle\int g\left(\beta_j\right) f\left(\hat\beta_j \mid \hat s_j, \beta_j \right) d\beta_j }\\ &=& \frac{ \displaystyle \sum\pi_k\sum w_l \frac{1}{\sigma_k} \varphi\left(\frac{\beta_j - \mu_k}{\sigma_k}\right) \frac{1}{\hat s_j} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) }{ \displaystyle \sum\pi_k\sum w_l \int \frac{1}{\sigma_k} \varphi\left(\frac{\beta_j - \mu_k}{\sigma_k}\right) \frac{1}{\hat s_j} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) d\beta_j } \ . \end{array} \]</span> The denominator <a href="ash_gd.html#normal_mixture_prior">readily has an analytic form</a> which is</p> <p><span class="math display">\[ \displaystyle \sum\pi_k \sum w_l \frac{\hat s_j^l}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) := \sum \pi_k \sum w_l f_{jkl} \ . \]</span></p> </div> <div id="posterior-mean" class="section level2"> <h2>Posterior mean</h2> <p>After algebra, the posterior mean is given by</p> <p><span class="math display">\[ E\left[\beta_j \mid \hat s_j, \hat \beta_j \right] = \int \beta_j f\left(\beta_j \mid \hat s_j, \hat\beta_j\right) d\beta_j = \displaystyle \frac{ \sum \pi_k \sum w_l m_{jkl} }{ \sum \pi_k \sum w_l f_{jkl} } \ , \]</span> where <span class="math inline">\(f_{jkl}\)</span> is defined as above and <span class="math display">\[ m_{jkl} = - \frac{\hat s_j^l \sigma_k^2}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+2}} \frac{1}{\sqrt{l!}} \varphi^{(l+1)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) + \frac{\hat s_j^l\mu_k}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) \ . \]</span></p> </div> <div id="local-fdr" class="section level2"> <h2>Local FDR</h2> <p>Assuming <span class="math inline">\(\mu_k \equiv 0\)</span>, the <code>lfdr</code> is given by <span class="math display">\[ p\left(\beta_j = 0\mid \hat s_j, \hat \beta_j\right) = \frac{ \pi_0 \sum w_l \frac{1}{\hat s_j} \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j}{\hat s_j}\right) }{ \sum \pi_k \sum w_l f_{jkl} } \ . \]</span></p> </div> <div id="local-fsr" class="section level2"> <h2>Local FSR</h2> <p>Right now the analytic form of <code>lfsr</code> using Gaussian derivatives is unavailable.</p> </div> <div id="simulation" class="section level2"> <h2>Simulation</h2> <p>The correlated <span class="math inline">\(N\left(0, 1\right)\)</span> <span class="math inline">\(z\)</span> scores are simulated from the GTEx/Liver data by the <a href="nullpipeline.html">null pipeline</a>. In order to get a better sense of the effectiveness of <code>GD-ASH</code> and <code>GD-Lik</code>, we are using data sets more distorted by correlation in the simulation. In particular, we are using an “inflation” batch, defined as the standard error of the correlated <span class="math inline">\(z\)</span> no less than <span class="math inline">\(1.2\)</span>, and a “deflation” batch, defined as that no greater than <span class="math inline">\(0.8\)</span>. Out of <span class="math inline">\(1000\)</span> simulated data sets, there are <span class="math inline">\(109\)</span> inflation ones and <span class="math inline">\(99\)</span> deflation ones.</p> <p>In order to create realistic heterskedastic estimated standard error, <span class="math inline">\(\hat s_j\)</span>’s are also simulated from the same <a href="nullpipeline.html">null pipeline</a>. Let <span class="math inline">\(\sigma^2 = \frac1n \sum\limits_{j = 1}^n \hat s_j^2\)</span> be the average strength of the heteroskedastic noise, and the true effects <span class="math inline">\(\beta_j\)</span>’s are simulated from <span class="math display">\[ 0.6\delta_0 + 0.3N\left(0, \sigma^2\right) + 0.1N\left(0, \left(2\sigma\right)^2\right) \ . \]</span></p> <p>Then let <span class="math inline">\(\hat\beta_j = \beta_j + \hat s_j z_j\)</span>. We are using <span class="math inline">\(\hat\beta_j\)</span>, <span class="math inline">\(\hat s_j\)</span>, along with <span class="math inline">\(\hat z_j = \hat\beta_j / \hat s_j\)</span>, <span class="math inline">\(\hat p_j= 2\left(1 - \Phi\left(\left|\hat z_j\right|\right)\right)\)</span>, as the summary statistics fed to <code>GD-ASH</code> and <code>GD-Lik</code>, as well as into <code>BH</code>, <code>qvalue</code>, <code>locfdr</code>, <code>ASH</code> for a comparison.</p> <pre class="r"><code>source("../code/gdash_lik.R")</code></pre> <pre class="r"><code>z.mat = readRDS("../output/z_null_liver_777.rds") se.mat = readRDS("../output/sebetahat_null_liver_777.rds")</code></pre> <pre class="r"><code>z.sd = apply(z.mat, 1, sd) inflation.index = (z.sd >= 1.2) deflation.index = (z.sd <= 0.8) z.inflation = z.mat[inflation.index, ] se.inflation = se.mat[inflation.index, ] ## Number of inflation data sets nrow(z.inflation)</code></pre> <pre><code>[1] 109</code></pre> <pre class="r"><code>z.deflation = z.mat[deflation.index, ] se.deflation = se.mat[deflation.index, ] ## Number of deflation data sets nrow(z.deflation)</code></pre> <pre><code>[1] 99</code></pre> </div> <div id="inflation-data-sets" class="section level2"> <h2>Inflation data sets</h2> <div id="some-examples-of-inflated-correlated-null-z-scores" class="section level3"> <h3>Some examples of inflated correlated null <span class="math inline">\(z\)</span> scores</h3> <p><img src="figure/gd_lik_2.rmd/inflation%20examples-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="hat-pi_0" class="section level3"> <h3><span class="math inline">\(\hat \pi_0\)</span></h3> <p><code>locfdr</code> overestimates, <code>ASH</code> underestimates, <code>GD-ASH</code> on target, <code>qvalue</code> surprisingly good.</p> <p><img src="figure/gd_lik_2.rmd/inflation%20pihat0-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="mse" class="section level3"> <h3>MSE</h3> <p><code>GD-Lik</code> clearly improves the posterior estimates of <code>GD-ASH</code>.</p> <p><img src="figure/gd_lik_2.rmd/inflation%20mse-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="auc" class="section level3"> <h3>AUC</h3> <p>For almost all methods, <code>.q</code> means using q values, and <code>.l</code> using local FDRs. <code>GD-Lik</code> is the best, yet even the vanilla <span class="math inline">\(p\)</span> values are not much worse. It indicates that all the methods based on summary statistics indeed make few changes to the order of original <span class="math inline">\(p\)</span> values. Worth noting is that <code>locfdr</code> doesn’t perform well, and <code>lfdr</code>’s give a drastically different result than q values do, probably due to some artifacts like <a href="auc_pvalue.html">ties</a>.</p> <p><img src="figure/gd_lik_2.rmd/inflation%20auc-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="q-values-and-positive-fdr-pfdr-calibration" class="section level3"> <h3>q values and positive FDR (pFDR) calibration</h3> <p>Dashed lines are <span class="math inline">\(y = x\)</span> and <span class="math inline">\(y = 2x\)</span>. <code>ASH</code> and <code>qvalue</code> are too liberal, and <code>locfdr</code> is too conservative. <code>GD-ASH</code> and <code>BH</code> give very similar results and not far off. <code>BH</code>’s calibrates pFDR relatively well, even though it’s only guaranteed to control FDR under independence. <code>GD-Lik</code> calibrates pFDR almost precisely.</p> <p><img src="figure/gd_lik_2.rmd/inflation%20qvalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gd_lik_2.rmd/inflation%20qvalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="s-values-and-positive-fsr-pfsr-calibration" class="section level3"> <h3>s values and positive FSR (pFSR) calibration</h3> <p>Both <code>ASH</code> and <code>GD-ASH</code> are too liberal, although <code>GD-ASH</code> is not too far off.</p> <p><img src="figure/gd_lik_2.rmd/inflation%20svalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gd_lik_2.rmd/inflation%20svalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p> </div> </div> <div id="deflation-data-sets" class="section level2"> <h2>Deflation data sets</h2> <div id="some-examples-of-deflated-correlated-null-z-scores" class="section level3"> <h3>Some examples of deflated correlated null <span class="math inline">\(z\)</span> scores</h3> <p><img src="figure/gd_lik_2.rmd/deflation%20examples-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="hat-pi_0-1" class="section level3"> <h3><span class="math inline">\(\hat \pi_0\)</span></h3> <p>Almost all methods except <code>GD-ASH</code> overestimate as expected. <code>GD-ASH</code> occasionally severely underestimates as <a href="simulation_real_se.html#global_null">seen before</a>.</p> <p><img src="figure/gd_lik_2.rmd/deflation%20pihat0-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="mse-1" class="section level3"> <h3>MSE</h3> <p><code>GD-Lik</code> does better than <code>ASH</code> and <code>GD-ASH</code> but not as significantly as in the inflation case.</p> <p><img src="figure/gd_lik_2.rmd/deflation%20mse-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="auc-1" class="section level3"> <h3>AUC</h3> <p>Similar story as in the inflation case, although this time <code>qvalue</code>’s <code>lfdr</code> behaves weirdly.</p> <p><img src="figure/gd_lik_2.rmd/deflation%20auc-1.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="q-values-and-positive-fdr-pfdr-calibration-1" class="section level3"> <h3>q values and positive FDR (pFDR) calibration</h3> <p>Essentially all methods successfully control pFDR. <code>GD-Lik</code> looks good although off a little. <code>qvalue</code> is the most conservative, followed by <code>ASH</code>, <code>GD-ASH</code>, and <code>locfdr</code>.</p> <p><img src="figure/gd_lik_2.rmd/deflation%20qvalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gd_lik_2.rmd/deflation%20qvalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="s-values-and-positive-fsr-pfsr-calibration-1" class="section level3"> <h3>s values and positive FSR (pFSR) calibration</h3> <p>Both <code>ASH</code> and <code>GD-ASH</code> seem too conservative, although <code>GD-ASH</code> is more powerful.</p> <p><img src="figure/gd_lik_2.rmd/deflation%20svalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/gd_lik_2.rmd/deflation%20svalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p> </div> </div> <div id="remarks" class="section level2"> <h2>Remarks</h2> <ol style="list-style-type: decimal"> <li>Would be nice to come up with a way to calculate <code>lfsr</code> in <code>GD-Lik</code>.</li> <li>Many methods are too liberal for inflation cases and too conservative for deflation cases, showing a lack of robustness against correlation. Although, on average they probably seem about right.</li> </ol> </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 10.12.5 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_2.1-19 Rmosek_7.1.3 PolynomF_0.94 cvxr_0.0.0.9009 [5] REBayes_0.62 Matrix_1.2-8 SQUAREM_2016.10-1 EQL_1.0-0 [9] ttutils_1.0-1 loaded via a namespace (and not attached): [1] Rcpp_0.12.11 knitr_1.16 magrittr_1.5 [4] MASS_7.3-45 pscl_1.4.9 doParallel_1.0.10 [7] lattice_0.20-34 foreach_1.4.3 stringr_1.2.0 [10] tools_3.3.3 parallel_3.3.3 grid_3.3.3 [13] git2r_0.18.0 iterators_1.0.8 htmltools_0.3.6 [16] yaml_2.1.14 rprojroot_1.2 digest_0.6.12 [19] codetools_0.2-15 evaluate_0.10 rmarkdown_1.5 [22] stringi_1.1.2 backports_1.0.5 truncnorm_1.0-7 </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>