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} </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">Marginal Distribution of <span class="math inline">\(z\)</span> Scores: Null</h1> <h4 class="author"><em>Lei Sun</em></h4> <h4 class="date"><em>2017-04-25</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> 2017-05-09</p> <!-- Insert the code version (Git commit SHA1) if Git repository exists and R package git2r is installed --> <p><strong>Code version:</strong> 6264529</p> <!-- Add your analysis here --> <p><strong>This simulation can be seen as an enhanced version of <a href="ExtremeOccurrence.html">a previous simulation</a>.</strong></p> <div id="introduction" class="section level2"> <h2>Introduction</h2> <p>An assumption of using <a href="gaussian_derivatives.html">Gaussian derivatives</a> to fit correlated null <span class="math inline">\(z\)</span> scores is that each of these <span class="math inline">\(z\)</span> scores should actually be null. That is, for <span class="math inline">\(n\)</span> <span class="math inline">\(z\)</span> scores <span class="math inline">\(z_1, \ldots, z_n\)</span>, although the correlation between <span class="math inline">\(z_i\)</span> and <span class="math inline">\(z_j\)</span> are not necessarily zero, the marginal distribution of <span class="math inline">\(z_i\)</span>, <span class="math inline">\(\forall i\)</span>, should be <span class="math inline">\(N\left(0, 1\right)\)</span>.</p> <p>However, in practice, it’s not easy to check whether these correlated <span class="math inline">\(z\)</span> scores are truly marginally <span class="math inline">\(N\left(0, 1\right)\)</span>. <a href="correlated_z.html">We’ve seen</a> that their historgram could be far from normal. Further more, <span class="math inline">\(z\)</span> scores in different data sets are distorted by different correlation structures. Therefore, we don’t have replicates here; that is, each data set is one single realization of a lot of random variables under correlation.</p> <p>For our data sets in particular, let <span class="math inline">\(Z = \left[z_{ij}\right]_{m \times n}\)</span> be the matrix of <span class="math inline">\(z\)</span> scores. Each <span class="math inline">\(z_{ij}\)</span> denotes the gene differential expression <span class="math inline">\(z\)</span> score for gene <span class="math inline">\(j\)</span> in the data set <span class="math inline">\(i\)</span>. Since all of these <span class="math inline">\(z\)</span> scores are obtained from the same tissue, theoretically they should all be marginally <span class="math inline">\(N\left(0, 1\right)\)</span>.</p> <p>Each row is a data set, consisting of <span class="math inline">\(10K\)</span> realized <span class="math inline">\(z\)</span> scores presumably marginally <span class="math inline">\(N\left(0, 1\right)\)</span>, whose empirical distribution distorted by correlation. If we plot the histogram of each row, it is grossly not <span class="math inline">\(N\left(0, 1\right)\)</span> due to correlation. Therefore, it’s not easy to verify that they are truly marginally <span class="math inline">\(N\left(0, 1\right)\)</span>.</p> <p>Here are two pieces of evidence that they are. Let’s take a look one by one, compared with the independent <span class="math inline">\(z\)</span> scores case.</p> <pre class="r"><code>z.null <- read.table("../output/z_null_liver_777.txt") n = ncol(z.null) m = nrow(z.null)</code></pre> <pre class="r"><code>set.seed(777) z.sim = matrix(rnorm(m * n), nrow = m, ncol = n)</code></pre> </div> <div id="row-wise-eleftf_nleftzrightright-phileftzright" class="section level2"> <h2>Row-wise: <span class="math inline">\(E\left[F_n\left(z\right)\right] = \Phi\left(z\right)\)</span></h2> <p>Let <span class="math inline">\(F_n^{R_i}\left(z\right)\)</span> be the empirical CDF of <span class="math inline">\(p\)</span> correlated <span class="math inline">\(z\)</span> scores in row <span class="math inline">\(i\)</span>. For any <span class="math inline">\(i\)</span>, <span class="math inline">\(F_n^{R_i}\left(z\right)\)</span> should be conspicuously different from <span class="math inline">\(\Phi\left(z\right)\)</span>, yet on average, <span class="math inline">\(E\left[F_n^{R_i}\left(z\right)\right]\)</span> should be equal to <span class="math inline">\(\Phi\left(z\right)\)</span>, if all <span class="math inline">\(z\)</span> scores are marginally <span class="math inline">\(N\left(0, 1\right)\)</span>.</p> <p>In order to check that, we can borrow <a href="index.html">Prof. Michael Stein’s insight</a> to look at the tail events, or empirical CDF.</p> <p>For each row, let <span class="math inline">\(\alpha\)</span> be a given probability level, <span class="math inline">\(z_\alpha = \Phi^{-1}\left(\alpha\right)\)</span> be the associated quantile, and we record a number <span class="math inline">\(R_i^\alpha\)</span> defined as follows.</p> <p>If <span class="math inline">\(\alpha \leq 0.5\)</span>, <span class="math inline">\(R_\alpha^i\)</span> is the number of <span class="math inline">\(z\)</span> scores in row <span class="math inline">\(i\)</span> that are smaller than <span class="math inline">\(z_\alpha\)</span>; otherwise, if <span class="math inline">\(\alpha > 0.5\)</span>, <span class="math inline">\(R_\alpha^i\)</span> is the number of <span class="math inline">\(z\)</span> scores in row <span class="math inline">\(i\)</span> that are larger than <span class="math inline">\(z_\alpha\)</span>.</p> <p>Defined this way, <span class="math inline">\(R_\alpha^i\)</span> should be a sample from <span class="math inline">\(n \times F_n^{R_i}\left(z_\alpha\right)\)</span> or <span class="math inline">\(n \times \left(1- F_n^{R_i}\left(z_\alpha\right)\right)\)</span>. We can check if <span class="math inline">\(E\left[F_n\left(z\right)\right] = \Phi\left(z\right)\)</span> by looking at if the average <span class="math display">\[ \bar R_\alpha \approx \begin{cases} n\Phi\left(z_\alpha\right) = n\alpha & \alpha \leq 0.5 \\ n\left(1-\Phi\left(z_\alpha\right)\right) = n\left(1 - \alpha\right) & \alpha > 0.5\end{cases} \ . \]</span> We may also compare the frequencies of <span class="math inline">\(R_\alpha^i\)</span> with their theoretical expected values <span class="math inline">\(m \times \text{Binomial}\left(n, \alpha\right)\)</span> (in blue) assuming <span class="math inline">\(z_{ij}\)</span> are independent.</p> <p><img src="figure/marginal_z.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-17.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-18.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-20.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-5-21.png" width="672" style="display: block; margin: auto;" /></p> <div id="independent-case-row-wise" class="section level3"> <h3>Independent case: row-wise</h3> <p><img src="figure/marginal_z.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-17.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-18.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-20.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-6-21.png" width="672" style="display: block; margin: auto;" /></p> </div> </div> <div id="column-wise-closer-to-nleft0-1right" class="section level2"> <h2>Column-wise: closer to <span class="math inline">\(N\left(0, 1\right)\)</span></h2> <p>Each column of <span class="math inline">\(z\)</span> should be seen as <span class="math inline">\(z\)</span> scores of a non-differentially expressed gene in different data sets. Therefore, column-wise, the empirical distribution <span class="math inline">\(F_m^{C_j}\left(z\right)\)</span> should be closer to <span class="math inline">\(\Phi\left(z\right)\)</span> than <span class="math inline">\(F_m^{R_i}\left(z\right)\)</span>.</p> <p>Similarly, we are plotting <span class="math inline">\(C_\alpha^i\)</span>, compared with their theoretical frequencies as follows.</p> <p><img src="figure/marginal_z.rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-8-17.png" width="672" style="display: block; margin: auto;" /></p> <div id="independent-case-column-wise" class="section level3"> <h3>Independent case: column wise</h3> <p><img src="figure/marginal_z.rmd/unnamed-chunk-9-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-10.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-12.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-14.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-16.png" width="672" style="display: block; margin: auto;" /><img src="figure/marginal_z.rmd/unnamed-chunk-9-17.png" width="672" style="display: block; margin: auto;" /></p> </div> </div> <div id="conclusion" class="section level2"> <h2>Conclusion</h2> <p>The empirical distribution and the indicated marginal distribution of the correlated null <span class="math inline">\(z\)</span> scores are behaving not different from the expectation.</p> <p>Row-wise, the number of tail observations averages to what would be expected from correlated marginally <span class="math inline">\(N\left(0, 1\right)\)</span> random samples, validating Prof. Stein’s intuition.</p> <p>Column-wise, the distribution of the number of tail observations is closer to normal, closer to what would be expected under corelated marginally <span class="math inline">\(N\left(0, 1\right)\)</span>. Moreover, the distribution seems unimodal, and peaked at <span class="math inline">\(m\alpha\)</span> when <span class="math inline">\(\alpha \leq0.5\)</span> or <span class="math inline">\(m\left(1-\alpha\right)\)</span> when <span class="math inline">\(\alpha\geq0.5\)</span>. It suggests that the marginal distribution of the null <span class="math inline">\(z\)</span> scores for a certain gene is usually centered at <span class="math inline">\(0\)</span>, and more often than not, close to <span class="math inline">\(N\left(0, 1\right)\)</span>.</p> </div> <div id="session-information" class="section level2"> <h2>Session information</h2> <!-- Insert the session information into the document --> <pre class="r"><code>sessionInfo()</code></pre> <pre><code>R version 3.3.3 (2017-03-06) Platform: x86_64-apple-darwin13.4.0 (64-bit) Running under: macOS Sierra 10.12.4 locale: [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8 attached base packages: [1] stats graphics grDevices utils datasets methods base loaded via a namespace (and not attached): [1] backports_1.0.5 magrittr_1.5 rprojroot_1.2 tools_3.3.3 [5] htmltools_0.3.5 yaml_2.1.14 Rcpp_0.12.10 stringi_1.1.2 [9] rmarkdown_1.3 knitr_1.15.1 git2r_0.18.0 stringr_1.2.0 [13] digest_0.6.11 evaluate_0.10 </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://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"; document.getElementsByTagName("head")[0].appendChild(script); })(); </script> </body> </html>