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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">Histogram of correlated <span class="math inline">\(z\)</span> scores, Part 1</h1> <h4 class="author"><em>Lei Sun</em></h4> <h4 class="date"><em>2017-03-06</em></h4> </div> <p><strong>Last updated:</strong> 2017-03-07</p> <p><strong>Code version:</strong> 03366d9</p> <div id="introduction" class="section level2"> <h2>Introduction</h2> <p><a href="http://amstat.tandfonline.com/doi/abs/10.1198/jasa.2010.tm09129">Efron 2010</a> and <a href="http://amstat.tandfonline.com/doi/abs/10.1198/jasa.2010.tm10237">Schwartzman’s comment</a> brings to the center the question “what’s the behavior of <span class="math inline">\(z\)</span> scores under correlation?” Schwartzman pointed out in theory that “the observed histogram is more likely to be narrow than wide, and that it cannot be too wide before it becomes bimodal.” Let’s take a look if this result holds true under our simulation scheme with GTex/Liver data.</p> </div> <div id="histograms-on-randomly-sampled-data-sets" class="section level2"> <h2>Histograms on randomly sampled data sets</h2> <pre class="r"><code>z = read.table("../output/z_null_liver_777.txt")</code></pre> <p>We randomly selected 20 data sets, each with <span class="math inline">\(10000\)</span> z scores, generated by the <a href="nullpipeline.html">null simulation pipeline</a>. Based on each data set we plot two histograms, one using default number of bins and the other <span class="math inline">\(100\)</span> bins. The red line indicates the density of <span class="math inline">\(N(0, 1)\)</span>.</p> <pre class="r"><code>set.seed(777) sample_z = sort(sample(dim(z)[1], 20)) x = seq(- 10, 10, 0.01) y = dnorm(x) for (i in sample_z) { cat("Data Set", i) hist(as.numeric(z[i, ]), xlab = "z scores", freq = FALSE, ylim = c(0, 0.45), main = "10000 z scores, default") lines(x, y, col = "red") hist(as.numeric(z[i, ]), xlab = "z scores", freq = FALSE, ylim = c(0, 0.45), nclass = 100, main = "10000 z scores, 100 bins") lines(x, y, col = "red") }</code></pre> <pre><code>Data Set 11</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-2.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 103</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-4.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 171</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-6.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 247</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-8.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 343</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-10.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 345</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-11.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-12.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 347</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-13.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-14.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 383</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-15.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-16.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 412</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-17.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-18.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 492</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-19.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-20.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 574</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-21.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-22.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 588</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-23.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-24.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 654</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-25.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-26.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 688</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-27.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-28.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 693</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-29.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-30.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 726</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-31.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-32.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 853</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-33.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-34.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 855</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-35.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-36.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 942</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-37.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-38.png" width="672" style="display: block; margin: auto;" /></p> <pre><code>Data Set 993</code></pre> <p><img src="figure/correlated_z.Rmd/unnamed-chunk-2-39.png" width="672" style="display: block; margin: auto;" /><img src="figure/correlated_z.Rmd/unnamed-chunk-2-40.png" width="672" style="display: block; margin: auto;" /></p> </div> <div id="session-information" class="section level2"> <h2>Session Information</h2> <pre class="r"><code>sessionInfo()</code></pre> <pre><code>R version 3.3.2 (2016-10-31) Platform: x86_64-apple-darwin13.4.0 (64-bit) Running under: macOS Sierra 10.12.3 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_3.3.2 [5] htmltools_0.3.5 yaml_2.1.14 Rcpp_0.12.9 stringi_1.1.2 [9] rmarkdown_1.3 knitr_1.15.1 git2r_0.18.0 stringr_1.1.0 [13] digest_0.6.9 workflowr_0.3.0 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'; 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