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<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>
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<p><strong>Last updated:</strong> 2018-05-15</p>
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<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>
<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
loaded via a namespace (and not attached):
[1] workflowr_1.0.1 Rcpp_0.12.16 digest_0.6.15
[4] rprojroot_1.3-2 R.methodsS3_1.7.1 backports_1.1.2
[7] git2r_0.21.0 magrittr_1.5 evaluate_0.10.1
[10] stringi_1.1.6 whisker_0.3-2 R.oo_1.21.0
[13] R.utils_2.6.0 rmarkdown_1.9 tools_3.4.3
[16] stringr_1.3.0 yaml_2.1.18 compiler_3.4.3
[19] htmltools_0.3.6 knitr_1.20 </code></pre>
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