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<h1 class="title toc-ignore">Creating Null</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-05-08</em></h4>
</div>
<p><strong>Last updated:</strong> 2017-12-21</p>
<p><strong>Code version:</strong> 6e42447</p>
<section id="introduction" class="level2">
<h2>Introduction</h2>
<p>In many of his seminal papers on simultaneous inference, for example, <a href="http://www.tandfonline.com/doi/abs/10.1198/016214501753382129">Efron et al 2001</a>, Efron used some techniques to create null from the raw data. One big concern of this approach is whether the created null could successfully remove the effects of interest whereas keep the distortion due to correlation. Let’s take a look.</p>
<p>Suppose in a simplified experiment, we have a data matrix <span class="math inline">\(X_{N \times 2n} = \left\{X_1, \ldots, X_n, X_{n + 1}, \ldots, X_{2n}\right\}\)</span>, <span class="math inline">\(n\)</span> columns of which, <span class="math inline">\(\left\{X_1, \ldots, X_n\right\}\)</span>, are controls and <span class="math inline">\(n\)</span> columns, <span class="math inline">\(\left\{X_{n + 1}, \ldots, X_{2n}\right\}\)</span>, are cases. We can obtain <span class="math inline">\(N\)</span> <span class="math inline">\(z\)</span> scores, called <span class="math inline">\(z^1\)</span>, from this experimental design.</p>
<p>On the other hand, if we re-label case and control, and let <span class="math inline">\(\left\{X_1, \ldots, X_{n/2}, X_{n + 1}, \ldots, X_{3n / 2}\right\}\)</span> be the controls and <span class="math inline">\(\left\{X_{n / 2 + 1}, \ldots, X_{n}, X_{3n / 2 + 1}, \ldots, X_{2n}\right\}\)</span> the cases, we can obtain another <span class="math inline">\(N\)</span> <span class="math inline">\(z\)</span> scores, called <span class="math inline">\(z^0\)</span>, from this shuffled experimental design.</p>
<p>Our hope is that <span class="math inline">\(z^0\)</span> will make the effects part in <span class="math inline">\(z^1\)</span> to be <span class="math inline">\(0\)</span>, and only keep the correlation-induced inflation or deflation parts.</p>
<p>Here is a way to check this. Suppose there is no difference between the conditions in cases and controls, then the effects part in <span class="math inline">\(z^1\)</span> should be <span class="math inline">\(0\)</span> anyway. Therefore, we may check if <span class="math inline">\(z^0\)</span> and <span class="math inline">\(z^1\)</span> have the same empirical distribution distorted by correlation.</p>
<pre class="r"><code>library(limma)
library(edgeR)
library(qvalue)
library(ashr)
r = read.csv("../data/liver.csv")
r = r[, -(1 : 2)] # remove gene name and description
#extract top g genes from G by n matrix X of expression
top_genes_index = function(g, X) {
return(order(rowSums(X), decreasing = TRUE)[1 : g])
}
lcpm = function (r) {
R = colSums(r)
t(log2(((t(r) + 0.5) / (R + 1)) * 10^6))
}
Y = lcpm(r)
subset = top_genes_index(10000, Y)
Y = Y[subset,]
r = r[subset,]</code></pre>
<pre class="r"><code>counts_to_z = function (counts, condition) {
design = stats::model.matrix(~condition)
dgecounts = edgeR::calcNormFactors(edgeR::DGEList(counts = counts, group = condition))
v = limma::voom(dgecounts, design, plot = FALSE)
lim = limma::lmFit(v)
r.ebayes = limma::eBayes(lim)
p = r.ebayes$p.value[, 2]
t = r.ebayes$t[, 2]
z = sign(t) * qnorm(1 - p/2)
return (z)
}</code></pre>
<p>We are looking at <span class="math inline">\(3\)</span> cases with different case-control sample sizes: <span class="math inline">\(n = 2\)</span>, <span class="math inline">\(n = 10\)</span>, <span class="math inline">\(n = 50\)</span>.</p>
</section>
<section id="vs-2" class="level2">
<h2>2 vs 2</h2>
<p><img src="figure/create_null.Rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-4-10.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="vs-10" class="level2">
<h2>10 vs 10</h2>
<p><img src="figure/create_null.Rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-6-10.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="vs-50" class="level2">
<h2>50 vs 50</h2>
<p><img src="figure/create_null.Rmd/unnamed-chunk-8-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-3.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-4.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-5.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-6.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-7.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-8.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-9.png" width="672" style="display: block; margin: auto;" /><img src="figure/create_null.Rmd/unnamed-chunk-8-10.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="conclusion" class="level2">
<h2>Conclusion</h2>
<p>Larger sample size doesn’t make the empirical distribution of correlated <span class="math inline">\(z\)</span> scores closer to <span class="math inline">\(N\left(0, 1\right)\)</span>.</p>
<p>Merely changing the labels of the null data could generate starkly different empirical distributions of <span class="math inline">\(z\)</span> scores, no matter what sample size <span class="math inline">\(n\)</span> is, although as <span class="math inline">\(n\)</span> gets larger, the difference in the empirical distributions of <span class="math inline">\(z^1\)</span> and <span class="math inline">\(z^0\)</span> seems to get smaller.</p>
<p>Therefore, creating null through various resampling schemes is basically impossible for this application.</p>
</section>
<section id="session-information" class="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 10.13.2
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] compiler_3.4.3 backports_1.1.2 magrittr_1.5 rprojroot_1.3-1
[5] tools_3.4.3 htmltools_0.3.6 yaml_2.1.16 Rcpp_0.12.14
[9] stringi_1.1.6 rmarkdown_1.8 knitr_1.17 git2r_0.20.0
[13] stringr_1.2.0 digest_0.6.13 workflowr_0.8.0 evaluate_0.10.1</code></pre>
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