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<h1 class="title toc-ignore">Thoughts on CorShrink</h1>
</div>
<p><strong>Last updated:</strong> 2018-09-15</p>
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<p><details> <summary> <strong style="color:blue;">✔</strong> <strong>Repository version:</strong> <a href="https://github.com/willwerscheid/MASHvFLASH/tree/0b8e07bf59108f26e6d19e2a132641ec16fa63ef" target="_blank">0b8e07b</a> </summary></p>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/0b8e07bf59108f26e6d19e2a132641ec16fa63ef/analysis/FLASHvCorShrink.Rmd" target="_blank">0b8e07b</a>
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<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>This analysis was prompted by <a href="https://www.biorxiv.org/content/early/2018/07/13/368316">Dey and Stephen’s</a> claim that data imputation (as performed by, for example, FLASH) “grossly distorts” correlation estimates.</p>
<p>My main thought is that data imputation produces de-noised estimates. With the GTEx data, FLASH tends to estimate missing data near the mean of the available data (or some multiple thereof). So of course the imputed data will show strong correlations.</p>
<p>This observation leads me to wonder whether CorShrink does not proceed from flawed principles. CorShrink operates on the noisy data itself, which is necessarily less correlated than the corresponding “true” effects (unless errors are strongly correlated, but this does not seem to be the case for the GTEx data). So in fact, it seems desirable to take an approach that <em>increases</em> the empirical correlations rather than shrinking them towards zero (or towards some other mean).</p>
</div>
<div id="example" class="section level2">
<h2>Example</h2>
<p>Take the case where effects are strongly correlated across all conditions, but effect sizes are small. (This is, of course, a quite realistic scenario.)</p>
<pre class="r"><code>n <- 100
p <- 25
# mean of effects is 0.5, sd is 0.2; correlation is 0.8
Sigma <- matrix(0.04 * 0.8, nrow=p, ncol=p)
diag(Sigma) <- 0.04
true_effects <- MASS::mvrnorm(n = n, mu = rep(0.5, p), Sigma = Sigma)
data <- true_effects + matrix(rnorm(n*p), nrow=n, ncol=p)</code></pre>
</div>
<div id="corshrink" class="section level2">
<h2>CorShrink</h2>
<p>CorShrink finds no correlations in such data.</p>
<pre class="r"><code>out <- CorShrink::CorShrinkData(data, image="both")</code></pre>
<p><img src="figure/FLASHvCorShrink.Rmd/corshrink-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>Allowing the mode to be non-zero does not greatly improve matters.</p>
<pre class="r"><code>out <- CorShrink::CorShrinkData(data, image="both",
ash.control = list(mode = "estimate"))</code></pre>
<p><img src="figure/FLASHvCorShrink.Rmd/corshrink2-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="flash" class="section level2">
<h2>FLASH</h2>
<p>FLASH, however, finds a rank-one structure. A truly rank-one structure would of course mean that the data was perfectly correlated, but I think that in this case FLASH comes much closer to the truth than CorShrink.</p>
<pre class="r"><code>devtools::load_all("~/GitHub/flashr/")</code></pre>
<pre><code>Loading flashr</code></pre>
<pre class="r"><code>devtools::load_all("~/GitHub/ebnm/")</code></pre>
<pre><code>Loading ebnm</code></pre>
<pre class="r"><code>fl <- flash(data, verbose=FALSE)
fl</code></pre>
<pre><code>Summary of flash object:
Number of factor/loading pairs: 1
Proportion of variance explained:
Factor/loading 1: 0.162
Value of objective function: -3688.319</code></pre>
<pre class="r"><code>barplot(fl$ldf$f[, 1], main="Factor values for factor/loading 1")</code></pre>
<p><img src="figure/FLASHvCorShrink.Rmd/flash-1.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 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6
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
other attached packages:
[1] ebnm_0.1-14 flashr_0.6-1
loaded via a namespace (and not attached):
[1] Rcpp_0.12.18 pillar_1.2.1 compiler_3.4.3
[4] git2r_0.21.0 plyr_1.8.4 workflowr_1.0.1
[7] R.methodsS3_1.7.1 R.utils_2.6.0 iterators_1.0.9
[10] tools_3.4.3 testthat_2.0.0 digest_0.6.15
[13] corrplot_0.84 tibble_1.4.2 evaluate_0.10.1
[16] memoise_1.1.0 gtable_0.2.0 lattice_0.20-35
[19] rlang_0.2.0 Matrix_1.2-12 foreach_1.4.4
[22] commonmark_1.4 CorShrink_0.1-6 yaml_2.1.17
[25] parallel_3.4.3 gridExtra_2.3 xml2_1.2.0
[28] roxygen2_6.0.1.9000 withr_2.1.1.9000 stringr_1.3.0
[31] knitr_1.20 devtools_1.13.4 rprojroot_1.3-2
[34] grid_3.4.3 glmnet_2.0-13 R6_2.2.2
[37] rmarkdown_1.8 ggplot2_2.2.1 reshape2_1.4.3
[40] corpcor_1.6.9 ashr_2.2-13 magrittr_1.5
[43] whisker_0.3-2 scales_0.5.0 backports_1.1.2
[46] codetools_0.2-15 htmltools_0.3.6 MASS_7.3-48
[49] softImpute_1.4 colorspace_1.3-2 stringi_1.1.6
[52] lazyeval_0.2.1 munsell_0.4.3 doParallel_1.0.11
[55] pscl_1.5.2 truncnorm_1.0-8 SQUAREM_2017.10-1
[58] R.oo_1.21.0 </code></pre>
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