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<h1 class="title toc-ignore">MASH v FLASH GTEx analysis: MASH v OHF</h1>
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<p><strong>Last updated:</strong> 2018-08-04</p>
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<p></details></p>
<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>In the <a href="MASHvFLASHgtex2.html">previous analysis</a>, I proposed several workflows for fitting MASH and FLASH objects to GTEx data. Here and in the next few analyses I examine differences among fits. I begin by comparing the MASH fit to the OHF fit.</p>
<p>For the code used to obtain the fits, see the <a href="MASHvFLASHgtex2.html#code">previous analysis</a>.</p>
<pre class="r"><code>library(mashr)</code></pre>
<pre><code>Loading required package: ashr</code></pre>
<pre class="r"><code>devtools::load_all("/Users/willwerscheid/GitHub/flashr/")</code></pre>
<pre><code>Loading flashr</code></pre>
<pre class="r"><code>gtex <- readRDS(gzcon(url("https://github.com/stephenslab/gtexresults/blob/master/data/MatrixEQTLSumStats.Portable.Z.rds?raw=TRUE")))
strong <- t(gtex$strong.z)
fpath <- "./output/MASHvFLASHgtex2/"
m_final <- readRDS(paste0(fpath, "m.rds"))
fl_final <- readRDS(paste0(fpath, "OHF.rds"))
m_lfsr <- t(get_lfsr(m_final))
m_pm <- t(get_pm(m_final))
fl_lfsr <- readRDS(paste0(fpath, "fllfsr.rds"))
fl_lfsr <- fl_lfsr[[1]]
fl_pm <- flash_get_fitted_values(fl_final)</code></pre>
</div>
<div id="introduction-to-plots" class="section level2">
<h2>Introduction to plots</h2>
<p>The main tool I will use in these analyses is a function that plots observed values vs. MASH or FLASH posterior means for a single test (over all 44 conditions).</p>
<p>The observed <span class="math inline">\(z\)</span>-scores are plotted as hollow circles. Posterior means are plotted as triangles if the effect is judged to be significant (LFSR <span class="math inline">\(\le .05\)</span>) and as squares otherwise. Larger triangles indicate lower LFSRs, with the largest triangles corresponding to “highly significant” effects (LFSR <span class="math inline">\(\le .01\)</span>). The posterior means are colored using the GTEx colors used in previous analyses.</p>
<pre class="r"><code>missing.tissues <- c(7, 8, 19, 20, 24, 25, 31, 34, 37)
gtex.colors <- read.table("https://github.com/stephenslab/gtexresults/blob/master/data/GTExColors.txt?raw=TRUE", sep = '\t', comment.char = '')[-missing.tissues, 2]
OHF.colors <- c("tan4", "tan3")
zero.colors <- c("black", gray.colors(19, 0.2, 0.9),
gray.colors(17, 0.95, 1))
plot_test <- function(n, lfsr, pm, method_name) {
plot(strong[, n], pch=1, col="black", xlab="", ylab="", cex=0.6,
ylim=c(min(c(strong[, n], 0)), max(c(strong[, n], 0))),
main=paste0("Test #", n, "; ", method_name))
size = rep(0.6, 44)
shape = rep(15, 44)
signif <- lfsr[, n] <= .05
shape[signif] <- 17
size[signif] <- 1.35 - 15 * lfsr[signif, n]
size <- pmin(size, 1.2)
points(pm[, n], pch=shape, col=as.character(gtex.colors), cex=size)
abline(0, 0)
}
plot_ohf_v_ohl_loadings <- function(n, ohf_fit, ohl_fit, ohl_name,
legend_pos = "bottomright") {
ohf <- abs(ohf_fit$EF[n, ] * apply(abs(ohf_fit$EL), 2, max))
ohl <- -abs(ohl_fit$EF[n, ] * apply(abs(ohl_fit$EL), 2, max))
data <- rbind(c(ohf, rep(0, length(ohl) - 45)),
c(ohl[1:45], rep(0, length(ohf) - 45),
ohl[46:length(ohl)]))
colors <- c("black",
as.character(gtex.colors),
OHF.colors,
zero.colors[1:(length(ohl) - 45)])
x <- barplot(data, beside=T, col=rep(colors, each=2),
main=paste0("Test #", n, " loadings"),
legend.text = c("OHF", ohl_name),
args.legend = list(x = legend_pos, bty = "n", pch="+-",
fill=NULL, border="white"))
text(x[2*(46:47) - 1], min(data) / 10,
labels=as.character(1:2), cex=0.4)
text(x[2*(48:ncol(data))], max(data) / 10,
labels=as.character(1:(length(ohl) - 45)), cex=0.4)
}
compare_methods <- function(lfsr1, lfsr2, pm1, pm2) {
res <- list()
res$first_not_second <- find_A_not_B(lfsr1, lfsr2)
res$lg_first_not_second <- find_large_A_not_B(lfsr1, lfsr2)
res$second_not_first <- find_A_not_B(lfsr2, lfsr1)
res$lg_second_not_first <- find_large_A_not_B(lfsr2, lfsr1)
res$diff_pms <- find_overall_pm_diff(pm1, pm2)
return(res)
}
# Find tests where many conditions are significant according to
# method A but not according to method B.
find_A_not_B <- function(lfsrA, lfsrB) {
select_tests(colSums(lfsrA <= 0.05 & lfsrB > 0.05))
}
# Find tests where many conditions are highly significant according to
# method A but are not significant according to method B.
find_large_A_not_B <- function(lfsrA, lfsrB) {
select_tests(colSums(lfsrA <= 0.01 & lfsrB > 0.05))
}
find_overall_pm_diff <- function(pmA, pmB, n = 4) {
pm_diff <- colSums((pmA - pmB)^2)
return(order(pm_diff, decreasing = TRUE)[1:4])
}
# Get at least four (or min_n) "top" tests.
select_tests <- function(colsums, min_n = 4) {
n <- 45
n_tests <- 0
while (n_tests < min_n && n > 0) {
n <- n - 1
n_tests <- sum(colsums >= n)
}
return(which(colsums >= n))
}</code></pre>
</div>
<div id="significant-for-ohf-not-mash" class="section level2">
<h2>Significant for OHF, not MASH</h2>
<p>By far the most common case is one in which FLASH finds a combination of a small equal effect and a large unique effect, while MASH only finds the unique effect to be significant. (Here, it is possibly relevant to recall that my <a href="MASHvFLASHsims2.html">simulation study</a> suggested that FLASH outperforms MASH when the “true” effect is a combination of an equal effect and a unique effect.) In such cases, FLASH usually (but not always) applies far less shrinkage to the unique effect. Some typical examples follow.</p>
<pre class="r"><code># interesting.tests <- compare_methods(fl_lfsr, m_lfsr, fl_pm, m_pm)
par(mfrow=c(1, 2))
identical.plus.unique <- c(15828, 1480, 15711, 10000)
for (n in identical.plus.unique) {
plot_test(n, fl_lfsr, fl_pm, "OHF")
plot_test(n, m_lfsr, m_pm, "MASH")
}</code></pre>
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<details> <summary><em>Expand here to see past versions of ohf_not_mash-2.png:</em></summary>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a91ab89034a324bfb9412b3142d4418ee3064b30/docs/figure/MASHvOHF.Rmd/ohf_not_mash-2.png" target="_blank">a91ab89</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-02
</td>
</tr>
</tbody>
</table>
<p></details> <img src="figure/MASHvOHF.Rmd/ohf_not_mash-3.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of ohf_not_mash-3.png:</em></summary>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a91ab89034a324bfb9412b3142d4418ee3064b30/docs/figure/MASHvOHF.Rmd/ohf_not_mash-3.png" target="_blank">a91ab89</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-02
</td>
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<p></details> <img src="figure/MASHvOHF.Rmd/ohf_not_mash-4.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of ohf_not_mash-4.png:</em></summary>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a91ab89034a324bfb9412b3142d4418ee3064b30/docs/figure/MASHvOHF.Rmd/ohf_not_mash-4.png" target="_blank">a91ab89</a>
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<td style="text-align:left;">
Jason Willwerscheid
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<td style="text-align:left;">
2018-08-02
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<p></details></p>
</div>
<div id="significant-for-mash-not-ohf" class="section level2">
<h2>Significant for MASH, not OHF</h2>
<p>To understand the typical situation where MASH declares effects to be significant but OHF does not, recall from my <a href="MASHvFLASHgtex2.html">analysis of GTEx workflows</a> that the MASH fit puts large mixture weights on data-driven covariance matrices (around 0.6 on “ED_tPCA” and 0.25 on “ED_PCA_2”) and a comparatively small weight (around 0.1) on the “equal effects” covariance structure. In contrast, the equal effects loading accounts for about 70% of the variance explained by the OHF fit. This, I think, is why OHF is so much more likely to find an equal effect in the tests above. MASH, on the contrary, is more likely to find a pattern of covariance that derives from the data-driven structures. Notice the similarity of the MASH estimates in the following tests (in the last example, signs are reversed):</p>
<pre class="r"><code>par(mfrow=c(1, 2))
mash.covar <- c(2868, 9716, 10368, 9716)
for (n in mash.covar) {
plot_test(n, fl_lfsr, fl_pm, "OHF")
plot_test(n, m_lfsr, m_pm, "MASH")
}</code></pre>
<p><img src="figure/MASHvOHF.Rmd/mash_not_ohf-1.png" width="672" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a91ab89034a324bfb9412b3142d4418ee3064b30/docs/figure/MASHvOHF.Rmd/mash_not_ohf-1.png" target="_blank">a91ab89</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-02
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</tbody>
</table>
<p></details> <img src="figure/MASHvOHF.Rmd/mash_not_ohf-2.png" width="672" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a91ab89034a324bfb9412b3142d4418ee3064b30/docs/figure/MASHvOHF.Rmd/mash_not_ohf-2.png" target="_blank">a91ab89</a>
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<td style="text-align:left;">
Jason Willwerscheid
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2018-08-02
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<p></details> <img src="figure/MASHvOHF.Rmd/mash_not_ohf-3.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of mash_not_ohf-3.png:</em></summary>
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<a href="https://github.com/willwerscheid/MASHvFLASH/blob/a91ab89034a324bfb9412b3142d4418ee3064b30/docs/figure/MASHvOHF.Rmd/mash_not_ohf-3.png" target="_blank">a91ab89</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-02
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</table>
<p></details> <img src="figure/MASHvOHF.Rmd/mash_not_ohf-4.png" width="672" style="display: block; margin: auto;" /></p>
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</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-02
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<p></details></p>
<p>At first, it seems bizarre that effects with posterior means near zero are judged to be significant. My guess is that in each of these cases, the observations follow a pattern that closely matches the data-driven covariance structures, so that MASH can be highly confident in the sign of the effect even if the effect is not very large.</p>
</div>
<div id="different-posterior-means" class="section level2">
<h2>Different posterior means</h2>
<p>I conclude with some examples of tests with a large mean squared difference in posterior means. As might be expected, these are all tests where effect sizes are large for many conditions. Interestingly, MASH applies little to no shrinkage in such cases (even for small effects). FLASH does a better job applying a reasonable amount of shrinkage to small and moderate effects, but can be overly aggressive in shrinking large effects.</p>
<p>In test #2857, for example, FLASH applies a huge amount of shrinkage to the large effects observed in visceral adipose omental tissue and in mammary tissue, shrinking the <span class="math inline">\(z\)</span>-scores from -9.0 to -2.3 and from -8.1 to -2.3, respectively. These are both tissues for which the prior on the unique effect is heavily concentrated near zero (see <a href="MASHvFLASHgtex2.html#priors_on_factors_(steps_2-3)">here</a>), so the results make sense given the FLASH fit, but the shrinkage still seems extreme. I think that this problem could be mitigated by training on a larger subset of random tests (so that some examples of large effects in these tissues get included).</p>
<p>In contrast, the MASH posterior mixture weights for test #2857 are heavily concentrated on the <code>simple_het</code> matrices, which posit only weak correlations among conditions.</p>
<pre class="r"><code>par(mfrow=c(1, 2))
diff.pms <- c(2857, 697, 384, 1680)
for (n in diff.pms) {
plot_test(n, fl_lfsr, fl_pm, "OHF")
plot_test(n, m_lfsr, m_pm, "MASH")
}</code></pre>
<p><img src="figure/MASHvOHF.Rmd/diff_pms-1.png" width="672" style="display: block; margin: auto;" /></p>
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</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-03
</td>
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</tbody>
</table>
<p></details> <img src="figure/MASHvOHF.Rmd/diff_pms-2.png" width="672" style="display: block; margin: auto;" /></p>
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</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-03
</td>
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</table>
<p></details> <img src="figure/MASHvOHF.Rmd/diff_pms-3.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of diff_pms-3.png:</em></summary>
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<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-03
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</table>
<p></details> <img src="figure/MASHvOHF.Rmd/diff_pms-4.png" width="672" style="display: block; margin: auto;" /></p>
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</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-03
</td>
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</tbody>
</table>
<p></details></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] flashr_0.5-12 mashr_0.2-7 ashr_2.2-10
loaded via a namespace (and not attached):
[1] Rcpp_0.12.17 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] tibble_1.4.2 gtable_0.2.0 evaluate_0.10.1
[16] memoise_1.1.0 lattice_0.20-35 rlang_0.2.0
[19] Matrix_1.2-12 foreach_1.4.4 commonmark_1.4
[22] yaml_2.1.17 parallel_3.4.3 ebnm_0.1-12
[25] mvtnorm_1.0-7 xml2_1.2.0 withr_2.1.1.9000
[28] stringr_1.3.0 knitr_1.20 roxygen2_6.0.1.9000
[31] devtools_1.13.4 rprojroot_1.3-2 grid_3.4.3
[34] R6_2.2.2 rmarkdown_1.8 rmeta_3.0
[37] ggplot2_2.2.1 magrittr_1.5 whisker_0.3-2
[40] scales_0.5.0 backports_1.1.2 codetools_0.2-15
[43] htmltools_0.3.6 MASS_7.3-48 assertthat_0.2.0
[46] softImpute_1.4 colorspace_1.3-2 stringi_1.1.6
[49] lazyeval_0.2.1 munsell_0.4.3 doParallel_1.0.11
[52] pscl_1.5.2 truncnorm_1.0-8 SQUAREM_2017.10-1
[55] R.oo_1.21.0 </code></pre>
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