MASH v FLASH GTEx analysis: MASH v OHF

Introduction

In the previous analysis, 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.

For the code used to obtain the fits, see the previous analysis.

library(mashr)

Loading required package: ashr

devtools::load_all("/Users/willwerscheid/GitHub/flashr/")

Loading flashr

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)

Introduction to plots

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).

The observed \(z\)-scores are plotted as hollow circles. Posterior means are plotted as triangles if the effect is judged to be significant (LFSR \(\le .05\)) and as squares otherwise. Larger triangles indicate lower LFSRs, with the largest triangles corresponding to “highly significant” effects (LFSR \(\le .01\)). The posterior means are colored using the GTEx colors used in previous analyses.

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))
}

Significant for OHF, not MASH

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 simulation study 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.

# 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")
}

Significant for MASH, not OHF

To understand the typical situation where MASH declares effects to be significant but OHF does not, recall from my analysis of GTEx workflows 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):

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")
}

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.

Different posterior means

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.

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 \(z\)-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 here), 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).

In contrast, the MASH posterior mixture weights for test #2857 are heavily concentrated on the simple_het matrices, which posit only weak correlations among conditions.

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")
}

Session information

sessionInfo()

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