Last updated: 2018-09-15

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Introduction

This analysis was prompted by Dey and Stephen’s claim that data imputation (as performed by, for example, FLASH) “grossly distorts” correlation estimates.

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.

This observation leads me to wonder whether CorShrink does not proceed from somewhat 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 increases the empirical correlations rather than shrinking them towards zero (or towards some other mean).

Example

Take the case where effects are strongly correlated across all conditions, but effect sizes are small. (This is, of course, a quite realistic scenario.)

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)

CorShrink

CorShrink finds no correlations in such data.

out <- CorShrink::CorShrinkData(data, image="both")

Expand here to see past versions of corshrink-1.png:
Version Author Date
5dac94e Jason Willwerscheid 2018-09-15

Allowing the mode to be non-zero does not greatly improve matters.

out <- CorShrink::CorShrinkData(data, image="both",
                                ash.control = list(mode = "estimate"))

Expand here to see past versions of corshrink2-1.png:
Version Author Date
5dac94e Jason Willwerscheid 2018-09-15

FLASH

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.

devtools::load_all("~/GitHub/flashr/")
Loading flashr
devtools::load_all("~/GitHub/ebnm/")
Loading ebnm
fl <- flash(data, verbose=FALSE)
fl
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
barplot(fl$ldf$f[, 1], main="Factor values for factor/loading 1")

Expand here to see past versions of flash-1.png:
Version Author Date
5dac94e Jason Willwerscheid 2018-09-15

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] 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

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