Last updated: 2021-04-13

Checks: 7 0

Knit directory: mr_mash_test/

This reproducible R Markdown analysis was created with workflowr (version 1.6.2). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.

R Markdown file: up-to-date

Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

Environment: empty

Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

Seed: set.seed(20200328)

The command set.seed(20200328) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

Session information: recorded

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Cache: none

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File paths: relative

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Repository version: bf711b8

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The results in this page were generated with repository version bf711b8. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.

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Untracked files:
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    Untracked:  code/20200502_Prepare_ED_prior.ipynb
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    Untracked:  dsc/mvreg_all_genes_prior_09_11_20_COPY.dsc

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These are the previous versions of the repository in which changes were made to the R Markdown (analysis/mrmash_larger_r_issue.Rmd) and HTML (docs/mrmash_larger_r_issue.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.

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Rmd bf711b8 fmorgante 2021-04-13 Add null data lfsr
html f3b730a fmorgante 2021-04-13 Build site.
Rmd c544bf2 fmorgante 2021-04-13 Add null data results
html 2034558 fmorgante 2021-04-13 Build site.
Rmd 0091cf6 fmorgante 2021-04-13 Add more mash results
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Rmd 5361380 fmorgante 2021-03-25 Add a few things
html cde23ac fmorgante 2021-03-24 Build site.
Rmd d5846a6 fmorgante 2021-03-24 Add investigation on the issue with larger r 
library(mr.mash.alpha)
library(glmnet)
library(mashr)

options(stringsAsFactors = FALSE)

###Functions to compute g-lasso coefficients
compute_coefficients_glasso <- function(X, Y, standardize, nthreads, version=c("Rcpp", "R")){
  
  version <- match.arg(version)
  
  n <- nrow(X)
  p <- ncol(X)
  r <- ncol(Y)
  Y_has_missing <- any(is.na(Y))
  
  if(Y_has_missing){
    ###Extract per-individual Y missingness patterns
    Y_miss_patterns <- mr.mash.alpha:::extract_missing_Y_pattern(Y)
    
    ###Initialize missing Ys
    muy <- colMeans(Y, na.rm=TRUE)
    for(l in 1:r){
      Y[is.na(Y[, l]), l] <- muy[l]
    }
    
    ###Compute V and its inverse
    V <- mr.mash.alpha:::compute_V_init(X, Y, matrix(0, p, r), method="flash")
    Vinv <- chol2inv(chol(V))
    
    ###Compute expected Y (assuming B=0)
    mu <- matrix(rep(muy, each=n), n, r)
    
    ###Impute missing Ys 
    Y <- mr.mash.alpha:::impute_missing_Y(Y=Y, mu=mu, Vinv=Vinv, miss=Y_miss_patterns$miss, non_miss=Y_miss_patterns$non_miss, 
                                          version=version)$Y
  }
  
  ##Fit group-lasso
  if(nthreads>1){
    doMC::registerDoMC(nthreads)
    paral <- TRUE
  } else {
    paral <- FALSE
  }
  
  cvfit_glmnet <- glmnet::cv.glmnet(x=X, y=Y, family="mgaussian", alpha=1, standardize=standardize, parallel=paral)
  coeff_glmnet <- coef(cvfit_glmnet, s="lambda.min")
  
  ##Build matrix of initial estimates for mr.mash
  B <- matrix(as.numeric(NA), nrow=p, ncol=r)
  
  for(i in 1:length(coeff_glmnet)){
    B[, i] <- as.vector(coeff_glmnet[[i]])[-1]
  }
  
  return(list(B=B, Y=Y))
}

In this website, we want to investigate an issue that happens with larger number of conditions than the 10 conditions we have used in the other investigations. The issue is that mr.mash fails to shrink to 0 coefficients that are actually 0. As we will see, some of them are pretty badly misestimated.

Let’s look at an example with n=150, p=400, r=25, PVE=0.2, p_causal=5, independent residuals, independent predictors, and equal effects across conditions from a single multivariate normal with variance 1.

set.seed(12)

n <- 200
ntest <- 50
p <- 400
p_causal <- 5
r <- 25
X_cor <- 0
pve <- 0.2
r_causal <- list(1:r)
B_cor <- 1
B_scale <- 1
w <- 1

###Simulate V, B, X and Y
out <- simulate_mr_mash_data(n, p, p_causal, r, r_causal, intercepts = rep(1, r),
                             pve=pve, B_cor=B_cor, B_scale=B_scale, w=w,
                             X_cor=X_cor, X_scale=1, V_cor=0)
colnames(out$Y) <- paste0("Y", seq(1, r))
rownames(out$Y) <- paste0("N", seq(1, n))
colnames(out$X) <- paste0("X", seq(1, p))
rownames(out$X) <- paste0("N", seq(1, n))

###Split the data in training and test sets
test_set <- sort(sample(x=c(1:n), size=ntest, replace=FALSE))
Ytrain <- out$Y[-test_set, ]
Xtrain <- out$X[-test_set, ]
Ytest <- out$Y[test_set, ]
Xtest <- out$X[test_set, ]

mr.mash is run standardizing X and dropping mixture components with weight less than \(1e-8\). However, the results look the same without dropping any component. The mixture prior consists only of canonical matrices, scaled by a grid computed as in the mash paper. group-LASSO is used to initialize mr.mash and as a comparison.

standardize <- TRUE
nthreads <- 4
w0_threshold <- 1e-8

S0_can <- compute_canonical_covs(ncol(Ytrain), singletons=TRUE, hetgrid=c(0, 0.25, 0.5, 0.75, 1))

####Complete data####
###Compute grid of variances
univ_sumstats <- compute_univariate_sumstats(Xtrain, Ytrain, standardize=standardize, 
                                             standardize.response=FALSE, mc.cores=nthreads)
grid <- autoselect.mixsd(univ_sumstats, mult=sqrt(2))^2

###Compute prior with only canonical matrices
S0 <- expand_covs(S0_can, grid, zeromat=TRUE)

###Fit mr.mash
out_coeff <- compute_coefficients_glasso(Xtrain, Ytrain, standardize=standardize, nthreads=nthreads, version="Rcpp")
prop_nonzero_glmnet <- sum(out_coeff$B[, 1]!=0)/p
w0 <- c((1-prop_nonzero_glmnet), rep(prop_nonzero_glmnet/(length(S0)-1), (length(S0)-1)))

fit_mrmash <- mr.mash(Xtrain, Ytrain, S0, w0=w0, update_w0=TRUE, update_w0_method="EM", tol=1e-2,
                      convergence_criterion="ELBO", compute_ELBO=TRUE, standardize=standardize, 
                      verbose=FALSE, update_V=TRUE, update_V_method="diagonal", e=1e-8,
                      w0_threshold=w0_threshold, nthreads=nthreads, mu1_init=out_coeff$B)

#####Coefficients plots######
layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                0, 3, 3, 0,
                0, 3, 3, 0), 4, 4, byrow = TRUE))

###Plot estimated vs true coeffcients
##mr.mash
plot(out$B, fit_mrmash$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)
##g-lasso
plot(out$B, out_coeff$B, main="g-lasso", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)

###Plot mr.mash vs g-lasso estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(fit_mrmash$mu1, out_coeff$B, main="mr.mash vs g-lasso", 
     xlab="mr.mash coefficients", ylab="g-lasso coefficients",
     col=colorz, cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version Author Date
cde23ac fmorgante 2021-03-24 
###Identify badly estimated variables
bad_est_1 <- which(out$B==0 & fit_mrmash$mu1 < -0.5, arr.ind = TRUE)
bad_est_2 <- which(out$B==0 & fit_mrmash$mu1 > 0.5, arr.ind = TRUE)
bad_est <- rbind(bad_est_1, bad_est_2)
cat("Bad variables:\n")
Bad variables:
bad_est
     row col
X274 274  10
X347 347  15
X183 183  10
X72   72  13

mr.mash is much better than group-LASSO at correctly estimating the non-zero coefficients. However, it does much worse at shrinking the true null coefficients to 0. In particular, there are a few coeffcients that are pretty badly misestimated. We are going to look at those cases more closely.

###Estimated coefficients by mr.mash for the bad variables
cat("mr.mash estimated coefficients:\n")
mr.mash estimated coefficients:
fit_mrmash$mu1[bad_est[, 1], ]
                Y1            Y2            Y3            Y4            Y5
X274 -2.642841e-05 -2.663867e-05 -2.335354e-05 -2.653733e-05 -2.613332e-05
X347  6.491175e-07  2.916934e-06  1.739985e-06  2.150244e-06  2.321851e-06
X183  1.133417e-04  1.284499e-04  1.203046e-04  1.115160e-04  1.203380e-04
X72   9.529781e-06 -4.624202e-06  1.235475e-06  1.068643e-06  2.780234e-06
                Y6            Y7            Y8            Y9           Y10
X274 -2.233829e-05 -2.511605e-05 -2.896884e-05 -2.870533e-05 -9.092047e-01
X347  1.451095e-06  6.299573e-06  3.501019e-06  6.391154e-07  6.646096e-05
X183  1.200349e-04  1.184511e-04  1.168999e-04  9.990242e-05  5.354676e-01
X72   1.060682e-05  1.326606e-05  7.372047e-06 -2.390740e-06 -6.613187e-04
               Y11           Y12           Y13           Y14           Y15
X274 -2.773517e-05 -2.442013e-05  8.829146e-05 -3.148464e-05  6.064451e-05
X347  5.231139e-06 -3.038223e-05 -2.331779e-04  1.595482e-06 -1.003837e+00
X183  1.290409e-04  1.330501e-04  3.121302e-04  1.422738e-04  3.496837e-04
X72   9.218040e-06 -3.617465e-05  6.008830e-01 -6.688684e-06  1.265758e-03
               Y16           Y17           Y18           Y19           Y20
X274 -2.554938e-05 -2.579409e-05 -2.871577e-05 -2.471882e-05 -2.455460e-05
X347  4.505555e-06  2.149654e-06  1.773809e-06  2.081464e-06 -8.069752e-08
X183  1.223263e-04  1.269347e-04  1.225920e-04  1.272094e-04  1.296489e-04
X72   5.131390e-06  3.996038e-06  2.568455e-06  4.814783e-06  9.623803e-06
               Y21           Y22           Y23           Y24           Y25
X274 -2.592537e-05  4.360263e-04 -2.732129e-05 -1.867228e-05 -2.871367e-05
X347  9.000980e-06 -9.735566e-07  2.812928e-06  3.894336e-06  8.448670e-07
X183  1.170115e-04 -1.310602e-04  1.225248e-04  1.238062e-04  9.362710e-05
X72  -1.681922e-05  3.996730e-04  4.902352e-06  9.337151e-06  1.645090e-05

We can see that the estimated coefficients for each one of those variables look like a singleton structure, where there is an effect in one condition and nowhere else. When we look at the prior weights at convergence, we can see that those singleton matrices get relatively large weight (see below).

###Estimated prior weights by mr.mash for the bad variables
cat("mr.mash estimated prior weights:\n")
mr.mash estimated prior weights:
head(sort(fit_mrmash$w0, decreasing=TRUE), 20)
              null     shared1_grid11 singleton13_grid10 
      0.9346759513       0.0086745867       0.0069429060 
singleton10_grid10 singleton22_grid10 singleton10_grid11 
      0.0065330516       0.0052718769       0.0051348432 
singleton15_grid11 singleton15_grid10     shared1_grid10 
      0.0049436750       0.0047741792       0.0034326125 
 singleton22_grid9  singleton13_grid9 singleton13_grid11 
      0.0030192287       0.0018686026       0.0014915698 
 singleton15_grid9  singleton10_grid9  singleton22_grid8 
      0.0004987422       0.0004802332       0.0003878599 
 singleton13_grid8  singleton12_grid8  singleton24_grid8 
      0.0003738464       0.0002453590       0.0001796324 
 singleton12_grid7 singleton22_grid11 
      0.0001623117       0.0001445351

To see why that is happening, the next step will be to look at the data. While the best thing to do would be to look at the summary statistics from a simple multivariate regression with the same residual covariance matrix as in mr.mash, we are going to look at the univariate summary statistics that are readily available.

###Estimated coefficients and Z-scores by univariate regression for the bad variables
rownames(univ_sumstats$Bhat) <- colnames(out$X)
colnames(univ_sumstats$Bhat) <- colnames(out$Y)
cat("SLR estimated coefficients:\n")
SLR estimated coefficients:
univ_sumstats$Bhat[bad_est[, 1], ]
             Y1          Y2          Y3         Y4         Y5         Y6
X274 -0.1325482 -0.09893288  0.61508065 -0.1165136 -0.1069066 0.35151305
X347 -0.2427989  0.28123165  0.06309551  0.1649556  0.3030400 0.02980773
X183 -0.2428503  0.49439918  0.20907556 -0.2292677  0.2192696 0.17487806
X72   0.3468332 -0.38077183 -0.36862902 -0.2403712 -0.2716020 0.18512322
            Y7           Y8          Y9        Y10        Y11        Y12
X274 0.1024990 -0.454970612 -0.25909065 -1.0881426 -0.2781424  0.0916439
X347 0.4888237  0.446450157 -0.08708036  0.2033015  0.6209967 -0.5022021
X183 0.1270435  0.003659657 -0.29134310  1.1836731  0.5654748  0.3028926
X72  0.2467956  0.150784203 -0.25086360 -0.1810409  0.2310031 -0.3425984
            Y13         Y14        Y15         Y16         Y17         Y18
X274  0.2435822 -0.39061436  0.2271793 0.071133592  0.01441381 -0.58419877
X347 -0.3747620  0.04851022 -1.1350130 0.640749645  0.19264130  0.07550262
X183  0.3618534  0.62647779  0.4556610 0.337172801  0.62552105  0.39018457
X72   1.1717008 -0.33024120  0.5026009 0.009030604 -0.08879668 -0.23306137
             Y19        Y20         Y21         Y22          Y23
X274  0.23015756  0.1516004  0.02275088  0.52279901 -0.226039596
X347  0.15321483 -0.1375850  0.51024287  0.07540335  0.296716744
X183  0.54839476  0.4148681  0.10061303 -0.04286237  0.308742629
X72  -0.01456058  0.1413060 -0.37970711  0.21641262 -0.006988736
            Y24        Y25
X274 0.32427352 -0.3004417
X347 0.19882637 -0.0776774
X183 0.22366030 -0.4267544
X72  0.06677394  0.3970696
###Estimated Z-scores by univariate regression for the bad variables
Zscores <- univ_sumstats$Bhat/univ_sumstats$Shat
cat("SLR estimated Z-scores:\n")
SLR estimated Z-scores:
Zscores[bad_est[, 1], ]
             Y1         Y2         Y3         Y4         Y5         Y6
X274 -0.4365736 -0.3252120  1.7394660 -0.3690672 -0.3517424 1.11406779
X347 -0.8009137  0.9267963  0.1766698  0.5227513  0.9999831 0.09408288
X183 -0.8010841  1.6391949  0.5860324 -0.7271807  0.7224003 0.55252082
X72   1.1466597 -1.2578544 -1.0357761 -0.7625325 -0.8956515 0.58496230
            Y7          Y8         Y9        Y10        Y11        Y12
X274 0.3267747 -1.44776342 -0.8270740 -3.3784413 -0.8473222  0.2965010
X347 1.5706941  1.42027984 -0.2774153  0.6090926  1.9102079 -1.6389006
X183 0.4051023  0.01156436 -0.9305960  3.7011173  1.7357935  0.9828482
X72  0.7881596  0.47683468 -0.8006966 -0.5422600  0.7031937 -1.1126963
           Y13        Y14       Y15        Y16         Y17        Y18
X274  0.732031 -1.1812598  0.696065 0.23658439  0.04131068 -1.9612099
X347 -1.129041  0.1460284 -3.621566 2.16389997  0.55268195  0.2503124
X183  1.089836  1.9086398  1.403033 1.12595782  1.81241787  1.3006204
X72   3.670433 -0.9973537  1.549786 0.03002948 -0.25454953 -0.7740532
             Y19        Y20        Y21        Y22         Y23       Y24
X274  0.71527253  0.4970719  0.0678337  1.7837366 -0.74185722 1.0726073
X347  0.47569908 -0.4510520  1.5332637  0.2546203  0.97512125 0.6560840
X183  1.71812707  1.3676601  0.3000724 -0.1447153  1.01491063 0.7383136
X72  -0.04517348  0.4632679 -1.1370125  0.7319304 -0.02289468 0.2200577
           Y25
X274 -0.932783
X347 -0.240511
X183 -1.328901
X72   1.235480
plot(fit_mrmash$mu1[bad_est[, 1], ], univ_sumstats$Bhat[bad_est[, 1], ], main="mr.mash vs SLR -- bad variables", 
     xlab="mr.mash coefficients", ylab="SLR coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)

Version Author Date
cde23ac fmorgante 2021-03-24

The data (wrongly) supports the singleton structure (i.e., large effect in one condition and smaller effects in the other conditions). mr.mash then estimates the coefficients accordingly, by shrinking to 0 the coefficients in the conditions with smaller effects and leaving the coefficent in the one condition almost unshrunken for two variables and a little shrunken for the other two variables.

If we do not include the singleton matrices in the prior, this behavior does not appear.

S0_can_nosing <- compute_canonical_covs(ncol(Ytrain), singletons=FALSE, hetgrid=c(0, 0.25, 0.5, 0.75, 1))

###Compute prior with only canonical matrices without singletons
S0_nosing <- expand_covs(S0_can_nosing, grid, zeromat=TRUE)

###Fit mr.mash
w0_nosing <- c((1-prop_nonzero_glmnet), rep(prop_nonzero_glmnet/(length(S0_nosing)-1), (length(S0_nosing)-1)))

fit_mrmash_nosing <- mr.mash(Xtrain, Ytrain, S0_nosing, w0=w0_nosing, update_w0=TRUE, update_w0_method="EM", tol=1e-2,
                              convergence_criterion="ELBO", compute_ELBO=TRUE, standardize=standardize, 
                              verbose=FALSE, update_V=TRUE, update_V_method="diagonal", e=1e-8,
                              w0_threshold=w0_threshold, nthreads=nthreads, mu1_init=out_coeff$B)

#####Coefficients plots######
layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                0, 3, 3, 0,
                0, 3, 3, 0), 4, 4, byrow = TRUE))

###Plot estimated vs true coeffcients
##mr.mash
plot(out$B, fit_mrmash_nosing$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)
##g-lasso
plot(out$B, out_coeff$B, main="g-lasso", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)

###Plot mr.mash vs g-lasso estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(fit_mrmash_nosing$mu1, out_coeff$B, main="mr.mash vs g-lasso", 
     xlab="mr.mash coefficients", ylab="g-lasso coefficients",
     col=colorz, cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version Author Date
cde23ac fmorgante 2021-03-24

When comparing the ELBO of the two models, we can see that the ELBO for the model with the singletons is a little larger.

###ELBO with singletons
cat("mr.mash ELBO with singletons:\n")
mr.mash ELBO with singletons:
fit_mrmash$ELBO
[1] -10083.11
###ELBO with singletons
cat("mr.mash ELBO without singletons:\n")
mr.mash ELBO without singletons:
fit_mrmash_nosing$ELBO
[1] -10085.82

However, at least in this example, the variable/condition combinations displaying this problem are not so many. Interestingly, a few conditions (10, 13, 15, 22) are the only ones creating this issue.

badd <- which(abs(fit_mrmash$mu1 - out$B) > 0.05, arr.ind = TRUE)
unique_vars <- unique(badd[,1])
unique_vars <- unique_vars[-unique(which(out$B[unique_vars, ] !=0, arr.ind = TRUE)[, 1])]
unique_vars_bad_y <- badd[which(badd[, 1] %in% unique_vars), ]

unique_vars_bad_y
     row col
X141 141  10
X183 183  10
X274 274  10
X72   72  13
X342 342  13
X357 357  13
X142 142  15
X147 147  15
X332 332  15
X347 347  15
X187 187  22
X192 192  22
X194 194  22
X232 232  22
X327 327  22
plot(out$B[unique_vars_bad_y], fit_mrmash$mu1[unique_vars_bad_y], xlab="True coefficents", ylab="Estimated coefficients", main="mr.mash vs truth\nmost bad variables/conditions")

Version Author Date
446489f fmorgante 2021-04-12
b2c3824 fmorgante 2021-03-25

Let’s what happens if we run mash on the summary statistics. Since predictors are independent, so we should roughly get the same results as mr.mash.

dat_mash <- mash_set_data(univ_sumstats$Bhat, univ_sumstats$Shat)
mash_obj <- mash(dat_mash, S0_can)
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.44 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.03 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.09 seconds.
mash_pm <- get_pm(mash_obj)

cat("mash estimated coefficients:\n")
mash estimated coefficients:
mash_pm[bad_est[, 1], ]
               Y1           Y2           Y3           Y4           Y5
X274 -0.023650792 -0.023650792 -0.023650792 -0.023650792 -0.023650792
X347  0.057239661  0.057239661  0.057239661  0.057239661  0.057239661
X183  0.213057913  0.213057913  0.213057913  0.213057913  0.213057913
X72   0.006652372  0.006652372  0.006652372  0.006652372  0.006652372
               Y6           Y7           Y8           Y9          Y10
X274 -0.023650792 -0.023650792 -0.023650792 -0.023650792 -0.023650792
X347  0.057239661  0.057239661  0.057239661  0.057239661  0.057239661
X183  0.213057913  0.213057913  0.213057913  0.213057913  0.213057913
X72   0.006652372  0.006652372  0.006652372  0.006652372  0.006652372
              Y11          Y12         Y13          Y14          Y15
X274 -0.023650792 -0.023650792 -0.02250090 -0.023650792 -0.023650792
X347  0.057239661  0.057239661  0.05573754  0.057239661  0.057239661
X183  0.213057913  0.213057913  0.21306298  0.213057913  0.213057913
X72   0.006652372  0.006652372  0.45452574  0.006652372  0.006652372
              Y16          Y17          Y18          Y19          Y20
X274 -0.023650792 -0.023650792 -0.023650792 -0.023650792 -0.023650792
X347  0.057239661  0.057239661  0.057239661  0.057239661  0.057239661
X183  0.213057913  0.213057913  0.213057913  0.213057913  0.213057913
X72   0.006652372  0.006652372  0.006652372  0.006652372  0.006652372
              Y21          Y22          Y23          Y24          Y25
X274 -0.023642208 -0.023650792 -0.023650792 -0.023650792 -0.023650792
X347  0.057577792  0.057239661  0.057239661  0.057239661  0.057239661
X183  0.213058005  0.213057913  0.213057913  0.213057913  0.213057913
X72   0.006514678  0.006652372  0.006652372  0.006652372  0.006652372

mash seems to have a different problem – it estimates that the effects of variables X274, X347, and X183 to be equal across conditions. However, similar behavior to mr.mash is shown for variable X72. Checking the prior weights, we can see that mash wrongly puts a lot of weight on equal effects components.

###Estimated prior weights by mr.mash for the bad variables
cat("mr.mash estimated prior weights:\n")
mr.mash estimated prior weights:
head(sort(get_estimated_pi(mash_obj, "all"), decreasing=TRUE), 20)
     shared1.6           null     shared1.10 singleton13.10     shared1.11 
  0.7600049178   0.2209967909   0.0074711314   0.0061764252   0.0046510891 
singleton21.11   singleton1.1   singleton2.1   singleton3.1   singleton4.1 
  0.0006996457   0.0000000000   0.0000000000   0.0000000000   0.0000000000 
  singleton5.1   singleton6.1   singleton7.1   singleton8.1   singleton9.1 
  0.0000000000   0.0000000000   0.0000000000   0.0000000000   0.0000000000 
 singleton10.1  singleton11.1  singleton12.1  singleton13.1  singleton14.1 
  0.0000000000   0.0000000000   0.0000000000   0.0000000000   0.0000000000

Let’s now look at the estimated effects from mash, and compare them to those from mr.mash.

#####Coefficients plots######
layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2), 2, 4, byrow = TRUE))
##mash
plot(out$B, mash_pm, main="mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)

##mr.mash vs mash
plot(fit_mrmash$mu1, mash_pm, main="mr.mash vs mash -- all variables", 
     xlab="mr.mash coefficients", ylab="mash coefficients",
     col=colorz, cex=2, cex.lab=2, cex.main=2, cex.axis=2)
abline(0, 1)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version Author Date
2034558 fmorgante 2021-04-13

In general, mash seems to have more difficulty to shrink true 0 coefficients.

Let’s now compare the two methods on completely null data.

nreps <- 4
mrmash_null_coeff <- vector("list", nreps)
mash_obj_null <- vector("list", nreps)

for(i in 1:nreps){
  ###Simulate null phenotypes
  Ytrain_null <- matrix(rnorm((n-ntest)*r), nrow=(n-ntest), ncol=r)

  ###Compute grid of variances
  univ_sumstats_null <- compute_univariate_sumstats(Xtrain, Ytrain_null, standardize=standardize, 
                                               standardize.response=FALSE, mc.cores=nthreads)
  grid_null <- autoselect.mixsd(univ_sumstats_null, mult=sqrt(2))^2

  ###Compute prior with only canonical matrices
  S0_null <- expand_covs(S0_can, grid_null, zeromat=TRUE)

  ###Fit mr.mash
  out_coeff_null <- compute_coefficients_glasso(Xtrain, Ytrain_null, standardize=standardize, nthreads=nthreads, version="Rcpp")
  prop_nonzero_glmnet_null <- sum(out_coeff_null$B[, 1]!=0)/p
  w0_null <- c((1-prop_nonzero_glmnet_null), rep(prop_nonzero_glmnet_null/(length(S0_null)-1), (length(S0_null)-1)))

  fit_mrmash_null <- mr.mash(Xtrain, Ytrain_null, S0_null, w0=w0_null, update_w0=TRUE, update_w0_method="EM", tol=1e-2,
                        convergence_criterion="ELBO", compute_ELBO=TRUE, standardize=standardize, 
                        verbose=FALSE, update_V=TRUE, update_V_method="diagonal", e=1e-8,
                        w0_threshold=w0_threshold, nthreads=nthreads, mu1_init=out_coeff_null$B)
  mrmash_null_coeff[[i]] <- fit_mrmash_null

  ###Fit mash
  dat_mash_null <- mash_set_data(univ_sumstats_null$Bhat, univ_sumstats_null$Shat)
  mash_obj_null[[i]] <- mash(dat_mash_null, S0_can)
}
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.41 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 4.28 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.17 seconds.
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.45 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.14 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.13 seconds.
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.41 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.70 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.13 seconds.
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.42 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.78 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.13 seconds.
#####Coefficients plots######
layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                3, 3, 4, 4,
                3, 3, 4, 4), 4, 4, byrow = TRUE))

for(i in 1:nreps){
  ##mr.mash vs mash
  plot(mrmash_null_coeff[[i]][[1]], get_pm(mash_obj_null[[i]]), 
       xlab="mr.mash coefficients", ylab="mash coefficients",
       col="red", cex=2, cex.lab=2, cex.main=2, cex.axis=2)
  abline(0, 1)
}

Version Author Date
f3b730a fmorgante 2021-04-13

It turns out that mr.mash does not seem to give more false pasitives than mash – it is actually the opposite. However, the lfsr might be informative about actual false positives in mash.

for(i in 1:nreps){
  print(summary(get_lfsr(mash_obj_null[[i]])))
}
       V1               V2               V3               V4        
 Min.   :0.1418   Min.   :0.1999   Min.   :0.1391   Min.   :0.1767  
 1st Qu.:0.9292   1st Qu.:0.9337   1st Qu.:0.9359   1st Qu.:0.9350  
 Median :0.9520   Median :0.9563   Median :0.9551   Median :0.9571  
 Mean   :0.9244   Mean   :0.9293   Mean   :0.9294   Mean   :0.9296  
 3rd Qu.:0.9627   3rd Qu.:0.9662   3rd Qu.:0.9670   3rd Qu.:0.9663  
 Max.   :0.9879   Max.   :0.9871   Max.   :0.9884   Max.   :0.9859  
       V5                V6               V7               V8        
 Min.   :0.08579   Min.   :0.2306   Min.   :0.1092   Min.   :0.2326  
 1st Qu.:0.88698   1st Qu.:0.9128   1st Qu.:0.9328   1st Qu.:0.9107  
 Median :0.92359   Median :0.9430   Median :0.9560   Median :0.9398  
 Mean   :0.88916   Mean   :0.9135   Mean   :0.9307   Mean   :0.9112  
 3rd Qu.:0.94137   3rd Qu.:0.9571   3rd Qu.:0.9672   3rd Qu.:0.9547  
 Max.   :0.98829   Max.   :0.9881   Max.   :0.9864   Max.   :0.9894  
       V9              V10              V11               V12        
 Min.   :0.1544   Min.   :0.1193   Min.   :0.08225   Min.   :0.2153  
 1st Qu.:0.9354   1st Qu.:0.9345   1st Qu.:0.93602   1st Qu.:0.9340  
 Median :0.9555   Median :0.9561   Median :0.95659   Median :0.9553  
 Mean   :0.9299   Mean   :0.9294   Mean   :0.93056   Mean   :0.9286  
 3rd Qu.:0.9669   3rd Qu.:0.9665   3rd Qu.:0.96674   3rd Qu.:0.9661  
 Max.   :0.9850   Max.   :0.9893   Max.   :0.98796   Max.   :0.9871  
      V13              V14              V15               V16         
 Min.   :0.2457   Min.   :0.1506   Min.   :0.02006   Min.   :0.07406  
 1st Qu.:0.9340   1st Qu.:0.8032   1st Qu.:0.93549   1st Qu.:0.92717  
 Median :0.9563   Median :0.8677   Median :0.95685   Median :0.95107  
 Mean   :0.9305   Mean   :0.8252   Mean   :0.93087   Mean   :0.92229  
 3rd Qu.:0.9657   3rd Qu.:0.8995   3rd Qu.:0.96693   3rd Qu.:0.96176  
 Max.   :0.9884   Max.   :0.9869   Max.   :0.98593   Max.   :0.98606  
      V17              V18               V19              V20        
 Min.   :0.2025   Min.   :0.03426   Min.   :0.2216   Min.   :0.2060  
 1st Qu.:0.9361   1st Qu.:0.92685   1st Qu.:0.9341   1st Qu.:0.9348  
 Median :0.9562   Median :0.95293   Median :0.9560   Median :0.9581  
 Mean   :0.9314   Mean   :0.92342   Mean   :0.9310   Mean   :0.9307  
 3rd Qu.:0.9661   3rd Qu.:0.96426   3rd Qu.:0.9671   3rd Qu.:0.9669  
 Max.   :0.9864   Max.   :0.98517   Max.   :0.9881   Max.   :0.9872  
      V21              V22              V23              V24        
 Min.   :0.2022   Min.   :0.2143   Min.   :0.1581   Min.   :0.1300  
 1st Qu.:0.9201   1st Qu.:0.9343   1st Qu.:0.9330   1st Qu.:0.9205  
 Median :0.9442   Median :0.9562   Median :0.9560   Median :0.9474  
 Mean   :0.9144   Mean   :0.9308   Mean   :0.9299   Mean   :0.9165  
 3rd Qu.:0.9577   3rd Qu.:0.9663   3rd Qu.:0.9664   3rd Qu.:0.9606  
 Max.   :0.9855   Max.   :0.9895   Max.   :0.9883   Max.   :0.9875  
      V25        
 Min.   :0.2032  
 1st Qu.:0.9352  
 Median :0.9559  
 Mean   :0.9300  
 3rd Qu.:0.9664  
 Max.   :0.9846  
       V1               V2               V3               V4        
 Min.   :0.0947   Min.   :0.8252   Min.   :0.6955   Min.   :0.8246  
 1st Qu.:0.9116   1st Qu.:0.9933   1st Qu.:0.9712   1st Qu.:0.9934  
 Median :0.9430   Median :0.9957   Median :0.9791   Median :0.9957  
 Mean   :0.9147   Mean   :0.9930   Mean   :0.9697   Mean   :0.9929  
 3rd Qu.:0.9565   3rd Qu.:0.9967   3rd Qu.:0.9844   3rd Qu.:0.9966  
 Max.   :0.9997   Max.   :1.0000   Max.   :0.9999   Max.   :1.0000  
       V5               V6               V7               V8        
 Min.   :0.8203   Min.   :0.8258   Min.   :0.8232   Min.   :0.8253  
 1st Qu.:0.9868   1st Qu.:0.9933   1st Qu.:0.9934   1st Qu.:0.9932  
 Median :0.9915   Median :0.9958   Median :0.9957   Median :0.9957  
 Mean   :0.9872   Mean   :0.9929   Mean   :0.9929   Mean   :0.9929  
 3rd Qu.:0.9934   3rd Qu.:0.9967   3rd Qu.:0.9966   3rd Qu.:0.9966  
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
       V9              V10              V11              V12        
 Min.   :0.8253   Min.   :0.8253   Min.   :0.8242   Min.   :0.7842  
 1st Qu.:0.9932   1st Qu.:0.9934   1st Qu.:0.9933   1st Qu.:0.9909  
 Median :0.9957   Median :0.9957   Median :0.9958   Median :0.9939  
 Mean   :0.9929   Mean   :0.9929   Mean   :0.9930   Mean   :0.9904  
 3rd Qu.:0.9966   3rd Qu.:0.9967   3rd Qu.:0.9967   3rd Qu.:0.9953  
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
      V13               V14              V15              V16        
 Min.   :0.07464   Min.   :0.8269   Min.   :0.8268   Min.   :0.8209  
 1st Qu.:0.96251   1st Qu.:0.9934   1st Qu.:0.9934   1st Qu.:0.9892  
 Median :0.97618   Median :0.9958   Median :0.9957   Median :0.9929  
 Mean   :0.96119   Mean   :0.9930   Mean   :0.9929   Mean   :0.9895  
 3rd Qu.:0.98213   3rd Qu.:0.9966   3rd Qu.:0.9967   3rd Qu.:0.9945  
 Max.   :0.99993   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
      V17              V18              V19              V20        
 Min.   :0.8242   Min.   :0.8260   Min.   :0.8254   Min.   :0.8249  
 1st Qu.:0.9934   1st Qu.:0.9933   1st Qu.:0.9934   1st Qu.:0.9934  
 Median :0.9957   Median :0.9958   Median :0.9957   Median :0.9957  
 Mean   :0.9929   Mean   :0.9930   Mean   :0.9929   Mean   :0.9930  
 3rd Qu.:0.9966   3rd Qu.:0.9966   3rd Qu.:0.9966   3rd Qu.:0.9967  
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
      V21                V22              V23              V24        
 Min.   :0.002521   Min.   :0.8237   Min.   :0.8254   Min.   :0.8251  
 1st Qu.:0.973019   1st Qu.:0.9933   1st Qu.:0.9933   1st Qu.:0.9935  
 Median :0.981149   Median :0.9957   Median :0.9957   Median :0.9957  
 Mean   :0.966744   Mean   :0.9929   Mean   :0.9929   Mean   :0.9930  
 3rd Qu.:0.986189   3rd Qu.:0.9967   3rd Qu.:0.9966   3rd Qu.:0.9966  
 Max.   :0.996424   Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  
      V25        
 Min.   :0.8245  
 1st Qu.:0.9933  
 Median :0.9958  
 Mean   :0.9929  
 3rd Qu.:0.9967  
 Max.   :1.0000  
       V1               V2                 V3               V4        
 Min.   :0.4998   Min.   :0.008582   Min.   :0.0923   Min.   :0.5022  
 1st Qu.:0.9514   1st Qu.:0.939807   1st Qu.:0.9442   1st Qu.:0.9310  
 Median :0.9658   Median :0.959550   Median :0.9602   Median :0.9478  
 Mean   :0.9524   Mean   :0.942474   Mean   :0.9460   Mean   :0.9327  
 3rd Qu.:0.9730   3rd Qu.:0.968575   3rd Qu.:0.9701   3rd Qu.:0.9593  
 Max.   :0.9996   Max.   :0.994603   Max.   :0.9997   Max.   :0.9995  
       V5               V6               V7               V8        
 Min.   :0.4986   Min.   :0.5173   Min.   :0.4560   Min.   :0.4758  
 1st Qu.:0.9527   1st Qu.:0.9530   1st Qu.:0.9207   1st Qu.:0.9478  
 Median :0.9660   Median :0.9667   Median :0.9447   Median :0.9639  
 Mean   :0.9530   Mean   :0.9538   Mean   :0.9260   Mean   :0.9501  
 3rd Qu.:0.9722   3rd Qu.:0.9729   3rd Qu.:0.9562   3rd Qu.:0.9711  
 Max.   :0.9997   Max.   :0.9996   Max.   :0.9995   Max.   :0.9996  
       V9              V10              V11              V12        
 Min.   :0.5261   Min.   :0.3935   Min.   :0.5017   Min.   :0.5258  
 1st Qu.:0.9511   1st Qu.:0.9399   1st Qu.:0.9505   1st Qu.:0.9499  
 Median :0.9665   Median :0.9580   Median :0.9664   Median :0.9661  
 Mean   :0.9530   Mean   :0.9423   Mean   :0.9528   Mean   :0.9527  
 3rd Qu.:0.9729   3rd Qu.:0.9657   3rd Qu.:0.9726   3rd Qu.:0.9733  
 Max.   :0.9997   Max.   :0.9995   Max.   :0.9997   Max.   :0.9996  
      V13              V14              V15              V16        
 Min.   :0.4916   Min.   :0.4014   Min.   :0.5037   Min.   :0.5218  
 1st Qu.:0.9500   1st Qu.:0.9282   1st Qu.:0.9497   1st Qu.:0.9516  
 Median :0.9659   Median :0.9524   Median :0.9657   Median :0.9665  
 Mean   :0.9528   Mean   :0.9340   Mean   :0.9525   Mean   :0.9532  
 3rd Qu.:0.9729   3rd Qu.:0.9620   3rd Qu.:0.9727   3rd Qu.:0.9731  
 Max.   :0.9996   Max.   :0.9995   Max.   :0.9996   Max.   :0.9997  
      V17              V18              V19              V20        
 Min.   :0.5375   Min.   :0.5035   Min.   :0.4859   Min.   :0.5072  
 1st Qu.:0.9475   1st Qu.:0.9501   1st Qu.:0.9502   1st Qu.:0.9503  
 Median :0.9623   Median :0.9658   Median :0.9662   Median :0.9671  
 Mean   :0.9499   Mean   :0.9534   Mean   :0.9526   Mean   :0.9530  
 3rd Qu.:0.9712   3rd Qu.:0.9729   3rd Qu.:0.9731   3rd Qu.:0.9728  
 Max.   :0.9997   Max.   :0.9997   Max.   :0.9996   Max.   :0.9996  
      V21              V22              V23              V24        
 Min.   :0.4975   Min.   :0.4823   Min.   :0.5121   Min.   :0.5127  
 1st Qu.:0.9510   1st Qu.:0.9498   1st Qu.:0.9513   1st Qu.:0.9504  
 Median :0.9665   Median :0.9659   Median :0.9656   Median :0.9661  
 Mean   :0.9532   Mean   :0.9534   Mean   :0.9526   Mean   :0.9531  
 3rd Qu.:0.9733   3rd Qu.:0.9733   3rd Qu.:0.9731   3rd Qu.:0.9725  
 Max.   :0.9997   Max.   :0.9996   Max.   :0.9996   Max.   :0.9997  
      V25        
 Min.   :0.5104  
 1st Qu.:0.9488  
 Median :0.9662  
 Mean   :0.9530  
 3rd Qu.:0.9731  
 Max.   :0.9996  
       V1               V2               V3               V4         
 Min.   :0.7763   Min.   :0.7834   Min.   :0.7880   Min.   :0.07595  
 1st Qu.:0.9137   1st Qu.:0.9197   1st Qu.:0.9167   1st Qu.:0.90631  
 Median :0.9351   Median :0.9378   Median :0.9354   Median :0.92710  
 Mean   :0.9285   Mean   :0.9319   Mean   :0.9297   Mean   :0.91690  
 3rd Qu.:0.9496   3rd Qu.:0.9510   3rd Qu.:0.9501   3rd Qu.:0.94315  
 Max.   :0.9839   Max.   :0.9819   Max.   :0.9818   Max.   :0.98285  
       V5               V6               V7               V8        
 Min.   :0.7734   Min.   :0.7562   Min.   :0.2977   Min.   :0.8238  
 1st Qu.:0.9110   1st Qu.:0.9143   1st Qu.:0.9129   1st Qu.:0.9162  
 Median :0.9321   Median :0.9352   Median :0.9337   Median :0.9361  
 Mean   :0.9254   Mean   :0.9281   Mean   :0.9250   Mean   :0.9304  
 3rd Qu.:0.9470   3rd Qu.:0.9498   3rd Qu.:0.9468   3rd Qu.:0.9503  
 Max.   :0.9847   Max.   :0.9816   Max.   :0.9856   Max.   :0.9850  
       V9              V10              V11              V12        
 Min.   :0.8145   Min.   :0.7682   Min.   :0.7521   Min.   :0.7331  
 1st Qu.:0.9169   1st Qu.:0.9143   1st Qu.:0.9162   1st Qu.:0.9187  
 Median :0.9367   Median :0.9363   Median :0.9349   Median :0.9348  
 Mean   :0.9312   Mean   :0.9287   Mean   :0.9300   Mean   :0.9303  
 3rd Qu.:0.9494   3rd Qu.:0.9495   3rd Qu.:0.9486   3rd Qu.:0.9499  
 Max.   :0.9860   Max.   :0.9856   Max.   :0.9827   Max.   :0.9855  
      V13              V14              V15              V16        
 Min.   :0.4343   Min.   :0.7583   Min.   :0.4274   Min.   :0.7685  
 1st Qu.:0.8873   1st Qu.:0.9167   1st Qu.:0.8618   1st Qu.:0.9177  
 Median :0.9143   Median :0.9362   Median :0.8942   Median :0.9364  
 Mean   :0.8980   Mean   :0.9300   Mean   :0.8812   Mean   :0.9298  
 3rd Qu.:0.9327   3rd Qu.:0.9502   3rd Qu.:0.9153   3rd Qu.:0.9497  
 Max.   :0.9824   Max.   :0.9858   Max.   :0.9820   Max.   :0.9860  
      V17              V18              V19              V20        
 Min.   :0.7392   Min.   :0.8013   Min.   :0.8169   Min.   :0.6024  
 1st Qu.:0.9170   1st Qu.:0.9184   1st Qu.:0.9173   1st Qu.:0.9069  
 Median :0.9351   Median :0.9360   Median :0.9372   Median :0.9309  
 Mean   :0.9303   Mean   :0.9305   Mean   :0.9315   Mean   :0.9223  
 3rd Qu.:0.9494   3rd Qu.:0.9499   3rd Qu.:0.9498   3rd Qu.:0.9452  
 Max.   :0.9825   Max.   :0.9842   Max.   :0.9835   Max.   :0.9826  
      V21              V22              V23              V24         
 Min.   :0.7855   Min.   :0.5693   Min.   :0.7650   Min.   :0.03912  
 1st Qu.:0.9183   1st Qu.:0.8637   1st Qu.:0.9188   1st Qu.:0.90507  
 Median :0.9345   Median :0.8996   Median :0.9364   Median :0.92537  
 Mean   :0.9295   Mean   :0.8830   Mean   :0.9301   Mean   :0.91677  
 3rd Qu.:0.9498   3rd Qu.:0.9224   3rd Qu.:0.9496   3rd Qu.:0.94356  
 Max.   :0.9817   Max.   :0.9803   Max.   :0.9829   Max.   :0.98370  
      V25        
 Min.   :0.7727  
 1st Qu.:0.9164  
 Median :0.9347  
 Mean   :0.9284  
 3rd Qu.:0.9482  
 Max.   :0.9849  

sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)

Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] mashr_0.2.41        ashr_2.2-51         glmnet_3.0         
[4] Matrix_1.2-18       mr.mash.alpha_0.2-6

loaded via a namespace (and not attached):
 [1] foreach_1.4.4      RcppParallel_4.4.3 rmeta_3.0         
 [4] assertthat_0.2.1   horseshoe_0.2.0    mixsqp_0.3-43     
 [7] yaml_2.2.1         ebnm_0.1-35        pillar_1.4.2      
[10] backports_1.1.4    lattice_0.20-38    quantreg_5.85     
[13] glue_1.4.2         digest_0.6.20      promises_1.0.1    
[16] colorspace_1.4-1   htmltools_0.3.6    httpuv_1.5.1      
[19] plyr_1.8.6         conquer_1.0.2      pkgconfig_2.0.3   
[22] invgamma_1.1       SparseM_1.81       purrr_0.3.4       
[25] mvtnorm_1.1-1      scales_1.1.0       whisker_0.3-2     
[28] later_0.8.0        MatrixModels_0.5-0 git2r_0.26.1      
[31] tibble_2.1.3       ggplot2_3.2.1      lazyeval_0.2.2    
[34] magrittr_1.5       crayon_1.4.1       mcmc_0.9-7        
[37] evaluate_0.14      fs_1.3.1           MASS_7.3-51.4     
[40] truncnorm_1.0-8    tools_3.6.1        doMC_1.3.7        
[43] softImpute_1.4     lifecycle_1.0.0    matrixStats_0.58.0
[46] stringr_1.4.0      MCMCpack_1.5-0     GIGrvg_0.5        
[49] trust_0.1-8        munsell_0.5.0      flashier_0.2.7    
[52] MBSP_1.0           irlba_2.3.3        compiler_3.6.1    
[55] rlang_0.4.10       grid_3.6.1         iterators_1.0.10  
[58] rmarkdown_1.13     gtable_0.3.0       codetools_0.2-16  
[61] abind_1.4-5        reshape2_1.4.3     flashr_0.6-7      
[64] R6_2.5.0           knitr_1.23         dplyr_0.8.3       
[67] workflowr_1.6.2    rprojroot_1.3-2    shape_1.4.4       
[70] stringi_1.4.3      parallel_3.6.1     SQUAREM_2021.1    
[73] Rcpp_1.0.6         tidyselect_1.1.0   xfun_0.8          
[76] coda_0.19-4