mr.mash issue with larger number of conditions

Last updated: 2021-04-13

Checks: 7 0

Knit directory: mr_mash_test/

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Repository version: c544bf2

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Untracked files:
    Untracked:  analysis/dense_issue_investigation2/
    Untracked:  code/20200502_Prepare_ED_prior.ipynb
    Untracked:  code/plot_test.R
    Untracked:  code/pred_fullV_LNP1.R
    Untracked:  dsc/mvreg_all_genes_prior_09_11_20_COPY.dsc

Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.

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.

File	Version	Author	Date	Message
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
html	446489f	fmorgante	2021-04-12	Build site.
Rmd	0fe4f50	fmorgante	2021-04-12	Add mash results
html	b2c3824	fmorgante	2021-03-25	Build site.
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 1.56 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.23 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.10 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_null_pm <- 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$B)
  mrmash_null_coeff[[i]] <- fit_mrmash_null$mu1

  ###Fit mash
  dat_mash_null <- mash_set_data(univ_sumstats_null$Bhat, univ_sumstats_null$Shat)
  mash_obj_null <- mash(dat_mash_null, S0_can)
  mash_null_pm[[i]] <- get_pm(mash_obj_null)

}

 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.51 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 4.44 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.17 seconds.
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.48 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.31 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.14 seconds.
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.56 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.89 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.14 seconds.
 - Computing 400 x 451 likelihood matrix.
 - Likelihood calculations took 1.46 seconds.
 - Fitting model with 451 mixture components.
 - Model fitting took 3.95 seconds.
 - Computing posterior matrices.
 - Computation allocated took 0.14 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]], mash_null_pm[[i]], 
       xlab="mr.mash coefficients", ylab="mash coefficients",
       col="red", cex=2, cex.lab=2, cex.main=2, cex.axis=2)
  abline(0, 1)
}

It turns out that mr.mash does not seem to give more false pasitives than mash – it is actually the opposite.

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

mr.mash issue with larger number of conditions

Fabio Morgante

April 13, 2021