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Rmd 465b10d fmorgante 2020-05-27 Add case study with only two causal tissues out of five 
library(mr.mash.alpha)
library(glmnet)
Loading required package: Matrix
Loading required package: foreach
Loaded glmnet 2.0-16
###Set options
options(stringsAsFactors = FALSE)

Let’s simulate data with n=600, p=1,000, p_causal=50, r=5, r_causal=2, PVE=0.5, shared effects, independent predictors, and lowly correlated residuals.

###Set seed
set.seed(123)

###Set parameters
n <- 600
p <- 1000
p_causal <- 50
r <- 5
r_causal <- 2

###Simulate V, B, X and Y
out <- mr.mash.alpha:::simulate_mr_mash_data(n, p, p_causal, r, r_causal, intercepts = rep(1, r),
                                             pve=0.5, B_cor=1, B_scale=0.8, X_cor=0, X_scale=0.8, V_cor=0.15)

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=round(n*0.2), replace=FALSE))
Ytrain <- out$Y[-test_set, ]
Xtrain <- out$X[-test_set, ]
Ytest <- out$Y[test_set, ]
Xtest <- out$X[test_set, ]

head(out$B[out$causal_variables, ])
            [,1]        [,2] [,3] [,4] [,5]
[1,]  0.07316118  0.07316118    0    0    0
[2,]  0.44056603  0.44056603    0    0    0
[3,]  0.31699557  0.31699557    0    0    0
[4,] -0.95215593 -0.95215593    0    0    0
[5,]  0.07038939  0.07038939    0    0    0
[6,] -0.92892961 -0.92892961    0    0    0

We build the mixture prior as usual including zero matrix, identity matrix, rank-1 matrices, low, medium and high heterogeneity matrices, shared matrix, each scaled by a grid from 0.1 to 2.1 in steps of 0.2.

grid <- seq(0.1, 2.1, 0.2)
S0 <- mr.mash.alpha:::compute_cov_canonical(ncol(Ytrain), singletons=TRUE, hetgrid=c(0, 0.25, 0.5, 0.75, 0.99), grid, zeromat=TRUE)

We run glmnet with \(\alpha=1\) to obtain an inital estimate for the regression coefficients to provide to mr.mash, and for comparison.

###Fit grop-lasso to initialize mr.mash
cvfit_glmnet <- cv.glmnet(x=Xtrain, y=Ytrain, family="mgaussian", alpha=1)
coeff_glmnet <- coef(cvfit_glmnet, s="lambda.min")
Bhat_glmnet <- matrix(as.numeric(NA), nrow=p, ncol=r)
for(i in 1:length(coeff_glmnet)){
  Bhat_glmnet[, i] <- as.vector(coeff_glmnet[[i]])[-1]
}

We run mr.mash with EM updates of the mixture weights, updating V, and initializing the regression coefficients with the estimates from glmnet.

###Fit mr.mash
fit_mrmash <- mr.mash(Xtrain, Ytrain, S0, update_w0=TRUE, update_w0_method="EM", 
                    compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                    e=1e-8, ca_update_order="consecutive", mu1_init=Bhat_glmnet)
Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 8.890438 minutes!

Let’s now compare the results.

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)
##glmnet
plot(out$B, Bhat_glmnet, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)

###Plot mr.mash vs glmnet 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(Bhat_glmnet, fit_mrmash$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=2, cex.main=2, cex.axis=2)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version Author Date
f2b6bf8 fmorgante 2020-05-27
7dad46c fmorgante 2020-05-27

As we can see, mr.mash performs pretty poorly. One possibility is that the prior used cannot capture the pattern of sharing present only among the first two tissues. Let’s now test this hypothesis by adding the correspondent prior matrices to the mixture and re-run mr.mash.

###Update the mixture prior
K <- length(S0)
matr <- matrix(c(1, 1, 0, 0, 0,
                 1, 1, 0, 0, 0,
                 0, 0, 0, 0, 0,
                 0, 0, 0, 0, 0,
                 0, 0, 0, 0, 0), 5, 5, byrow=T)

S0_1 <- S0
for(i in 1:length(grid)){
  S0_1[[K+i]] <- matr * grid[i]
}
print(paste("Length of the original mixture:", K))
[1] "Length of the original mixture: 111"
print(paste("Length of the updated mixture:", length(S0_1)))
[1] "Length of the updated mixture: 122"
###Fit mr.mash
fit_mrmash_updated <- mr.mash(Xtrain, Ytrain, S0_1, update_w0=TRUE, update_w0_method="EM", 
                              compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                              e=1e-8, ca_update_order="consecutive", mu1_init=Bhat_glmnet)
Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 5.530349 minutes!

Let’s compare the results again.

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$B, fit_mrmash_updated$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients", cex=2, cex.lab=2, cex.main=2, cex.axis=2)
##glmnet
plot(out$B, Bhat_glmnet, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=2, cex.main=2, cex.axis=2)

###Plot mr.mash vs glmnet 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(Bhat_glmnet, fit_mrmash_updated$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=2, cex.main=2, cex.axis=2)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version Author Date
7dad46c fmorgante 2020-05-27

Now the results look much better!


sessionInfo()
R version 3.5.1 (2018-07-02)
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] glmnet_2.0-16        foreach_1.4.4        Matrix_1.2-15       
[4] mr.mash.alpha_0.1-72

loaded via a namespace (and not attached):
 [1] MBSP_1.0           Rcpp_1.0.4.6       compiler_3.5.1    
 [4] later_0.7.5        git2r_0.26.1       workflowr_1.6.1   
 [7] iterators_1.0.10   tools_3.5.1        digest_0.6.25     
[10] evaluate_0.12      lattice_0.20-38    GIGrvg_0.5        
[13] yaml_2.2.1         SparseM_1.77       mvtnorm_1.0-12    
[16] coda_0.19-3        stringr_1.4.0      knitr_1.20        
[19] fs_1.3.1           MatrixModels_0.4-1 rprojroot_1.3-2   
[22] grid_3.5.1         glue_1.4.0         R6_2.4.1          
[25] rmarkdown_1.10     mixsqp_0.3-43      irlba_2.3.3       
[28] magrittr_1.5       whisker_0.3-2      codetools_0.2-15  
[31] backports_1.1.5    promises_1.0.1     htmltools_0.3.6   
[34] matrixStats_0.55.0 mcmc_0.9-6         MASS_7.3-51.1     
[37] httpuv_1.4.5       quantreg_5.36      stringi_1.4.3     
[40] MCMCpack_1.4-4