Last updated: 2020-06-01

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Rmd ceda278 fmorgante 2020-06-01 Add case study with dense scenario 
library(mr.mash.alpha)
library(glmnet)
Loading required package: Matrix
Loading required package: foreach
Loaded glmnet 2.0-16
###Set options
options(stringsAsFactors = FALSE)

###Function to compute accuracy
compute_accuracy <- function(Y, Yhat) {
  bias <- rep(as.numeric(NA), ncol(Y))
  r2 <- rep(as.numeric(NA), ncol(Y))
  mse <- rep(as.numeric(NA), ncol(Y))
  
  for(i in 1:ncol(Y)){
    fit  <- lm(Y[, i] ~ Yhat[, i])
    bias[i] <- coef(fit)[2] 
    r2[i] <- summary(fit)$r.squared
    mse[i] <- mean((Y[, i] - Yhat[, i])^2)
  }
  
  return(list(bias=bias, r2=r2, mse=mse))
}

Here we investigate why mr.mash does not perform as well in denser scenarios. Let’s simulate data with n=600, p=1,000, p_causal=500, r=5, r_causal=5, PVE=0.5, shared effects, independent predictors, and independent residuals. The models will be fitted to the training data (80% of the full data).

###Set seed
set.seed(123)

###Set parameters
n <- 600
p <- 1000
p_causal <- 500
r <- 5
r_causal <- r
pve <- 0.5
B_cor <- 1
X_cor <- 0
V_cor <- 0

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

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, ]

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, 1), 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 group-lasso
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)
ahat_glmnet <- vector("numeric", r)
for(i in 1:length(coeff_glmnet)){
  Bhat_glmnet[, i] <- as.vector(coeff_glmnet[[i]])[-1]
  ahat_glmnet[i] <- as.vector(coeff_glmnet[[i]])[1]
}
Yhat_test_glmnet <- drop(predict(cvfit_glmnet, newx=Xtest, s="lambda.min"))

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, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                     mu1_init = Bhat_glmnet, w0_threshold=1e-8)
Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 2.450516 minutes!
Yhat_test_mrmash <- predict(fit_mrmash, Xtest)

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=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out$B, Bhat_glmnet, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###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=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2) 

print("Prediction performance with glmnet")
[1] "Prediction performance with glmnet"
compute_accuracy(Ytest, Yhat_test_glmnet)
$bias
[1] 0.7973441 0.7171240 1.1201318 0.9992656 0.9375002

$r2
[1] 0.12560024 0.10044653 0.14357016 0.13204939 0.09516285

$mse
[1] 446.4747 472.2412 509.8068 511.4644 642.4144
print("Prediction performance with mr.mash")
[1] "Prediction performance with mr.mash"
compute_accuracy(Ytest, Yhat_test_mrmash)
$bias
[1] 2.269604 2.196994 3.438804 2.832561 3.030053

$r2
[1] 0.1310534 0.1127552 0.2346756 0.1833728 0.1537069

$mse
[1] 460.6600 471.2208 525.7587 522.0806 649.4204
print("True V (full data)")
[1] "True V (full data)"
out$V
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 294.59   0.00   0.00   0.00   0.00
[2,]   0.00 294.59   0.00   0.00   0.00
[3,]   0.00   0.00 294.59   0.00   0.00
[4,]   0.00   0.00   0.00 294.59   0.00
[5,]   0.00   0.00   0.00   0.00 294.59
print("Estimated V (training data)")
[1] "Estimated V (training data)"
fit_mrmash$V
         Y1       Y2       Y3       Y4       Y5
Y1 534.0861 241.6130 189.3974 175.6404 206.0833
Y2 241.6130 517.0130 217.6493 216.6763 227.4030
Y3 189.3974 217.6493 482.4770 224.0472 212.9069
Y4 175.6404 216.6763 224.0472 473.4047 190.4193
Y5 206.0833 227.4030 212.9069 190.4193 459.0911

It looks mr.mash shrinks the coeffcients too much and the residual covariance is absorbing part of the effects. Let’s now try to fix the residual covariance to the truth (although it is going to be a little different since we are fitting the model to the training set only).

###Fit mr.mash
fit_mrmash_fixV <- mr.mash(Xtrain, Ytrain, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=FALSE,
                     mu1_init = Bhat_glmnet, w0_threshold=1e-8, V=out$V)
Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 3.078888 minutes!
Yhat_test_mrmash_fixV <- predict(fit_mrmash_fixV, Xtest)

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$B, fit_mrmash_fixV$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out$B, Bhat_glmnet, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###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_fixV$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2) 

print("Prediction performance with glmnet")
[1] "Prediction performance with glmnet"
compute_accuracy(Ytest, Yhat_test_glmnet)
$bias
[1] 0.7973441 0.7171240 1.1201318 0.9992656 0.9375002

$r2
[1] 0.12560024 0.10044653 0.14357016 0.13204939 0.09516285

$mse
[1] 446.4747 472.2412 509.8068 511.4644 642.4144
print("Prediction performance with mr.mash")
[1] "Prediction performance with mr.mash"
compute_accuracy(Ytest, Yhat_test_mrmash_fixV)
$bias
[1] 0.3667682 0.4788130 0.5380369 0.5509452 0.4881568

$r2
[1] 0.06928235 0.11834339 0.12732200 0.13692969 0.08741368

$mse
[1] 576.2796 535.1912 571.8264 567.9749 716.4623

Prediction performance looks worse in terms of MSE and \(r^2\) while bias, although still off, is closer to 1. However, looking at the plots of the coefficients, we can see that many coefficients are no longer overshrunk.

For comparison let’s try to repeat the analysis above (updating V) on a sparser scenario, i.e., p_causal=50.

###Set parameters
n <- 600
p <- 1000
p_causal <- 50
r <- 5
r_causal <- r
pve <- 0.5
B_cor <- 1
X_cor <- 0
V_cor <- 0

###Simulate V, B, X and Y
out50 <- mr.mash.alpha:::simulate_mr_mash_data(n, p, p_causal, r, r_causal=r, intercepts = rep(1, r),
                                             pve=pve, B_cor=B_cor, B_scale=0.8, X_cor=X_cor, X_scale=0.8, V_cor=V_cor)

colnames(out50$Y) <- paste0("Y", seq(1, r))
rownames(out50$Y) <- paste0("N", seq(1, n))
colnames(out50$X) <- paste0("X", seq(1, p))
rownames(out50$X) <- paste0("N", seq(1, n))

###Split the data in training and test sets
test_set50 <- sort(sample(x=c(1:n), size=round(n*0.2), replace=FALSE))
Ytrain50 <- out50$Y[-test_set50, ]
Xtrain50 <- out50$X[-test_set50, ]
Ytest50 <- out50$Y[test_set50, ]
Xtest50 <- out50$X[test_set50, ]

###Fit group-lasso
cvfit_glmnet50 <- cv.glmnet(x=Xtrain50, y=Ytrain50, family="mgaussian", alpha=1)
coeff_glmnet50 <- coef(cvfit_glmnet50, s="lambda.min")
Bhat_glmnet50 <- matrix(as.numeric(NA), nrow=p, ncol=r)
ahat_glmnet50 <- vector("numeric", r)
for(i in 1:length(coeff_glmnet50)){
  Bhat_glmnet50[, i] <- as.vector(coeff_glmnet50[[i]])[-1]
  ahat_glmnet50[i] <- as.vector(coeff_glmnet50[[i]])[1]
}
Yhat_test_glmnet50 <- drop(predict(cvfit_glmnet50, newx=Xtest50, s="lambda.min"))

###Fit mr.mash
fit_mrmash50 <- mr.mash(Xtrain50, Ytrain50, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                     mu1_init = Bhat_glmnet50, w0_threshold=1e-8)
Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 0.9045867 minutes!
Yhat_test_mrmash50 <- predict(fit_mrmash50, Xtest50)

Let’s look at 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(out50$B, fit_mrmash50$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out50$B, Bhat_glmnet50, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###Plot mr.mash vs glmnet estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out50$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(Bhat_glmnet50, fit_mrmash50$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2) 

print("Prediction performance with glmnet")
[1] "Prediction performance with glmnet"
compute_accuracy(Ytest50, Yhat_test_glmnet50)
$bias
[1] 1.1055781 1.2342390 0.8929952 1.2093933 1.1997154

$r2
[1] 0.3720343 0.4451268 0.2709590 0.3909355 0.4424055

$mse
[1] 50.11301 49.98023 54.17894 53.96508 46.36995
print("Prediction performance with mr.mash")
[1] "Prediction performance with mr.mash"
compute_accuracy(Ytest50, Yhat_test_mrmash50)
$bias
[1] 0.9295387 1.1205934 0.8382439 0.9741666 1.0034935

$r2
[1] 0.4227190 0.5544520 0.3641726 0.4142314 0.4747928

$mse
[1] 46.04314 39.62819 48.00426 51.00798 42.98684
print("True V (full data)")
[1] "True V (full data)"
out50$V
       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 43.262  0.000  0.000  0.000  0.000
[2,]  0.000 43.262  0.000  0.000  0.000
[3,]  0.000  0.000 43.262  0.000  0.000
[4,]  0.000  0.000  0.000 43.262  0.000
[5,]  0.000  0.000  0.000  0.000 43.262
print("Estimated V (training data)")
[1] "Estimated V (training data)"
fit_mrmash50$V
           Y1        Y2         Y3        Y4         Y5
Y1 37.0783688  2.693322  2.7229456  3.426152 -0.6935028
Y2  2.6933223 42.243137  2.0293575  1.966294  2.6486737
Y3  2.7229456  2.029357 37.9331012 -2.456907 -0.2269764
Y4  3.4261520  1.966294 -2.4569072 42.752140 -3.0717702
Y5 -0.6935028  2.648674 -0.2269764 -3.071770 43.5576891

The results of this simulation look very good in terms of prediction performance, estimation of the coefficients, and estimation of the residual covariance.

Just as an additional test let’s now try to maintain the same signal-to-noise ratio as the first simulation but with fewer total variables, i.e., p=500 and p_causal=250.

###Set parameters
n <- 600
p <- 500
p_causal <- 250
r <- 5
r_causal <- r
pve <- 0.5
B_cor <- 1
X_cor <- 0
V_cor <- 0

###Simulate V, B, X and Y
out500 <- mr.mash.alpha:::simulate_mr_mash_data(n, p, p_causal, r, r_causal=r, intercepts = rep(1, r),
                                             pve=pve, B_cor=B_cor, B_scale=0.8, X_cor=X_cor, X_scale=0.8, V_cor=V_cor)

colnames(out500$Y) <- paste0("Y", seq(1, r))
rownames(out500$Y) <- paste0("N", seq(1, n))
colnames(out500$X) <- paste0("X", seq(1, p))
rownames(out500$X) <- paste0("N", seq(1, n))

###Split the data in training and test sets
test_set500 <- sort(sample(x=c(1:n), size=round(n*0.2), replace=FALSE))
Ytrain500 <- out500$Y[-test_set500, ]
Xtrain500 <- out500$X[-test_set500, ]
Ytest500 <- out500$Y[test_set500, ]
Xtest500 <- out500$X[test_set500, ]

###Fit group-lasso
cvfit_glmnet500 <- cv.glmnet(x=Xtrain500, y=Ytrain500, family="mgaussian", alpha=1)
coeff_glmnet500 <- coef(cvfit_glmnet500, s="lambda.min")
Bhat_glmnet500 <- matrix(as.numeric(NA), nrow=p, ncol=r)
ahat_glmnet500 <- vector("numeric", r)
for(i in 1:length(coeff_glmnet500)){
  Bhat_glmnet500[, i] <- as.vector(coeff_glmnet500[[i]])[-1]
  ahat_glmnet500[i] <- as.vector(coeff_glmnet500[[i]])[1]
}
Yhat_test_glmnet500 <- drop(predict(cvfit_glmnet500, newx=Xtest500, s="lambda.min"))

###Fit mr.mash
fit_mrmash500 <- mr.mash(Xtrain500, Ytrain500, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                     mu1_init = Bhat_glmnet500, w0_threshold=1e-8)
Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 1.160227 minutes!
Yhat_test_mrmash500 <- predict(fit_mrmash500, Xtest500)

Let’s look at 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(out500$B, fit_mrmash500$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out500$B, Bhat_glmnet500, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###Plot mr.mash vs glmnet estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out500$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(Bhat_glmnet500, fit_mrmash500$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2) 

print("Prediction performance with glmnet")
[1] "Prediction performance with glmnet"
compute_accuracy(Ytest500, Yhat_test_glmnet500)
$bias
[1] 0.7460558 0.8379627 0.8827708 1.0150154 0.8988217

$r2
[1] 0.1566151 0.1754140 0.2246133 0.2126951 0.2354353

$mse
[1] 262.5938 307.5913 280.3553 259.2769 265.2623
print("Prediction performance with mr.mash")
[1] "Prediction performance with mr.mash"
compute_accuracy(Ytest500, Yhat_test_mrmash500)
$bias
[1] 0.8007963 1.0543346 0.9437460 0.9812104 1.0269983

$r2
[1] 0.1917870 0.2740736 0.2402338 0.2626540 0.2816563

$mse
[1] 249.0113 269.0810 268.4814 242.8198 247.6815
print("True V (full data)")
[1] "True V (full data)"
out500$V
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 168.3364   0.0000   0.0000   0.0000   0.0000
[2,]   0.0000 168.3364   0.0000   0.0000   0.0000
[3,]   0.0000   0.0000 168.3364   0.0000   0.0000
[4,]   0.0000   0.0000   0.0000 168.3364   0.0000
[5,]   0.0000   0.0000   0.0000   0.0000 168.3364
print("Estimated V (training data)")
[1] "Estimated V (training data)"
fit_mrmash500$V
           Y1         Y2        Y3        Y4         Y5
Y1 190.385218   4.826421  31.00999  22.18042   5.743413
Y2   4.826421 181.999035  16.65374   8.40475   9.590063
Y3  31.009989  16.653736 198.77957  39.23005  35.494926
Y4  22.180421   8.404750  39.23005 209.04480  25.167324
Y5   5.743413   9.590063  35.49493  25.16732 194.837000

In this last simulation, we can see that mr.mash performs pretty well, even in a denser scenario with fewer variables.


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_0.1-75

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