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options(stringsAsFactors = FALSE)

accuracy <- function(Y, Yhat) {
  bias <- rep(NA, ncol(Y))
  r2 <- rep(NA, ncol(Y))
  mse <- rep(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))
}

Simulation 1 – Shared effects, independent variables

dat1 <- readRDS("output/fit_mr_mash_n600_p1000_p_caus50_r5_pve0.5_sigmaoffdiag1_sigmascale0.8_gammaoffdiag0_gammascale0.8_Voffdiag0.2_Vscale0_updatew0TRUE_updatew0TRUE_updatew0methodmixsqp_updateVTRUE.rds")
n1 <- dat1$params$n
p1 <- dat1$params$p
p_causal1 <- dat1$params$p_causal
r1 <- dat1$params$r
k1 <- length(dat1$fit$w0)
pve1 <- dat1$params$pve
prop_testset1 <- dat1$params$prop_testset
B1 <- dat1$inputs$B
V1 <- dat1$inputs$V
Sigma1 <- dat1$inputs$Sigma
Gamma1 <- dat1$inputs$Gamma
Ytrain1 <- dat1$Ytrain
Ytest1 <- dat1$Ytest
mu11 <- dat1$fit$mu1
fitted1 <- dat1$fit$fitted
Yhat_test1 <- dat1$Yhat_test

The results below are based on simulation with 600 samples, 1000 variables of which 50 were causal, 5 responses with a per-response proportion of variance explained (PVE) of 0.5. Variables, X, were drawn from MVN(0, Gamma), causal effects, B, were drawn from MVN(0, Sigma). The responses, Y, were drawn from MN(XB, I, V).

cat("Gamma (First 5 elements)")
Gamma (First 5 elements)
Gamma1[1:5, 1:5]
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.0  0.0  0.0  0.0
[2,]  0.0  0.8  0.0  0.0  0.0
[3,]  0.0  0.0  0.8  0.0  0.0
[4,]  0.0  0.0  0.0  0.8  0.0
[5,]  0.0  0.0  0.0  0.0  0.8
cat("Sigma")
Sigma
Sigma1
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.8  0.8  0.8  0.8
[2,]  0.8  0.8  0.8  0.8  0.8
[3,]  0.8  0.8  0.8  0.8  0.8
[4,]  0.8  0.8  0.8  0.8  0.8
[5,]  0.8  0.8  0.8  0.8  0.8
cat("V")
V
V1
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 25.55836  0.00000  0.00000  0.00000  0.00000
[2,]  0.00000 25.55836  0.00000  0.00000  0.00000
[3,]  0.00000  0.00000 25.55836  0.00000  0.00000
[4,]  0.00000  0.00000  0.00000 25.55836  0.00000
[5,]  0.00000  0.00000  0.00000  0.00000 25.55836

mr.mash was fitted to the training data (80% of the data) updating V and updating the prior weights using mixSQP. Then, responses were predicted on the test data (20% of the data). The mixture prior consisted of 101 components.

In the plots below, each color/symbol defines a diffrent response.

Here, we compare the estimated effects with the true effects.

plot(B1[, 1], mu11[, 1], xlab="True effects", ylab="Estimated effects", main="True vs Estimated Effects", pch=1, cex.lab=1.5, cex.axis=1.5)
points(B1[, 2], mu11[, 2], col="blue", pch=2)
points(B1[, 3], mu11[, 3], col="red", pch=3)
points(B1[, 4], mu11[, 4], col="green", pch=4)
points(B1[, 5], mu11[, 5], col="yellow", pch=8)

Version Author Date
8621d7d fmorgante 2020-03-30
4e9aad6 fmorgante 2020-03-30
7911e81 fmorgante 2020-03-30
e1707b2 fmorgante 2020-03-30

Then, we compare the predicted responses with the true responses in the training data (left panel) and test data (right panel).

par(mfrow=c(1,2))
plot(Ytrain1[, 1], fitted1[, 1], xlab="True responses", ylab="Fitted values", main="True vs Fitted values \nTraining data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytrain1[, 2], fitted1[, 2], col="blue", pch=2)
points(Ytrain1[, 3], fitted1[, 3], col="red", pch=3)
points(Ytrain1[, 4], fitted1[, 4], col="green", pch=4)
points(Ytrain1[, 5], fitted1[, 5], col="yellow", pch=8)
abline(0, 1)

plot(Ytrain1[, 1], fitted1[, 1], xlab="True responses", ylab="Predicted responses", main="True vs Predicted Responses \nTest data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytest1[, 2], Yhat_test1[, 2], col="blue", pch=2)
points(Ytest1[, 3], Yhat_test1[, 3], col="red", pch=3)
points(Ytest1[, 4], Yhat_test1[, 4], col="green", pch=4)
points(Ytest1[, 5], Yhat_test1[, 5], col="yellow", pch=8)
abline(0, 1)

Version Author Date
8621d7d fmorgante 2020-03-30
4e9aad6 fmorgante 2020-03-30
7911e81 fmorgante 2020-03-30 
par(mfrow=c(1,1))

r2_train1 <- round(accuracy(Ytrain1, fitted1)$r2, 4)
r2_test1 <- round(accuracy(Ytest1, Yhat_test1)$r2, 4)
bias_train1 <- round(accuracy(Ytrain1, fitted1)$bias, 4)
bias_test1 <- round(accuracy(Ytest1, Yhat_test1)$bias, 4)
mse_train1 <- round(accuracy(Ytrain1, fitted1)$mse, 4)
mse_test1 <- round(accuracy(Ytest1, Yhat_test1)$mse, 4)

acc1 <- rbind(r2_train1, r2_test1, bias_train1, bias_test1, mse_train1, mse_test1)
colnames(acc1) <- paste0("Y", seq(1, r1))
part_metric1 <- c("Training data r2", "Test data r2", "Training data bias", "Test data bias" , "Training data MSE" , "Test data MSE")
res1 <- data.frame(part_metric1, acc1)
colnames(res1)[1] <- c("Partition_metric")
rownames(res1) <- NULL
print(res1)
    Partition_metric      Y1      Y2      Y3      Y4      Y5
1   Training data r2  0.5338  0.5418  0.4904  0.5239  0.5620
2       Test data r2  0.4591  0.4476  0.4630  0.4523  0.4237
3 Training data bias  1.0606  1.0586  0.9935  1.0013  1.0971
4     Test data bias  0.9892  1.0660  1.0666  1.0905  0.9659
5  Training data MSE 24.6985 23.8540 25.3913 22.4603 23.9106
6      Test data MSE 24.5054 29.4890 27.1642 29.6734 26.7496

Simulation 2 – Independent effects, independent variables

dat2 <- readRDS("output/fit_mr_mash_n600_p1000_p_caus50_r5_pve0.5_sigmaoffdiag0_sigmascale0.8_gammaoffdiag0_gammascale0.8_Voffdiag0.2_Vscale0_updatew0TRUE_updatew0TRUE_updatew0methodmixsqp_updateVTRUE.rds")
n2 <- dat2$params$n
p2 <- dat2$params$p
p_causal2 <- dat2$params$p_causal
r2 <- dat2$params$r
k2 <- length(dat2$fit$w0)
pve2 <- dat2$params$pve
prop_testset2 <- dat2$params$prop_testset
B2 <- dat2$inputs$B
V2 <- dat2$inputs$V
Sigma2 <- dat2$inputs$Sigma
Gamma2 <- dat2$inputs$Gamma
Ytrain2 <- dat2$Ytrain
Ytest2 <- dat2$Ytest
mu12 <- dat2$fit$mu1
fitted2 <- dat2$fit$fitted
Yhat_test2 <- dat2$Yhat_test

The results below are based on simulation with 600 samples, 1000 variables of which 50 were causal, 5 responses with a per-response proportion of variance explained (PVE) of 0.5. Variables, X, were drawn from MVN(0, Gamma), causal effects, B, were drawn from MVN(0, Sigma). The responses, Y, were drawn from MN(XB, I, V).

cat("Gamma (First 5 elements)")
Gamma (First 5 elements)
Gamma2[1:5, 1:5]
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.0  0.0  0.0  0.0
[2,]  0.0  0.8  0.0  0.0  0.0
[3,]  0.0  0.0  0.8  0.0  0.0
[4,]  0.0  0.0  0.0  0.8  0.0
[5,]  0.0  0.0  0.0  0.0  0.8
cat("Sigma")
Sigma
Sigma2
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.0  0.0  0.0  0.0
[2,]  0.0  0.8  0.0  0.0  0.0
[3,]  0.0  0.0  0.8  0.0  0.0
[4,]  0.0  0.0  0.0  0.8  0.0
[5,]  0.0  0.0  0.0  0.0  0.8
cat("V")
V
V2
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 39.87305  0.00000  0.00000  0.00000  0.00000
[2,]  0.00000 24.41042  0.00000  0.00000  0.00000
[3,]  0.00000  0.00000 27.69452  0.00000  0.00000
[4,]  0.00000  0.00000  0.00000 25.53166  0.00000
[5,]  0.00000  0.00000  0.00000  0.00000 29.00472

mr.mash was fitted to the training data (80% of the data) updating V and updating the prior weights using mixSQP. Then, responses were predicted on the test data (20% of the data). The mixture prior consisted of 101 components.

In the plots below, each color/symbol defines a diffrent response.

Here, we compare the estimated effects with the true effects.

plot(B2[, 1], mu12[, 1], xlab="True effects", ylab="Estimated effects", main="True vs Estimated Effects", pch=1, cex.lab=1.5, cex.axis=1.5)
points(B2[, 2], mu12[, 2], col="blue", pch=2)
points(B2[, 3], mu12[, 3], col="red", pch=3)
points(B2[, 4], mu12[, 4], col="green", pch=4)
points(B2[, 5], mu12[, 5], col="yellow", pch=8)

Version Author Date
4f5c291 fmorgante 2020-03-30

Then, we compare the predicted responses with the true responses in the training data (left panel) and test data (right panel).

par(mfrow=c(1,2))
plot(Ytrain2[, 1], fitted2[, 1], xlab="True responses", ylab="Fitted values", main="True vs Fitted values \nTraining data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytrain2[, 2], fitted2[, 2], col="blue", pch=2)
points(Ytrain2[, 3], fitted2[, 3], col="red", pch=3)
points(Ytrain2[, 4], fitted2[, 4], col="green", pch=4)
points(Ytrain2[, 5], fitted2[, 5], col="yellow", pch=8)
abline(0, 1)

plot(Ytrain2[, 1], fitted2[, 1], xlab="True responses", ylab="Predicted responses", main="True vs Predicted Responses \nTest data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytest2[, 2], Yhat_test2[, 2], col="blue", pch=2)
points(Ytest2[, 3], Yhat_test2[, 3], col="red", pch=3)
points(Ytest2[, 4], Yhat_test2[, 4], col="green", pch=4)
points(Ytest2[, 5], Yhat_test2[, 5], col="yellow", pch=8)
abline(0, 1)

Version Author Date
4f5c291 fmorgante 2020-03-30 
par(mfrow=c(1,1))

r2_train2 <- round(accuracy(Ytrain2, fitted2)$r2, 4)
r2_test2 <- round(accuracy(Ytest2, Yhat_test2)$r2, 4)
bias_train2 <- round(accuracy(Ytrain2, fitted2)$bias, 4)
bias_test2 <- round(accuracy(Ytest2, Yhat_test2)$bias, 4)
mse_train2 <- round(accuracy(Ytrain2, fitted2)$mse, 4)
mse_test2 <- round(accuracy(Ytest2, Yhat_test2)$mse, 4)

acc2 <- rbind(r2_train2, r2_test2, bias_train2, bias_test2, mse_train2, mse_test2)
colnames(acc2) <- paste0("Y", seq(1, r2))
part_metric2 <- c("Training data r2", "Test data r2", "Training data bias", "Test data bias" , "Training data MSE" , "Test data MSE")
res2 <- data.frame(part_metric2, acc2)
colnames(res2)[1] <- c("Partition_metric")
rownames(res2) <- NULL
print(res2)
    Partition_metric      Y1      Y2      Y3      Y4      Y5
1   Training data r2  0.5373  0.5341  0.5146  0.5716  0.5570
2       Test data r2  0.4600  0.3854  0.4395  0.4473  0.4948
3 Training data bias  1.1737  1.1069  1.1264  1.1019  1.1270
4     Test data bias  0.9847  0.9370  1.1124  1.1859  0.9449
5  Training data MSE 38.1690 22.6568 25.8972 21.0610 26.7705
6      Test data MSE 41.8959 28.1329 34.3800 34.3338 30.6942

Simulation 3 – Shared effects, correlated variables

dat3 <- readRDS("output/fit_mr_mash_n600_p1000_p_caus50_r5_pve0.5_sigmaoffdiag1_sigmascale0.8_gammaoffdiag0.5_gammascale0.8_Voffdiag0.2_Vscale0_updatew0TRUE_updatew0TRUE_updatew0methodmixsqp_updateVTRUE.rds")
n3 <- dat3$params$n
p3 <- dat3$params$p
p_causal3 <- dat3$params$p_causal
r3 <- dat3$params$r
k3 <- length(dat3$fit$w0)
pve3 <- dat3$params$pve
prop_testset3 <- dat3$params$prop_testset
B3 <- dat3$inputs$B
V3 <- dat3$inputs$V
Sigma3 <- dat3$inputs$Sigma
Gamma3 <- dat3$inputs$Gamma
Ytrain3 <- dat3$Ytrain
Ytest3 <- dat3$Ytest
mu13 <- dat3$fit$mu1
fitted3 <- dat3$fit$fitted
Yhat_test3 <- dat3$Yhat_test

The results below are based on simulation with 600 samples, 1000 variables of which 50 were causal, 5 responses with a per-response proportion of variance explained (PVE) of 0.5. Variables, X, were drawn from MVN(0, Gamma), causal effects, B, were drawn from MVN(0, Sigma). The responses, Y, were drawn from MN(XB, I, V).

cat("Gamma (First 5 elements)")
Gamma (First 5 elements)
Gamma3[1:5, 1:5]
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.4  0.4  0.4  0.4
[2,]  0.4  0.8  0.4  0.4  0.4
[3,]  0.4  0.4  0.8  0.4  0.4
[4,]  0.4  0.4  0.4  0.8  0.4
[5,]  0.4  0.4  0.4  0.4  0.8
cat("Sigma")
Sigma
Sigma3
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.8  0.8  0.8  0.8
[2,]  0.8  0.8  0.8  0.8  0.8
[3,]  0.8  0.8  0.8  0.8  0.8
[4,]  0.8  0.8  0.8  0.8  0.8
[5,]  0.8  0.8  0.8  0.8  0.8
cat("V")
V
V3
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 13.98626  0.00000  0.00000  0.00000  0.00000
[2,]  0.00000 13.98625  0.00000  0.00000  0.00000
[3,]  0.00000  0.00000 13.98625  0.00000  0.00000
[4,]  0.00000  0.00000  0.00000 13.98625  0.00000
[5,]  0.00000  0.00000  0.00000  0.00000 13.98625

mr.mash was fitted to the training data (80% of the data) updating V and updating the prior weights using mixSQP. Then, responses were predicted on the test data (20% of the data). The mixture prior consisted of 101 components.

In the plots below, each color/symbol defines a diffrent response.

Here, we compare the estimated effects with the true effects.

plot(B3[, 1], mu13[, 1], xlab="True effects", ylab="Estimated effects", main="True vs Estimated Effects", pch=1, cex.lab=1.5, cex.axis=1.5)
points(B3[, 2], mu13[, 2], col="blue", pch=2)
points(B3[, 3], mu13[, 3], col="red", pch=3)
points(B3[, 4], mu13[, 4], col="green", pch=4)
points(B3[, 5], mu13[, 5], col="yellow", pch=8)

Version Author Date
b48d34d fmorgante 2020-03-30
7a2afe7 fmorgante 2020-03-30

Then, we compare the predicted responses with the true responses in the training data (left panel) and test data (right panel).

par(mfrow=c(1,2))
plot(Ytrain3[, 1], fitted3[, 1], xlab="True responses", ylab="Fitted values", main="True vs Fitted values \nTraining data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytrain3[, 2], fitted3[, 2], col="blue", pch=2)
points(Ytrain3[, 3], fitted3[, 3], col="red", pch=3)
points(Ytrain3[, 4], fitted3[, 4], col="green", pch=4)
points(Ytrain3[, 5], fitted3[, 5], col="yellow", pch=8)
abline(0, 1)

plot(Ytrain3[, 1], fitted3[, 1], xlab="True responses", ylab="Predicted responses", main="True vs Predicted Responses \nTest data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytest3[, 2], Yhat_test3[, 2], col="blue", pch=2)
points(Ytest3[, 3], Yhat_test3[, 3], col="red", pch=3)
points(Ytest3[, 4], Yhat_test3[, 4], col="green", pch=4)
points(Ytest3[, 5], Yhat_test3[, 5], col="yellow", pch=8)
abline(0, 1)

Version Author Date
7a2afe7 fmorgante 2020-03-30 
par(mfrow=c(1,1))

r2_train3 <- round(accuracy(Ytrain3, fitted3)$r2, 4)
r2_test3 <- round(accuracy(Ytest3, Yhat_test3)$r2, 4)
bias_train3 <- round(accuracy(Ytrain3, fitted3)$bias, 4)
bias_test3 <- round(accuracy(Ytest3, Yhat_test3)$bias, 4)
mse_train3 <- round(accuracy(Ytrain3, fitted3)$mse, 4)
mse_test3 <- round(accuracy(Ytest3, Yhat_test3)$mse, 4)

acc3 <- rbind(r2_train3, r2_test3, bias_train3, bias_test3, mse_train3, mse_test3)
colnames(acc3) <- paste0("Y", seq(1, r3))
part_metric3 <- c("Training data r2", "Test data r2", "Training data bias", "Test data bias" , "Training data MSE" , "Test data MSE")
res3 <- data.frame(part_metric3, acc3)
colnames(res3)[1] <- c("Partition_metric")
rownames(res3) <- NULL
print(res3)
    Partition_metric      Y1      Y2      Y3      Y4      Y5
1   Training data r2  0.4892  0.5037  0.4710  0.4907  0.5390
2       Test data r2  0.4148  0.4341  0.4220  0.4639  0.4251
3 Training data bias  1.0358  1.0376  0.9876  0.9817  1.0887
4     Test data bias  1.0015  1.0979  1.0552  1.1736  1.0079
5  Training data MSE 14.4128 14.0238 14.5096 12.8917 13.6560
6      Test data MSE 14.0750 15.9244 15.3882 15.9627 14.0499

Simulation 4 – Independent effects, correlated variables

dat4 <- readRDS("output/fit_mr_mash_n600_p1000_p_caus50_r5_pve0.5_sigmaoffdiag0_sigmascale0.8_gammaoffdiag0.5_gammascale0.8_Voffdiag0.2_Vscale0_updatew0TRUE_updatew0TRUE_updatew0methodmixsqp_updateVTRUE.rds")
n4 <- dat4$params$n
p4 <- dat4$params$p
p_causal4 <- dat4$params$p_causal
r4 <- dat4$params$r
k4 <- length(dat4$fit$w0)
pve4 <- dat4$params$pve
prop_testset4 <- dat4$params$prop_testset
B4 <- dat4$inputs$B
V4 <- dat4$inputs$V
Sigma4 <- dat4$inputs$Sigma
Gamma4 <- dat4$inputs$Gamma
Ytrain4 <- dat4$Ytrain
Ytest4 <- dat4$Ytest
mu14 <- dat4$fit$mu1
fitted4 <- dat4$fit$fitted
Yhat_test4 <- dat4$Yhat_test

The results below are based on simulation with 600 samples, 1000 variables of which 50 were causal, 5 responses with a per-response proportion of variance explained (PVE) of 0.5. Variables, X, were drawn from MVN(0, Gamma), causal effects, B, were drawn from MVN(0, Sigma). The responses, Y, were drawn from MN(XB, I, V).

cat("Gamma (First 5 elements)")
Gamma (First 5 elements)
Gamma4[1:5, 1:5]
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.4  0.4  0.4  0.4
[2,]  0.4  0.8  0.4  0.4  0.4
[3,]  0.4  0.4  0.8  0.4  0.4
[4,]  0.4  0.4  0.4  0.8  0.4
[5,]  0.4  0.4  0.4  0.4  0.8
cat("Sigma")
Sigma
Sigma4
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.8  0.0  0.0  0.0  0.0
[2,]  0.0  0.8  0.0  0.0  0.0
[3,]  0.0  0.0  0.8  0.0  0.0
[4,]  0.0  0.0  0.0  0.8  0.0
[5,]  0.0  0.0  0.0  0.0  0.8
cat("V")
V
V4
         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 31.75545  0.00000  0.00000  0.00000  0.00000
[2,]  0.00000 31.47091  0.00000  0.00000  0.00000
[3,]  0.00000  0.00000 14.55202  0.00000  0.00000
[4,]  0.00000  0.00000  0.00000 42.12604  0.00000
[5,]  0.00000  0.00000  0.00000  0.00000 15.37456

mr.mash was fitted to the training data (80% of the data) updating V and updating the prior weights using mixSQP. Then, responses were predicted on the test data (20% of the data). The mixture prior consisted of 101 components.

In the plots below, each color/symbol defines a diffrent response.

Here, we compare the estimated effects with the true effects.

plot(B4[, 1], mu14[, 1], xlab="True effects", ylab="Estimated effects", main="True vs Estimated Effects", pch=1, cex.lab=1.5, cex.axis=1.5)
points(B4[, 2], mu14[, 2], col="blue", pch=2)
points(B4[, 3], mu14[, 3], col="red", pch=3)
points(B4[, 4], mu14[, 4], col="green", pch=4)
points(B4[, 5], mu14[, 5], col="yellow", pch=8)

Version Author Date
378e278 fmorgante 2020-03-30

Then, we compare the predicted responses with the true responses in the training data (left panel) and test data (right panel).

par(mfrow=c(1,2))
plot(Ytrain4[, 1], fitted4[, 1], xlab="True responses", ylab="Fitted values", main="True vs Fitted values \nTraining data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytrain4[, 2], fitted4[, 2], col="blue", pch=2)
points(Ytrain4[, 3], fitted4[, 3], col="red", pch=3)
points(Ytrain4[, 4], fitted4[, 4], col="green", pch=4)
points(Ytrain4[, 5], fitted4[, 5], col="yellow", pch=8)
abline(0, 1)

plot(Ytrain4[, 1], fitted4[, 1], xlab="True responses", ylab="Predicted responses", main="True vs Predicted Responses \nTest data", pch=1, cex.lab=1.5, cex.axis=1.5)
points(Ytest4[, 2], Yhat_test4[, 2], col="blue", pch=2)
points(Ytest4[, 3], Yhat_test4[, 3], col="red", pch=3)
points(Ytest4[, 4], Yhat_test4[, 4], col="green", pch=4)
points(Ytest4[, 5], Yhat_test4[, 5], col="yellow", pch=8)
abline(0, 1)

Version Author Date
378e278 fmorgante 2020-03-30 
par(mfrow=c(1,1))

r2_train4 <- round(accuracy(Ytrain4, fitted4)$r2, 4)
r2_test4 <- round(accuracy(Ytest4, Yhat_test4)$r2, 4)
bias_train4 <- round(accuracy(Ytrain4, fitted4)$bias, 4)
bias_test4 <- round(accuracy(Ytest4, Yhat_test4)$bias, 4)
mse_train4 <- round(accuracy(Ytrain4, fitted4)$mse, 4)
mse_test4 <- round(accuracy(Ytest4, Yhat_test4)$mse, 4)

acc4 <- rbind(r2_train4, r2_test4, bias_train4, bias_test4, mse_train4, mse_test4)
colnames(acc4) <- paste0("Y", seq(1, r4))
part_metric4 <- c("Training data r2", "Test data r2", "Training data bias", "Test data bias" , "Training data MSE" , "Test data MSE")
res4 <- data.frame(part_metric4, acc4)
colnames(res4)[1] <- c("Partition_metric")
rownames(res4) <- NULL
print(res4)
    Partition_metric      Y1      Y2      Y3      Y4      Y5
1   Training data r2  0.4107  0.4811  0.3574  0.5330  0.4272
2       Test data r2  0.3990  0.2959  0.3766  0.3298  0.3542
3 Training data bias  1.1022  1.0512  1.0676  1.0356  1.0803
4     Test data bias  1.0478  0.7843  1.1576  0.9455  0.9463
5  Training data MSE 36.2095 32.5375 17.8351 39.5427 18.2162
6      Test data MSE 36.5641 38.1858 20.4419 62.0936 19.4614

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     

loaded via a namespace (and not attached):
 [1] workflowr_1.6.1 Rcpp_1.0.3      digest_0.6.23   later_0.7.5    
 [5] rprojroot_1.3-2 R6_2.4.1        backports_1.1.5 git2r_0.26.1   
 [9] magrittr_1.5    evaluate_0.12   stringi_1.4.3   fs_1.3.1       
[13] promises_1.0.1  whisker_0.3-2   rmarkdown_1.10  tools_3.5.1    
[17] stringr_1.4.0   glue_1.3.1      httpuv_1.4.5    yaml_2.2.0     
[21] compiler_3.5.1  htmltools_0.3.6 knitr_1.20