Prediction accuracy

Last updated: 2020-03-30

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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
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")

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).

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
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

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")

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