##
Main code and output

npts <- 10000
for (i in 1:2) {
for (j in 1:5) {
C <- c1[i,j]
da <- d_a1[i,j]
ret <- GetLimits(da,C)
LL <- ret$LL;UL <- ret$UL
vc <- seq (LL, UL, length.out = npts)
TPF <- TruePositiveFraction (vc, da, C)
FPF <- FalsePositiveFraction (vc, da, C)
FPF <- rev(FPF);TPF <- rev(TPF)
df2 <- data.frame(FPF = FPF, TPF = TPF)
# do integral numerically
numAuc <- trapz(FPF, TPF)
# Implement Eqn. 36 from Metz-Pan paper
rho <- -(1-C^2)/(1+C^2);sigma <- rbind(c(1, rho), c(rho, 1))
lower <- rep(-Inf,2);upper <- c(-da/sqrt(2),0)
aucProproc <- pnorm(da/sqrt(2)) + 2 * pmvnorm(lower, upper, sigma = sigma)
aucProproc <- as.numeric(aucProproc)
cat("i = ", i,"j = ", j,"C = ", C, ", da = ", da, "aucProproc =", aucProproc, "Norm. Diff. = ", (aucProproc-numAuc)/aucProproc,"\n")
}
}
#> i = 1 j = 1 C = -0.1322804 , da = 1.197239 aucProproc = 0.8014164 Norm. Diff. = 3.520017e-08
#> i = 1 j = 2 C = -0.08696513 , da = 1.771176 aucProproc = 0.8947898 Norm. Diff. = 4.741875e-08
#> i = 1 j = 3 C = -0.1444419 , da = 1.481935 aucProproc = 0.8526605 Norm. Diff. = 3.515431e-08
#> i = 1 j = 4 C = 0.08046016 , da = 1.513757 aucProproc = 0.8577776 Norm. Diff. = 4.971428e-08
#> i = 1 j = 5 C = 0.2225588 , da = 1.740157 aucProproc = 0.8909392 Norm. Diff. = 2.699855e-08
#> i = 2 j = 1 C = -0.08174248 , da = 0.6281251 aucProproc = 0.6716574 Norm. Diff. = 2.801793e-08
#> i = 2 j = 2 C = 0.04976448 , da = 0.9738786 aucProproc = 0.7544739 Norm. Diff. = 5.275242e-08
#> i = 2 j = 3 C = -0.1326126 , da = 1.155871 aucProproc = 0.7931787 Norm. Diff. = 3.472577e-08
#> i = 2 j = 4 C = 0.1182226 , da = 1.620176 aucProproc = 0.8740274 Norm. Diff. = 3.922161e-08
#> i = 2 j = 5 C = 0.0781033 , da = 0.8928816 aucProproc = 0.7360989 Norm. Diff. = 3.798459e-08

##
Discussion

Note the close correspondence between the formula, Eqn. 36 in the Metz-Pan paper and the numerical estimate. As a historical note, Eqn. 31 and Eqn. 36 (they differ only in parameterizations) in the referenced publication are provided without proof – it was probably obvious to Prof Metz or he wanted to leave it to us “mere mortals” to figure it out, as a final parting gesture of his legacy. The author once put a significant effort into proving it and even had a bright graduate student from the biostatistics department work on it to no avail. The author has observed that these equations always yield very close to the numerical estimates, to within numerical precisions, so the theorem is correct empirically, but he has been unable to prove it analytically. It is left as an exercise for a gifted reader to prove/disprove these equations.