Estimate ROC curve using binormal distribuiton I am conducting a meta-analysis on diagnostic studies but for each study I have only mean and standard deviation reported. How can I estimate the ROC curve using the binormal assumption in R ?
Tks,
 A: The answer is remarkably simple, assuming that you can be sure that the bi-normal model can be applied. You can find the necessary formulas in the useful NCSS document One ROC Curve and Cutoff Analysis (https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/One_ROC_Curve_and_Cutoff_Analysis.pdf), section 'AUC of a Binormal ROC Curve'. An R function based on these formulas is:
plotROC <- function(mu0, sd0, mu1, sd1){
   b<-sd0/sd1
   a<-(mu1-mu0)/sd1
   x<-seq(0,1 ,length=500)
   y<-1-pnorm(b*qnorm(1-x)-a)
   plot(x, y, lty=2, xlab="FPR", ylab="Sensitivity", 
     main="ROC curve for continuous data", type="l")
   segments(0,0,1,1)
   cp=seq(min(mu0-3*sd0, mu1-3*sd1), max(mu0+3*sd0, mu1+3*sd1), length.out=500)
   Sp = 1-pnorm((mu0-cp)/sigma0)
   Se = pnorm((mu1-cp)/sigma1)
   return(data.frame(cp, Sp, Se))
}

# example
res=plotROC(mu0=0,sd0=1,mu1=2,sd1=1)
head(res)
# and the same curve
plot(1-res$Sp, res$Se, xlab="FPR", ylab="Sensitivity", 
   main="ROC curve for continuous data", type="l")
# max Youden:
(J = max(res$Sp+res$Se-1))
# max Youden threshold:
res$cp[which.max(res$Sp+res$Se-1)]

I hope it is still of any use to you ...     
P.S. Be aware that the bi-normal ROC plot can cross the diagonal chance line and can be considered as improper. See for instance: Hillis, S. L., & Berbaum, K. S. (2011). Using the mean-to-sigma ratio as a measure of the improperness of binormal ROC curves. Academic radiology, 18(2), 143-154.
A: You can use the ROCit package that does it. Use method = "bin" in the rocit function.
https://cran.r-project.org/web/packages/ROCit/index.html
