For large values of \(n_Y\) and \(n_{\bar{Y}}\),
the distribution of \(TPR(c)\) at
\(FPR(c)\) can be approximated as a normal distribution
with following mean and variance:
$$\mu_{TPR(c)}=\sum_{i=1}^{n_Y}I(D_{Y_i}\geq c)/n_Y$$
$$V ( TPR(c) )= \frac{ TPR(c) ( 1- TPR(c)) }{n_Y}
+ ( \frac{g(c^*)}{f(c^*) } )^2 * K $$
where \(K=\frac{ FPR(c) (1-FPR(c))}{n_{\bar{Y}} } \), \(g\)
and \(f\) are the probability distribution functions of
the diagnostic variable in positive and negative groups
(with corresponding cumulative distribution functions \(G\) and \(F\)),
\(c^*=S^{-1}_{D_{\bar{ Y}}}( FPR(c) )\), and \(S\) is the survival
function given by: \(S(t)=P(T>t)=1-F(t)\). density and
approxfun were used to approximate PDF and CDF
of the diagnostic score in the two groups and the inverse survival
of the diagnostic in the negative responses.
For "binomial" type, variance of \(A+BZ_x\) is given by
\(V(A)+Z_x^2V(B)+2Z_xCov(A, B)\). Bootstrap method was used to estimate
\(V(A)\), \(V(B)\) and \(Cov{A,B}\). The lower and upper limit of
\(A+BZ_x\) are inverse probit transformed to obtain the confidence interval
of the ROC curve.