# Asymptotic Confidence Interval for Rh

In this section we will follow the ideas of Kent (1983) to construct an asymptotic confidence interval for 21 (а, в). An asymptotic confidence interval for *Rh _{t}* can then be constructed using a proper transformation. If the model describing the association between в and

*a*is correctly specified, then Kent (1983) showed that

where *a* can be estimated by a = *2nl (а,* в), with n the total number of trials, p =1, and 1(а,в) = G^{2}/n calculated as in (9.3). Therefore,

The previous expressions and the tables for the non-central x^{2} distribution can now be used to construct confidence intervals. Let us define K_{1:a}(a) and

?i:a(a) by

If P(x2(0) > a) = a/2, then we set K_{1:a}/_{2}(a) = 0. A two-sided 1 — a asymptotic confidence interval for 21 *(T, S*) is then given by the closed interval

Notice further than under the null hypothesis of no association *H _{0}* : I(а, в) = 0 classical maximum likelihood theory indicates that, asymptotically,

Using the previous distribution, a test can be constructed for evaluating the validity of the surrogate at the trial level, i.e., the hypothesis *H _{0}* : = 0-