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 Rht 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) = G2/n calculated as in (9.3). Therefore,

The previous expressions and the tables for the non-central x2 distribution can now be used to construct confidence intervals. Let us define K1:a(a) and

?i:a(a) by

If P(x2(0) > a) = a/2, then we set K1: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 H0 : 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 H0 : = 0-

 
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