By Henry Bradford Smith

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13), we obtain an asymptotic 100(1 − α) percent conﬁdence interval for of [max{ ˆ (u) − Zα/2 Var( ˆ (u) ), −1}, min{ ˆ (u) + Zα/2 Var( ˆ (u) ), −1}]. 37) Consider testing H0 : = 0 versus Ha : = 0 . 38) ˆ 0 ) denotes the restricted MLE of T, given a ﬁxed 0 , and χ 2 is the where T( α upper 100αth percentile of the central χ 2 distribution with one degree of freedom. 38). Details of the doubly-iterative procedure for ﬁnding these two roots of 0 can be found in the Appendix of Lui (1999). 39). In fact, when Xi = xi and ni − Xi = Yi , the asymptotic likelihood ratio test-based conﬁdence intervals under these two different samplings are identical.

G. and Kuo, L. (1996) An interval estimate for the intraclass correlation in beta-binomial sampling. Biometrics, 52, 412–425. Lui, K. , Mayer, J. A. and Eckhardt, L. (2000) Conﬁdence intervals for the risk ratio under cluster sampling based on the beta-binomial model. Statistics in Medicine, 19, 2933–2942. Mak, T. K. (1988) Analysing intraclass correlation for dichotomous variables. Applied Statistics, 37, 344–352. Mayer, J. , Slymen, D. , et al. (1997) Reducing ultraviolet radiation exposure in children.

5)/(mi. + 1). Following Beal (1987), we may also consider using Tˆ 2 to estimate T regardless of whether πˆ i = 0 or 1. Thus, we obtain an asymptotic 100(1 − α) percent conﬁdence interval for given by [ ˆ l (T2 ), ˆ u (T2 )]. 7) to account for the intraclass correlation between responses within clusters. We obtain an asymptotic 100(1 − α) percent conﬁdence interval for given by [max{ ˆ − Zα/2 f (m1 , ρˆ1 )l1 (1 − l1 )/m1. + f (m0 , ρˆ0 )u0 (1 − u0 )/m0. , −1}, min{ ˆ + Zα/2 f (m1 , ρˆ1 )u1 (1 − µ1 )/m1.