Likelihood-based confidence intervals for functions of many parameters

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SUMMARY Methods described first by Madansky (1965) and revived more recently by Cox & Oakes (1984, pp. 51-2) are extended to incorporate the calculation of likelihood-based confidence intervals for functions of many parameters. Examples involving failure time data are used to illustrate the techniques. This paper concerns the derivation of confidence intervals for complicated functions of many binomial parameters. Examples include the probability-of-being-in-response function introduced by Temkin (1978), and the cumulative incidence functions that arise in the context of multivariate failure time data with competing risks. Although point estimates of these 'summary functions' are available, their standard errors, given by the multivariate 8-method, may be algebraically complicated, so that confidence intervals may not always be easily obtained. In ? 2 of this paper we describe a likelihood-based solution to the problem of deriving 1 - a confidence intervals for complicated functions of many parameters. Because the intervals are based directly on the likelihood function, they are invariant under 1-1 transformations of the parameters. The description of the solution is set in the context of k independent binomial samples since the two examples presented in ? 3 arise in this context. However, the method does not appear to be intrinsically limited by the binomial framework, and could yield useful results in many other parametric and nonparametric settings.