The likelihood ratio test for non-standard hypotheses near the boundary of the null – with application to the assessment of non-inferiority

Abstract

Abstract We consider a class of testing problems where the null space is the union of k-1 subgraphs of the form h j (θ j )≤θ k , with j=1,…,k-1, (θ 1,…,θ k ) the unknown parameter, and h j given increasing functions. The data consist of k independent samples, assumed to be drawn from a distribution with parameter θ j , j=1,…,k, respectively. An important class of examples covered by this setting is that of non-inferiority hypotheses, which have recently become important in the evaluation of drugs or therapies. When the true parameter approaches the boundary at a 1/√n rate, we give the explicit form of the asymptotic distribution of the log-likelihood ratio statistic. This extends previous work on the distribution of likelihood ratio statistics to local alternatives. We consider the prominent example of binomial data and illustrate the theory for k=2 and 3 samples. We explain how this can be used for planning a non-inferiority trial. To this end we calculate the optimal sample ratios yielding the maximal power in a binomial non-inferiority trial.