Tests for noninferiority trials with binomial endpoints: A guide to modern and quasi-exact methods for biomedical researchers.

Applied statisticians and pharmaceutical researchers are frequently involved in the design and analysis of clinical trials where at least one of the outcomes is binary. Treatments are judged by the probability of a positive binary response. A typical example is the noninferiority trial, where it is tested whether a new experimental treatment is practically not inferior to an active comparator with a prespecified margin δ. Except for the special case of δ = 0, no exact conditional test is available although approximate conditional methods (also called second-order methods) can be applied. However, in some situations, the approximation can be poor and the logical argument for approximate conditioning is not compelling. The alternative is to consider an unconditional approach. Standard methods like the pooled z-test are already unconditional although approximate. In this article, we review and illustrate unconditional methods with a heavy emphasis on modern methods that can deliver exact, or near exact, results. For noninferiority trials based on either rate difference or rate ratio, our recommendation is to use the so-called E-procedure, based on either the score or likelihood ratio statistic. This test is effectively exact, computationally efficient, and respects monotonicity constraints in practice. We support our assertions with a numerical study, and we illustrate the concepts developed in theory with a clinical example in pulmonary oncology; R code to conduct all these analyses is available from the authors.

Accurate confidence limits for stratified clinical trials.

For stratified 2 × 2 tables, standard approximate confidence limits can perform poorly from a strict frequentist perspective, even for moderate-sized samples, yet they are routinely used. In this paper, I show how to use importance sampling to compute highly accurate limits in reasonable time. The methodology is very general and simple to implement, and orders of magnitude are faster than existing alternatives.

Unconditional efficient one-sided confidence limits for the odds ratio based on conditional likelihood.

We compare various one-sided confidence limits for the odds ratio in a 2 x 2 table. The first group of limits relies on first-order asymptotic approximations and includes limits based on the (signed) likelihood ratio, score and Wald statistics. The second group of limits is based on the conditional tilted hypergeometric distribution, with and without mid-P correction. All these limits have poor unconditional coverage properties and so we apply the general transformation of Buehler (J. Am. Statist. Assoc. 1957; 52:482-493) to obtain limits which are unconditionally exact. The performance of these competing exact limits is assessed across a range of sample sizes and parameter values by looking at their mean size. The results indicate that Buehler limits generated from the conditional likelihood have the best performance, with a slight preference for the mid-P version. This confidence limit has not been proposed before and is recommended for general use, especially when the underlying probabilities are not extreme.