Correct and logical inference on efficacy in subgroups and their mixture for binary outcomes.

Targeted therapies are becoming more common. In targeted therapy development, suppose its companion diagnostic test divides patients into a marker-positive subgroup and its complementary marker-negative subgroup. To find the right patient population for the therapy to target, inference on efficacy in the marker-positive and marker-negative subgroups as well as efficacy in the overall mixture population are all of interest. Depending on the type of clinical endpoints, inference on mixture population can be nontrivial and commonly used efficacy measures may not be suitable for a mixture population. Correlations among estimates of efficacy in the marker-positive, marker-negative, and overall mixture population play a crucial role in using an earlier phase study to inform on the design of a confirmatory study (e.g., determination of sample size). This article first shows that when the clinical endpoint is binary (such as respond or not), odds ratio is inappropriate as an efficacy measure in this setting, but relative response (RR) is appropriate. We show a safe way of calculating estimated correlations is to consider mixing subgroup response probabilities within each treatment arm first, and then derive the joint distribution of RR estimates. We also show, if one calculates RR within each subgroup first, how wrong the correlations can be if the Delta method derivation fails to take randomness of estimating the mixing coefficient into account.

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