Testing for a treatment effect in a selected subgroup

There is a growing interest in clinical trials that investigate how patients may respond differently to an experimental treatment depending on the basis of some biomarker measured on a continuous scale, and in particular to identify some threshold value for the biomarker above which a positive treatment effect can be considered to have been demonstrated. This can be statistically challenging when the same data are used both to select the threshold and to test the treatment effect in the subpopulation that it defines. This paper describes a hierarchical testing framework to give familywise type I error rate control in this setting and proposes two specific tests that can be used within this framework. One, a simple test based on the estimated value from a linear regression model with treatment by biomarker interaction, is powerful but can lead to type I error rate inflation if the assumptions of the linear model are not met. The other is more robust to these assumptions, but can be slightly less powerful when the assumptions hold.