Abstract: In medical studies, the minimum p-value method is often used to determine a cutpoint of a continuous biomarker for predictive classification and to assess whether a subset of patients may have a different treatment effect than that of other patients. However, this method suffers from type-I error inflation when the estimated cutpoint is treated as known. In this paper, we propose bootstrap-based procedures to obtain the valid p-value for the minimum p-value test statistic when the treatment effect is measured by a continuous outcome under both random and fixed designs, regardless of whether the cutpoint is identifiable. In the fixed design case, the test statistic is the supremum of a noncentered random process, the mean function (i.e., bias) of which diverges as the sample size goes to infinity, even under the null hypothesis. The proposed bootstrap statistic matches the diverging bias asymptotically, and we apply the high-dimensional Gaussian approximation results to establish the asymptotic size validity and the power consistency under local alternatives. The proposed method is applied to a data set from a clinical trial on advanced colorectal cancer.

Key words and phrases: High-dimensional Gaussian approximation, minimum p-value method, multiplier residual bootstrap, non-centered process.