Some baseline patient factors, such as biomarkers, are useful in predicting patients’ responses to a new therapy.
Identification of such factors is important in enhancing treatment outcomes, avoiding potentially toxic therapy that is
destined to fail and improving the cost-effectiveness of treatment. Many of the biomarkers, such as gene expression, are measured on a
continuous scale. A threshold of the biomarker is often needed to define a sensitive subset for making easy clinical decisions.
A novel hierarchical Bayesian method is developed to make statistical inference simultaneously on the threshold and the treatment effect
restricted on the sensitive subset defined by the biomarker threshold. In the proposed method, the threshold parameter is treated as
a random variable that takes values with a certain probability distribution. The observed data are used to estimate parameters in the
prior distribution for the threshold, so that the posterior is less dependent on the prior assumption. The proposed Bayesian method is
evaluated through simulation studies. Compared to the existing approaches such as the profile likelihood method, which makes inferences
about the threshold parameter using the bootstrap, the proposed method provides better finite sample properties in terms of the coverage
probability of a 95% credible interval. The proposed method is also applied to a clinical trial of prostate cancer with the serum prostatic
acid phosphatase (AP) biomarker.