Post-selection inference in Cox Proportional Hazards models

Variable selection causes the distributions of parameter estimators to be unknown and difficult to determine. To do inference after selection, conditional distributions for parameter estimators given the selected model are needed. Taylor and Tibshirani (2018) call this post-selection inference and describe an estimator of regression parameters along with the corresponding conditional distribution, making post-selection inference possible. The Polyhedral Lemma (Lee et al., 2016) is used to determine the conditional distribution of this estimator given the model selected - a truncated normal distribution. We implement Taylor and Tibshirani's (2018) method in the Cox Proportional Hazards Regression setting and do a Monte Carlo study. The results are analyzed. The method controls the level of tests and coverage of confidence intervals well – much better than unadjusted Cox Proportional Hazards techniques. Numerical difficulties in the Cox Proportional Hazards software are identified and addressed in the post-selection inference context.