On the Bootstrap and Monotone Likelihood in the Cox Proportional Hazards Regression Model

Recent literature has provided encouragement for using the bootstrap for inference on regression parameters in the Cox proportional hazards (PH) model. However, generating and performing the necessary partial likelihood computations on multitudinous bootstrap samples greatly increases the chances of incurring problems with monotone likelihood at some point in the analysis. The only symptom of monotone likelihood may be a failure to converge in the numerical maximization procedure, and so the problem might naively be dismissed by deleting the offending data set and replacing it with a new one. This strategy is shown to lead to potentially high selection biases in the subsequent summary statistics. This note discusses the importance of keeping track of these monotone likelihood cases and provides recommendations for their use in interpreting bootstrap findings, and for avoiding unwanted biases that may result from high rates of occurrence. In many cases, high monotone likelihood rates indicate that a more highly-specified model may be preferred. Special consideration is given to the problem of high monotone likelihood incidence in Monte Carlo studies of the bootstrap.