Hazard and rate regression in the presence of differential selection or termination probability

We study unrepresentative observational data in survival analysis.;The first paper focuses on proportional hazards regression when observed subjects have different selection probabilities. We develop methods which are applicable when the selection probabilities are unknown but estimated using auxiliary information. With a two-stage method, first, a logistic model is fitted and selection probabilities are estimated. Second, a proportional hazards model is fitted to the biased sample employing the estimated inverse selection probabilities as weights. The asymptotic properties of the proposed estimators are derived and evaluated in finite samples through simulation. In applying this method to renal transplant data, the effect on transplant failure of Expanded Criteria Donor (ECD) kidneys is estimated using the proposed methods and found to be of considerably greater magnitude than that implied by previous (unweighted) analyses.;In the second paper, we develop hypothesis testing procedures for contrasting parameters from weighted and unweighted proportional hazards models. Comprehensive statistics are proposed for both regression parameter estimators and baseline hazard function. Asymptotic properties of the test statistics are derived, while the empirical significance level and power are examined in numerical studies. Various patient characteristics are found to have significantly different effects in transplanted patients versus wait-list candidates.;The third paper considers recurrent events with a terminating event. The method involves fitting a proportional hazards model for the terminating event and an additive model for the recurrent event rate conditional on survival, then integrating over time. Two methods are proposed to compare the mean number of events between treatments. The first method factors out differences in the survival distributions between treatments, while the second method features treatment-specific survival functions. The estimators of both proposed measures are proved to be consistent with explicit covariance functions. Asymptotic properties are evaluated in moderate-size samples and the methods are found to be robust to unadjusted predictors. The motivating example is repeated hospitalizations after kidney transplant, where the effect of ECD transplant compared to non-ECD transplant on the mean number of hospitalizations is of interest. We found although ECD transplant patients tend to die sooner than non-ECD patients, they experience a significantly more hospitalizations.