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 regressionwhen observed subjects have different selection probabilities. Wedevelop methods which are applicable when the selectionprobabilities 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 sampleemploying the estimated inverse selection probabilities as weights.The asymptotic properties of the proposed estimators are derived andevaluated in finite samples through simulation. In applying thismethod to renal transplant data, the effect on transplant failure ofExpanded Criteria Donor (ECD) kidneys is estimated using the proposedmethods and found to be of considerably greater magnitude than thatimplied by previous (unweighted) analyses.In the second paper, we develop hypothesis testingprocedures for contrasting parameters from weighted and unweightedproportional hazards models. Comprehensive statistics are proposedfor both regression parameter estimators and baseline hazardfunction. Asymptotic properties of the test statistics arederived, while the empirical significance level and power areexamined in numerical studies. Various patient characteristics are found to havesignificantly different effects in transplanted patients versuswait-list candidates.The third paper considers recurrent events with a terminating event. The method involves fitting a proportional hazards modelfor the terminating event and an additive model for the recurrent eventrate conditional on survival, then integrating over time. Twomethods are proposed to compare the mean number of events betweentreatments. The first method factors out differences in the survivaldistributions between treatments, while the second method featurestreatment-specific survival functions. The estimators of bothproposed measures are proved to be consistent with explicit covariance functions. Asymptotic properties are evaluated in moderate-size samples and themethods are found to be robust to unadjusted predictors. The motivating example is repeated hospitalizations after kidney transplant, where the effect of ECDtransplant compared to non-ECD transplant on the mean number ofhospitalizations is of interest. We found although ECD transplant patients tend to die sooner than non-ECD patients, they experience a significantly more hospitalizations.