| In this dissertation, three problems related to the analysis of time-to-event data are considered: (1) a new partial likelihood for the Cox proportional hazards model; (2) bias reduction for the Cox proportional hazards model; (3) robust testing of survival differences using global testing procedures.;The Cox (1972) proportional hazards model (PH) is based on the concept of the partial likelihood, which is essentially a product of conditional probabilities. We show that under the proportional odds assumption, the partial likelihood given by Cox (1972) is indeed a product of conditional probabilities, and through conditioning, the infinite dimensional baseline hazard function is removed from the estimation of regression parameters with censored data. However, under the PH assumption, Cox's partial likelihood cannot be interpreted as a product of conditional probabilities; rather it is based on the approximation of relative risks using odds ratios. We develop a new partial likelihood for the PH model and show that for finite samples, the efficiency of the parameter estimates can be greatly improved. Asymptotically, the new partial likelihood converges to Cox's partial likelihood.;For the Cox PH model, the parameter estimate from Cox's partial likelihood is biased away from 0, namely over-estimation. The bias is notable when the sample size is small or moderate. The parameter estimate based on the new partial likelihood is also biased, but toward 0, namely under-estimation. We propose a new synthetic estimator which is almost unbiased, and more efficient than Cox's partial likelihood estimate as measured by the Mean Squared Error.;The log-rank test is optimal for assessing difference between two survival distributions when the corresponding hazard functions are proportional. However, if the PH assumption is not tenable, a weighted log-rank test is often used in which the pre-specified weight function assigns more weight to early, middle or late events. Since the performance of a single weighted log-rank statistic is sensitive to the choice of the weight function, we propose the use of multiple weighted log-rank statistics. The null hypothesis of equality of the two survival distributions is tested using a pre-specified global testing procedure.;Key words. Partial Likelihood; Cox Proportional Hazards Model; Proportional Odds Model; Conditional Probability; Counting Process; Bias; Robust Testing; Weighted Log-rank Statistics. |
