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Survival Methods: Additional Topics

Survival analysis concerns outcome variables that are the times to the occurrence of events such as myocardial infarction, hospitalization for heart failure, or death. Rao and Schoenfeld1 give an excellent account of frequently used techniques in survival analysis, including Kaplan–Meier estimation of a single survival distribution, the log-rank test for comparing 2 survival distributions, and Cox’s proportional-hazards model (PHM) for assessing the affect of multiple predictors on survival.

Our article delves more deeply into the methodology, emphasizing the use of time-dependent covariates both as devices for assessing the fit of a model and as risk predictors in their own right. We review the accelerated life model as an important alternative to the PHM. We address the analysis of event-time data when individuals may experience multiple events and the use of combined end points. We describe the use of cumulative incidence curves as an alternative to Kaplan–Meier estimation in situations involving competing risks. Finally, we mention some issues related to the analysis of quality of life and cost-effectiveness data in survival studies.

Recall that the hazard function h(t) for an event at time t is the instantaneous event rate among t subjects who have not yet experienced the event. It is related to the survivor function (probability of not yet having experienced the event), S(t), by the expression h(t)=−[1/S(t)]dS(t)/dt (Rao and Schoenfeld1). Cox’s model states that the hazard function for an individual with covariates x1, x2, …, x3 takes the form equation ![Formula][1] 

where h(t) is the so-called baseline (or reference) hazard and the xj are covariates. Model 1 implies that the ratio of the hazard functions for 2 individuals is constant over time because the term h(t) cancels from the ratio and the other terms are free of t. …

 [1]: /embed/graphic-1.gif