On fitting Cox's proportional hazards models to survey data

Binder (1992) proposed a method of fitting Cox's proportional hazards models to survey data with complex sampling designs. He defined the regression parameter of interest as the solution to the partial likelihood score equation based on all the data values of the survey population under study, and developed heuristically a procedure to estimate the regression parameter and the corresponding variance. In this paper, we provide a formal justification of Binder's method. Furthermore, we present an alternative approach which regards the survey population as a random sample from an infinite universe and accounts for this randomness in the statistical inference. Under the alternative approach, the regression parameter retains its original interpretation as the log hazard ratio, and the statistical conclusion applies to other populations. The related problem of survival function estimation is also studied.