ABSTRACT
The usual censoring mechanism considered for survival models is one where the censoring is independent of the failure time given the covariate value. Sometimes a stronger assumption is needed in which we require independence between the censoring and failure times. Under the weaker assumption, the partial likelihood estimator for the regression coefficient β in the proportional hazards model was first shown to be consistent by Cox (1975) and later, more formally, by Tsiatis (1981). Consistency can be established using a martingale approach (Andersen and Gill, 1982). The two ways in which the main model assumptions may fail are that where the covariate specification is incorrect or that where the constancy of regression effect, β, is not respected by the observations. Either of these will result in a dependency of the usual estimates on the censoring mechanism. In certain simplified cases, say a simple binary grouping variable, the only thing we need consider is the time dependency. This is because any transformation of the grouping variable that still allows us to distinguish the groups will not change the model essentially; the only inadequacy we need consider is then that of the constancy of regression effect. Again, this argument extends immediately, to p groups, represented by p − 1 binary indicator variables. Moving away from this situation to that of ordinal variables, or even continuous variables, then the covariate representation can be crucial. Some transformation, for example using normal order statistics, or using a uniform transformation of the scale, can produce a model that may be more robust to small
assumptions. If so, we would expect the impact of the censoring to be weak if not almost absent.
