Tests for Hazard Transformation

The semiparametric Cox proportional hazards model is routinely adopted to model time-to-event data. Proportionality is a strong assumption, especially when follow-up time, or study duration, is long. Zeng and Lin (J. R. Stat. Soc., Ser. B, 69:1–30, 2007) proposed a useful generalisation through a family of transformation models which allow hazard ratios to vary over time. In this paper we explore a variety of tests for the need for transformation, arguing that the Cox model is so ubiquitous that it should be considered as the default model, to be discarded only if there is good evidence against the model assumptions. Since fitting an alternative transformation model is more complicated than fitting the Cox model, especially as procedures are not yet incorporated in standard software, we focus mainly on tests which require a Cox fit only. A score test is derived, and we also consider performance of omnibus goodness-of-fit tests based on Schoenfeld residuals. These tests can be extended to compare different transformation models. In addition we explore the consequences of fitting a misspecified Cox model to data generated under a true transformation model. Data on survival of 1043 leukaemia patients are used for illustration.