The impact of time-dependent bias in proportional hazards modelling

In the clinical literature, time-dependent exposure status has regularly been analysed as if known at time origin. Although statisticians agree that such an analysis yields biased results when analysing the effect on the time until some endpoint of interest, this paper is the first to study in detail the bias arising in a proportional hazards analysis. We show that the biased hazard ratio estimate will always be less than the unbiased one; this leads to either an inflated or a damped effect of exposure, depending on the sign of the correct log hazard ratio estimate. We find an explicit formula of the asymptotic bias based on generalized rank estimators, and we investigate the role of censoring, which may prevent an individual from being considered as being baseline exposed in the biased analysis. We illustrate our results with data on hospital infection status and different censoring patterns.