Modelling time to event with observations made at arbitrary times

In many time-to-event studies, particularly in epidemiology, the time of the first observation or study entry is arbitrary in the sense that this is not a time of risk modification. We present a formal argument that, in these situations, it is not advisable to take the first observation as the time origin, either in accelerated failure time or proportional hazards models. Instead, we advocate using birth as the time origin. We use a two-stage process to account for the fact that baseline observations may be made at different ages in different subjects. First, we marginally regress any potentially age-varying covariates against age, retaining the residuals. These residuals are then used as covariates in fitting an accelerated failure time or proportional hazards model - we call the procedures residual accelerated failure time regression and residual proportional hazards regression, respectively. We compare residual accelerated failure time regression with the standard approach, demonstrating superior predictive ability of the residual method in realistic examples and potentially higher power of the residual method. This highlights flaws in current approaches to communicating risks from epidemiological evidence to support clinical and health policy decisions.