  We introduce new methods of analysing time to event data via extended
versions of the proportional hazards and accelerated failure time (AFT) models.
In many time to event studies, the time of first observation is arbitrary, in
the sense that no risk modifying event occurs. This is particularly common in
epidemiological studies. We show formally that, in these situations, it is not
sensible to take the first observation as the time origin, either in AFT or
proportional hazards type models. Instead, we advocate using age of the subject
as the time scale. We account for the fact that baseline observations may be
made at different ages in different patients via a two stage procedure. First,
we marginally regress any potentially age-varying covariates against age,
retaining the residuals. These residuals are then used as covariates in the
fitting of either an AFT model or a proportional hazards model. We call the
procedures residual accelerated failure time (RAFT) regression and residual
proportional hazards (RPH) regression respectively. We compare standard AFT
with RAFT, and demonstrate superior predictive ability of RAFT in real
examples. In epidemiology, this has real implications in terms of risk
communication to both patients and policy makers.
