Accelerated failure time regression for backward recurrence times and current durations

Keiding, Fine, Hansen and Slama have a new paper in Statistics & Probability Letters. This considers regression models for time to event data in which only cross-sectional data is available. In this situation the process is assumed to be a stationary renewal process. The observed times are then taken to be backward recurrence times (i.e. time since last renewal to a given observation time). Here it is noted that when the inter-arrival times f(x) are subject to an accelerated failure time model, the same accelerated failure time model will apply to the backward recurrence times (a result apparently first found by Yamaguchi in the social science literature). As a consequence, an AFT model can be fitted to the observed backward recurrence times to give estimates of the AFT model for the inter-arrival times.

Keiding et al consider modelling time-to-pregnancy using data on the current duration spent attempting to become pregnant by modelling the backward recurrence times as Pareto or generalised Gamma distributed within an accelerated failure time model with frequency of sexual intercourse as a covariate in the AFT model.

The equivalence of backward and forward recurrence times (the latter being the time to next event given observation from some fixed time) means that the same approach could be applied to prevalent cohort studies with an unknown initiation time.