Omnibus tests of the martingale assumption in the analysis of recurrent failure time data

The Andersen-Gill multiplicative intensity (MI) model is well-suited to the analysis of recurrent failure time data. The fundamental assumption of the MI model is that the process Mi(t) for subjects i = 1, ..., n, defined to be the difference between a subject's counting process and compensator, i.e., Ni(t) - Ai(t); t > 0, is a martingale with respect to some filtration. We propose omnibus procedures for testing this assumption. The methods are based on transformations of the estimated martingale residual process Mi(t) a function of consistent estimates of the log-intensity ratios and the baseline cumulative hazard. Under a correctly specified model, the expected value of Mi(t) is approximately equal to zero with approximately uncorrelated increments. These properties are exploited in the proposed testing procedures. We examine the effects of censoring and covariate effects on the operating characteristics of the proposed methods via simulation. The procedures are most sensitive to the omission of a time-varying continuous covariate. We illustrate use of the methods in an analysis of data from a clinical trial involving patients with chronic granulatomous disease.