Testing independence between two sequential gap times in the presence of covariates

In the risk analysis of sequential events, the successive gap times are often correlated, e.g. as a result of an individual heterogeneity. Correlation is usually accounted for by using a shared gamma-frailty model, where the variance φ of the random individual effect quantifies the correlation between gap times. This method is known to yield satisfactory estimates of covariate effects, but underestimates φ, which could result in a lack of power of the test of independence. We propose a new test of independence between two sequential gap times where the first is the time elapsed from the origin. The test is based on an approximation of the hazard of the second event given the first gap time in a frailty model, with a frailty distribution belonging to the power variance function family. Simulation results show an increased power of the new test compared with the test derived from the gamma-frailty model. In the realistic case where hazards are event specific, and using event-specific approaches, the proposed estimation of the variance of the frailty is less biased than the gamma-frailty based estimation for a wide range of values (φ < 2.5 with the set of parameters considered), and similar for higher values. As an illustration, the methods are applied to a previously analysed asthma prevention trial with results showing a significant positive association between the successive times to asthmatic events. We also analyse data from a cohort of HIV-seropositive patients in order to assess the effect of risk factors on the occurrence of two successive markers of progression of the HIV disease. The results demonstrate the ability of the proposed model to account for negative correlations between gap times.