Cause-specific quantile regression on inactivity time

In time-to-event analysis, the traditional summary measures have been based on the hazard function, survival function, quantile event time, restricted mean event time, and residual lifetime. Under competing risks, furthermore, typical summary measures have been the cause-specific hazard function and cumulative incidence function. Recently inactivity time has recaptured attention in the literature, being interpreted as life lost. In this paper, we further interpret it as quality of life reduced and time period after transition to a drug, and propose a quantile regression model to associate the inactivity time with potential predictors under competing risks. We define the proper cumulative distribution function of the inactivity time distribution for each specific event type among those subjects who experience the same type of events during a follow-up period. A score function-type estimating equation is developed and asymptotic properties of the regression coefficient estimators are derived by assuming that competing events are censored at their occurrence times as in the cause-specific hazard analysis. The proposed approach reduces to a regular quantile regression on the inactivity time without competing risks when all types of competing events are collapsed into the same type. Due to difficulty in estimating the improper probability density function of the cause-specific inactivity distribution to evaluate the variance of the quantiles, a computationally efficient perturbation method is adopted to infer the regression coefficients. Simulation results show that our proposed method works well under the assumed finite sample settings. The proposed method is illustrated with a real dataset from a breast cancer study.