Survival Analysis with Time-Varying Covariates Measured at Random Times by Design

Ecological momentary assessment (EMA) is a method for collecting real-time data in subjects' environments. It often uses electronic devices to obtain information on psychological state through administration of questionnaires at times selected from a probability-based sampling design. This information can be used to model the impact of momentary variation in psychological state on the lifetimes to events such as smoking lapse. Motivated by this, a probability-sampling framework is proposed for estimating the impact of time-varying covariates on the lifetimes to events. Presented as an alternative to joint modeling of the covariate process as well as event lifetimes, this framework calls for sampling covariates at the event lifetimes and at times selected according to a probability-based sampling design. A design-unbiased estimator for the cumulative hazard is substituted into the log likelihood, and the resulting objective function is maximized to obtain the proposed estimator. This estimator has two quantifiable sources of variation, that due to the survival model and that due to sampling the covariates. Data from a nicotine patch trial are used to illustrate the proposed approach.