Stochastically changing covariates may influence survival. They may be observed, unobserved or partly observed. We review the properties of hazard models explicitly representing the effects of unobserved, and partially observed, stochastic covariates. Such models will increase in importance as new longitudinal population studies, and longitudinal surveys of high dimensional failure processes in humans, become available--many are now in progress. It is shown that marginal survival distributions and likelihoods generated in analytically closed form make such parametrically detailed models computationally tractable. Several ways of defining the marginal distribution of the data for constructing a likelihood function are considered. The most complete models can handle both continuously and discretely evolving covariates. Parameters can be estimated from multiple data sets to retrospectively and prospectively evaluate covariate trajectories. Such methods will both extract more information from a longitudinal study and use it in a parametric structure that is logically consistent with the behavior of the underlying processes of substantive interest.
"Effects of unobserved and partially observed covariate processes on system failure: a review of models and estimation strategies." Statist. Sci. 12 (1) 20 - 34, February 1997. https://doi.org/10.1214/ss/1029963259