Discrete-Time Threshold Regression for Survival Data with Time-Dependent Covariates
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A natural approach to survival analysis in many settings is to model the subject's ``health'' status as a latent stochastic process, where the terminal event is represented by the first time that the process crosses a threshold. ``Threshold regression'' models the covariate effects on the latent process. Much of the literature on threshold regression assumes that the process is a one-dimensional Wiener process, where crossing times have a tractable inverse Gaussian distribution but where the process characteristics are fixed at baseline. This framework is not easily extended to incorporate time-varying covariates or dependent competing risks. We introduce a novel approach for performing threshold regression with time-dependent covariates in a discrete time setting, where the process is a Gaussian random walk, with time-varying drift as a parameterized function of time-varying covariates. This model is then extended to consider dual correlated competing risks. We present methods for estimating model parameters, including an EM algorithm, and outline numerical algorithms for efficiently evaluating the observed and complete data likelihoods and score functions and for estimating standard errors. We discuss results of applying these methods to both simulated data and to the Freddie Mac residential mortgage data set. In the latter case we quantify associations between baseline borrower characteristics and time-varying macroeconomic conditions versus time to mortgage default and prepayment events.
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