Modeling event times with multiple outcomes using the Wiener process with drift.

## Mathematics

### Journal of Biometrics & Biostatistics

Author(s): Horrocks J, Thompson ME

Abstract Length of stay in hospital (LOS) is a widely used outcome measure in Health Services research, often acting as a surrogate for resource consumption or as a measure of efficiency. The distribution of LOS is typically highly skewed, with a few large observations. An interesting feature is the presence of multiple outcomes (e.g. healthy discharge, death in hospital, transfer to another institution). Health Services researchers are interested in modeling the dependence of LOS on covariates, often using administrative data collected for other purposes, such as calculating fees for doctors. Even after all available covariates have been included in the model, unexplained heterogeneity usually remains. In this article, we develop a parametric regression model for LOS that addresses these features. The model is based on the time, T, that a Wiener process with drift (representing an unobserved health level process) hits one of two barriers, one representing healthy discharge and the other death in hospital. Our approach to analyzing event times has many parallels with competing risks analysis (Kalbfleisch and Prentice, The Statistical Analysis of Failure Time Data, New York: John Wiley and Sons, 1980)), and can be seen as a way of formalizing a competing risks situation. The density of T is an infinite series, and we outline a proof that the density and its derivatives are absolutely and uniformly convergent, and regularity conditions are satisfied. Expressions for the expected value of T, the conditional expectation of T given outcome, and the probability of each outcome are available in terms of model parameters. The proposed regression model uses an approximation to the density formed by truncating the series, and its parameters are estimated by maximum likelihood. An extension to allow a third outcome (e.g. transfers out of hospital) is discussed, as well as a mixture model that addresses the issue of unexplained heterogeneity. The model is illustrated using administrative data.

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