Double barrier models for length of stay in hospital

Abstract

Length of stay (LOS) in hospital is a widely used outcome measure in Health Services research, often acting as a surrogate for resource consumption or as a measure of efficiency. Recent activity in the field focuses on modeling the dependence of LOS on covariates, using administrative data collected for the purpose of calculating fees for doctors, or data extracted from medical charts. This problem is a challenging one, due to the high skewness of the distribution of LOS, the presence of multiple destinations (healthy discharge, death in hospital, transfer to another institution) and the unexplained heterogeneity which remains even after all available covariates have been included in the model.

In this thesis, we develop parametric models for LOS that accommodate the skewness of the distribution and allow for multiple destinations. The models are based on the time, T, until a Wiener process with drift (representing a health level process) hits one of two barriers, one representing healthy discharge, the other representing death in hospital. The model is parameterized in terms of the barrier levels and drift, which are allowed to depend on covariates. The parameters of the model are estimated using the method of maximum likelihood. We show bow to estimate expected LOS and probability of discharge, and discuss ways of testing hypotheses of interest. An interesting feature of the variable T is that the density and distribution functions are infinite series. We show that the density and its derivatives are absolutely and uniformly convergent, and that regularity conditions are satisfied in the zero drift case for iid observations.

The models can easily be extended to allow the drift parameter to have a mixing distribution, thereby partially addressing the issue of unexplained heterogeneity. While mixture models often require numerical integration in order to estimate parameters, we show that, if the mixing distribution is normal, numerical integration is unnecessary for these models. Also, an extension to handle transfers out of hospital is implemented. Since the decision to transfer is at least partially based on the health level of the patient, transfers cannot be treated as independently censored observations. We develop a model in which patients are transferred with probability p when their health level reaches an intermediate decision barrier. We can then model pas a function of co-variates. As before, the parameters of these models are estimated using maximum likelihood, and we show how to estimate expected LOS and probability of healthy discharge.

This approach to analyzing LOS has many parallels with competing risks analysis, and can be seen as a way of formalizing a competing risk situation. Further work will explore incorporation of time-varying covariates, different distributions for the health level process, and formal measures of goodness of fit.