# Stochastic Process and Health Data: A Full Maximum Likelihood Method to Hospital Charge and Length of Stay Data

## TL;DR

This paper develops a full maximum likelihood method for jointly modeling hospital charges and length of stay, addressing endogeneity and improving fit over existing models using real data.

## Contribution

It derives an explicit joint density function for charge and LOS, enabling efficient estimation and better model fitting compared to phase-type models.

## Key findings

- Our method provides more efficient fit to real hospital data.
- It effectively resolves endogeneity between charge and LOS.
- The joint model outperforms marginal models like phase-type in accuracy.

## Abstract

We extend the model used in Gardiner et al. (2002) and Polverejan et al. (2003) through deriving an explicit expression for the joint probability density function of hospital charge and length of stay (LOS) under a general class of conditions. Using this joint density function, we can apply the full maximum likelihood method (FML) to estimate the effect of covariates on charge and LOS. By FML, the endogeneity issues arisen from the dependence between charge and LOS can be efficiently resolved. As an illustrative example, we apply our method to real charge and LOS data sampled from New York State Statewide Planning and Research Cooperative System 2013 (SPARCS 2013). We compare our fitting result with the fitting to the marginal LOS data generated by the widely used Phase-Type model, and conclude that our method is more efficient in fitting.