Experimenting with the Coxian Phase-Type Distribution to Uncover Suitable Fits

In the past few decades, Coxian phase-type distributions have become increasingly more popular as a means of representing survival times. In healthcare, they are considered suitable for modelling the length of stay of patients in hospital and more recently for modelling the patient waiting times in Accident and Emergency Departments. The Coxian phase-type distribution has not only been shown to provide a good representation of real survival data, but its interpretation seems reasonably initiative to the medical experts. The drawback, however, is fitting the distribution to the data. There have been many attempts at accurately estimating the Coxian phase-type parameters. This paper wishes to examine the most promising of the approaches reported in the literature to determine the most accurate. Three performance measures are introduced to assess the fitting process of the algorithms along with the likelihood values and AIC to examine the goodness of fit and complexity of the model. Previous research suggests that the fitting process is strongly influenced by the initial parameter estimates and the data itself being quite variable. To overcome this, one experiment in this research paper will use the same initial parameter values for each estimation and perform the fits on the data simulated from a Coxian phase-type distribution with known parameters.

The Conditional Phase-type distribution: Graphical Models for Continuous Non-Normal Data:The Conditional Phase-type (C-Ph) distribution:

Conditional Gaussian (CG) distributions allow the inclusion of both discrete and continuous variables in a model assuming that the continuous variable is normally distributed. However, the CG distributions have proved to be unsuitable for survival data which tends to be highly skewed. A new method of analysis is required to take into account continuous variables which are not normally distributed. The aim of this paper is to introduce the more appropriate conditional phase-type (C-Ph) distribution for representing a continuous non-normal variable while also incorporating the causal information in the form of a Bayesian network.

The Coxian phase-type distribution as a contribution to the multilevel model of in-hospital mortality

This paper presents multilevel models that utilize the Coxian phase-type distribution in order to be able to include a survival component in the model. The approach is demonstrated by modeling patient length of stay and in-hospital mortality in geriatric wards in Italy. The multilevel model is used to provide a means of controlling for the existence of possible intra-ward correlations, which may make patients within a hospital more alike in terms of experienced outcome than patients coming from different hospitals, everything else being equal. Within this multilevel model we introduce the use of the Coxian phase-type distribution to create a covariate that represents patient length of stay or stage (of hospital care). Results demonstrate that the use of the multilevel model for representing the in-patient mortality is successful and further enhanced by the inclusion of the Coxian phase-type distribution variable (stage covariate).

The Coxian Phase-type distribution with a Hidden Node

Healthcare providers are under increased pressure to ensure that the quality
of care delivered to patients are off the highest standard. Modelling quality of
care is difficult due to the many ways of defining it. This paper introduces a potential
model which could be used to take quality of care into account when modelling
length of stay. The Coxian phase-type distribution is used to model length of stay
and quality of care incorporated into this using a Hidden Markov model. This model
is then applied to

Computational learning of the conditional phase-type (C-Ph) distribution: Learning C-Ph distributions

This paper presents a new algorithm for learning the structure of a special type of Bayesian network. The conditional phase-type (C-Ph) distribution is a Bayesian network that models the probabilistic causal relationships between a skewed continuous variable, modelled by the Coxian phase-type distribution, a special type of Markov model, and a set of interacting discrete variables. The algorithm takes a dataset as input and produces the structure, parameters and graphical representations of the fit of the C-Ph distribution as output.The algorithm, which uses a greedy-search technique and has been implemented in MATLAB, is evaluated using a simulated data set consisting of 20,000 cases. The results show that the original C-Ph distribution is recaptured and the fit of the network to the data is discussed.