# Phase-Type Survival Trees to Model a Delayed Discharge and Its Effect in a Stroke Care Unit
## 2 Materials and Methods  
The discharge destinations play a crucial role as a factor in discharge delays. Therefore, we also model the discharge delays based on discharge destinations and their impact in terms of the extra beds required, additional cost, and bed blocking (overcrowding).  
In our previous work [50; 51; 52; 53; 54; 55; 56; 57; 58], we proposed and validated the phase-type survival tree (PHTST)-based analysis to cluster patients into clinically meaningful patient groups and, in an extension of this approach, to examine the relationship between the length of stay in hospital and destination on discharge. This paper describes how PHTST-based clustering can be used for modelling delayed discharge and its effects in a stroke care unit. The PHTST length of stay distribution of each patient group (each PHTST node) stratified by demographic factors and their conditions diagnosed is separately modelled as a finite state continuous-time Markov chain using Coxian-phase-type distributions (C-PHDs). Delayed discharge patients waiting for discharge are modelled as a 'blocking state', an additional state of the Markov chain. For PHTST construction, we used splitting criteria based on the weighted-average information criterion [53].  
We can use this model to recognise the association between demographic and other factors and discharge delays and their effects on the average length of stay, cost of care, etc. We can also utilise the model to identify groups of patients who require attention to resolve the most common delays and prevent them from happening again. Each patient cluster can reveal unique issues. The approach is illustrated using five years of retrospective data [59; 60] for 1985 stroke patients (hemorrhagic stroke, cerebral infarction, transient ischaemic attack TIA, and unspecified stroke). The patients were admitted to the Belfast City Hospital between January 2003 and December 2007 and discharged between 9 January 2003 and 11 March 2008. No information that identified individual patients was supplied. Patients were aged between 24 years and 101 years. The length of stay (LOS) range was 0 to 1425 days, the mean LOS was 29.01 days, and the standard deviation was 52.84 days [59; 60]. The underpinning phase-type model-based clusters were previously validated for these data using statistical log-rank tests for equality between survival distributions for different discharge clusters [54].  
We have used our model to understand the effect of delays in discharge from the stroke unit of the Belfast City Hospital. Patients can be discharged to either private nursing homes or other destinations such as their usual residences. In addition, some patients can eventually die during their treatment. The estimated daily cost of care is calculated using estimates from [61], adjusted from 2005. We attach unit costs of \(\varepsilon\)164.80 per day for stay-in acute care (phase 1) and \(\varepsilon\)114.80 per day for stay-in rehabilitative care or long-stay care (phases 2-4).  
The following section describes the methodology used, including the description of the phase-type distributions, phase-type survival trees, and their mathematical foundation. The third section elaborates on the results, followed by a discussion and the conclusion sections.