Conditional phase-type distributions for modelling patient length of stay in hospital

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1 Intl. Trans. in Op. Res. 10 (2003) INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH Conditional phase-type distributions for modelling patient length of stay in hospital A. H. Marshall a and S. I. McClean b a Department of Applied Mathematics and Theoretical Physics, Queen s University of Belfast, Northern Ireland b School of Computing and Information Engineering, Faculty of Engineering, University of Ulster, Northern Ireland a.h.marshall@qub.ac.uk[marshall] Received 17 June 2002; received in revised form 18 March 2003; accepted 10 April 2003 Abstract The proportion of elderly in the population is continuing to increase, placing additional demands on highly competitive medical budgets. The management of the care of the elderly within hospitals can be assisted by the accurate modelling of the length of stay of patients in hospital. This paper uses conditional phase-type distributions for modelling the length of stay of a group of elderly patients in hospital. The model incorporates the use of Bayesian belief networks with Coxian phase-type distributions, a special type of Markov model that describes the duration of stay in hospital as a process consisting of a sequence of latent phases. The incorporation of the Bayesian belief network in the model permits the inclusion of additional patient information which may provide a better understanding of the system, in particular the incorporation of any potential causal information that may exist in the data. Keywords: Markov processes; stochastic processes; artificial intelligence; statistics: distributions; health services; resource allocation Introduction The proportion of elderly in the population is continuing to increase, placing an added strain on medical resources. In particular, there has been an increased demand for hospital services, leading to resource utilisation problems such as bed allocation and staffing within the geriatric wards. These demands have placed an increased emphasis on the benefits that may be achieved through modelling hospital wards, for instance to assess the movement of patients throughout the wards, their anticipated length of stay in hospital, and the resources they require. The specialty of geriatric medicine has already witnessed several attempts to model patient behaviour, which have unfortunately been considered too complex and impractical to use. Not only is it important to develop modelling approaches which are easy to use and cost efficient, but it is essential that the models do not have a black box structure. This is a common approach r 2003 International Federation of Operational Research Societies. Published by Blackwell Publishing Ltd.

2 566 adopted in the past when measuring activity in hospitals whereby decision-makers focus attention on information collected outside the system under study, omitting information on the process of care itself. It is desirable to have models which provide a better insight into the understanding of how the system works so that a clinician or hospital manager can view the whole care process. This in turn will instil more trust or value in the results of a model. The modelling of hospital wards and patient activity can be addressed by focusing on the modelling techniques for patient duration of stay in hospital. In particular, the study of duration of stay of geriatric patients in hospital has led to the modelling of survival data using a twotermed mixed exponential distribution (McClean and Millard, 1993a). Further work has resulted in the representation of this distribution as a compartmental model and the final presentation of a phase-type distribution. This paper briefly discusses some of the previously developed models of patient duration of stay considering in particular phase-type distributions and their integration into a new methodology the conditional phase-type model. The conditional phase-type (C-Ph) model incorporates the use of Bayesian belief networks with Coxian phase-type distributions, a special type of Markov model that describes the duration of stay in hospital as a process consisting of a sequence of latent phases. A C-Ph model is developed to represent the patient length of stay of a group of elderly patients in hospital. Such a model may be used to aid clinicians in resource allocation. All of the analysis in this paper is based on the Clinics data set. Clinics data set The Clinics database is a clinical computer system that was used between 1994 and 1997 for the management of patients in the Department of Geriatric Medicine. It contains 4722 patient records including patient personal details, admission reasons, discharge details, outcome, and duration of stay (Marshall et al., 2001). Variables include; personal information such as age, gender, marital status, next of kin, lives alone; admission reasons such as stroke, fall, confusion; Barthel scores on feeding, grooming, bathing, mobility, and outcome details; destination on departure from hospital, and duration of stay in hospital. Patient length of stay in hospital has an average of 85 days and a median value of 17 days. This indicates a heavily skewed distribution, a typical characteristic of survival data. The distribution of the duration of stay of elderly patients in hospital generally tends to be highly skewed in nature where there is a large peak in the distribution at the start which then gradually tails off as duration increases. Previously developed models A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) The following models have been developed to represent the length of stay of elderly patients in hospital: two-term mixed exponential model; compartmental model; phase-type model; conditional phase-type model. Model development has progressed from the utilisation of the two-term mixed exponential distribution to its incorporation into a phase-type distribution in the more recently developed conditional phase-type distribution. These are described in more detail in the following section.

3 A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) Two-term mixed exponential model It was observed that the pattern of bed occupancy in departments of geriatric medicine could be expressed using mixed exponential distributions (Millard, 1989). In particular, the two-term mixed exponential model with probability density function ( pdf ) of the form f ðtþ ¼pl 1 e l 1t þð1 pþl 2 e l 2t 0opo1; l 1 ; l 2 X0 ð1þ has shown to give a good fit for durations of occupancy of geriatric beds (McClean and Millard, 1993a; Millard, 1989). In general, the model also gives a reasonable approximation to the numbers of patients departing each week from hospital and has thus led to a new method of estimating the usage of hospital beds. McClean and Millard (1993b) use the fits of the mixed exponential models to provide a method for predicting future behaviour of patients and identifying where there is a change in patterns. Compartmental models Godfrey (1983) defines compartmental systems as those consisting of a finite number of homogeneous, well-mixed, lumped subsystems, called compartments, which exchange with each other and with the environment so that the quantity or concentration of material within each compartment may be described by a first-order differential equation. In more recent years, compartmental models have been applied to the movement of patients throughout hospital systems. Markov models (Bartholomew, 1982; Iosifescu, 1980) are often used to represent stochastic compartmental models whereby the process assumes a probabilistic behaviour of patients moving around the system. The compartments in the model can be regarded as states and the probabilities of patients moving within those states calculated. P ij (t) represents the probability that a patient who is in state S i at time zero will be in state S j at time t and P(t) 5 {P ij (t)} the transition matrix of probabilities of patient movement throughout the model. Irvine, McClean and Millard (1994), Taylor, McClean and Millard (1998) and McClean and Millard (1998) have all described the movement of patients through geriatric hospitals using Markov models with differing numbers of compartments. They conclude by allowing the number of compartments in the model to be governed by the data so to obtain a model that gives the best representation of the data. Phase-type models Phase-type ( Ph) distributions (Neuts, 1981) describe the time to absorption of a finite Markov chain in continuous time, where there is a single absorbing state and the stochastic process starts in a transient state (Faddy, 1994; Faddy and McClean, 1999). The models describe time in terms of a process or sequence of latent phases, which terminates when the process has reached the absorbing state, when a certain event occurs. In fact the assumptions of the distributions state that the 1,y, n states are all transient, so absorption into state (n11), from any initial state is certain. The distributions are considered a highly versatile class of probability distributions that exhibit the ability to represent certain qualitative features of data, and may serve as convenient numerical

4 568 A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) approximations to other families of probability distributions (Neuts, 1981). In most instances, phase-type distributions can be generalised to include almost all continuous distributions such as the exponential, which will only have one phase, the Erlang, and mixed exponential distributions, therefore making them appealing to use. Although they are very similar in nature, there is one key difference in the Erlang and phasetype distributions. The movement between all the transient stages and the absorbing phase can occur in the phase-type distribution, whereas in the case of the Erlang transitions movement can only occur between sequential phases. In other words, the phase-type distributions allow a patient to leave the system completely at any stage and move directly into the absorbing state. This is not the case for the Erlang distributions. However, the generality of the phase-type distributions makes it difficult to estimate all the parameters. To overcome this problem the following Coxian phase-type distributions were introduced. Coxian phase-type distributions Coxian phase-type distributions (Cox, 1955) are a special subclass employed to describe the probability P(t) that the process is still active at time t. They differ from general phase-type distributions in that the transient states (or phases) of the model are ordered. The process begins in the first phase and may either progress through the phases sequentially or enter into the absorbing state (the terminating event phase n11). Such phases may then be used to describe stages of a process which terminates at some stage. For example, in the case of the duration of stay of the elderly patients in hospital, transitions through the ordered transient states could correspond to various stages in the patients stay in hospital such as stages of diagnosis, assessment, rehabilitation, and long-stay care where patients eventually discharge, transfer or die (Faddy, 1994). A Coxian phase-type distribution {X(t); tx0} may be defined as a (latent) Markov chain in continuous time with states {1, 2,y,n, n11}, X(0) 5 1, and for i 5 1, 2,y,n 1 probfxðt þ dtþ ¼i þ 1jXðtÞ ¼ig ¼l i dt þ oðdtþ ð2þ and for i 5 1, 2,y,n probfxðt þ dtþ ¼n þ 1jXðtÞ ¼ig ¼m i dt þ oðdtþ: ð3þ Here states {1, 2,y,n} are latent (transient) states of the process and state n11 is the (absorbing) state. l i represents the transition from state i to state (i11) and m i the transition from state i to the absorbing state (n11) (Fig. 1). Cox and Miller (1965) develop the theory of Markov chains such as those defined by (2) and (3). The Coxian phase-type distribution is defined as having a transition matrix Q of the following form,

5 A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) Phase 1 Phase 2 Phase n λ 1 λ 2 µ 1 µ 2 µ n Fig. 1. An illustration of phase-type distributions. 0 Q ¼ ðl 1 þ m 1 Þ l 1 0 ::: ðl 2 þ m 2 Þ l 2 ::: 0 0 : : : : : : : : : : : : : : : ::: ðl n 1 þ m n 1 Þ l n ::: 0 m n 1 ; ð4þ C A where the l i s and m i s are from (2) and (3). The survival probability, the probability that an individual survives longer than t, where X(t) 5 1, 2, y, n is given by SðtÞ ¼p expfqtg1; ð5þ where p ¼ð Þ; ð6þ and 1 is a column vector of 1s and Q is the matrix of transition rates between states, as shown in (4) (Faddy, 1994). The pdf of T then follows by differentiation: f ðtþ ¼p expfqtgq where q ¼ Q1 ¼ðm 1 ; m 2 ;...; m n Þ T : ð7þ ð8þ Faddy and McClean (1999) found Coxian phase-type distributions to be ideal for measuring the lengths of stay of geriatric patients in hospital, as the ordered phases have an interpretable structure corresponding to increasing amounts of time in care. The previously described two-term mixed exponential model (McClean and Millard, 1993b) can be regarded as a two-phase distribution where the patients are split into two groups, acute and long-stay, according to their length of stay in geriatric hospital. In addition to the Coxian phase-type distribution for representing length of stay, the inclusion of additional patient variables would provide a better understanding of the patient data. This has inspired the development of the conditional phase-type (C-Ph) approach which uses Bayesian belief networks to represent potential causal relationships, along with Coxian phase-type distributions for modelling patient duration of stay.

6 570 Bayesian belief networks A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) Bayesian belief networks (BBNs) are special cases of probabilistic graphical models that use directed arcs exclusively to form a directed acyclic graph (DAG) and Bayes Theorem to represent the probabilistic relationships between variables (Buntine, 1996; Cox and Wermuth, 1996). The structure of the BBN is formed by nodes and arrows which represent the variables and relationships respectively in a DAG. An arrow or directed arc, in which two nodes are connected by an edge, indicates that one directly influences the other. Attached to the nodes in the graphical model are probabilities of various events occurring given that some previous event has already taken place. One of the key features of BBNs is the ability to describe potentially causal relations. There has been much discussion and concern over the use of the term causality. In the context of BBNs, such use of the word causal is in the representation of relationships which have the potential to be causal in nature, when one variable directly influences another. Cox and Wermuth (1996) explain this reasoning by stating that it is rare that firm conclusions about causality can be drawn from one study but rather the objective is to provide representations of data that are potentially causal; those which are consistent with or suggestive of causal interpretation. The extension of Bayesian belief networks to include continuous variables has been considered by Lauritzen and Wermuth (1989) who introduce Conditional Gaussian (CG) distributions. However the CG distributions are not appropriate for modelling the survival of patients in hospital because of the skewed nature of the data set. An alternative distribution is that of the conditional phase-type model capable of representing skewed patient survival while also incorporating the causal patient information in the form of a BBN. Conditional phase-type distribution The conditional phase-type (C-Ph) distribution is a new approach which incorporates a causal network of interrelated variables and probabilities along with a dependent continuous survival variable. One of the benefits of using the C-Ph models rather than the CG distributions is its appropriateness for skewed data. This methodology uses a Coxian phase-type distribution conditioned on a BBN where the phase-type distribution represents the continuous variable, the duration of time until a particular event occurs, and the BBN represents the network of interrelated variables. The C-Ph model could be used to model the duration of stay of elderly patients in hospital using a causal network of patient variables, for example, age, gender, and admission method. Figure 2 illustrates the model as consisting of these two components; a BBN of interrelated causal nodes which precede and predetermine (in a probabilistic sense), the second component, the effect node(s) which constitute the process. The effect node(s) here is (are) characterised by a continuous positive random variable(s), the duration, described by a Coxian phase-type distribution. The conditional phase-type model is defined as consisting of Causal Nodes C 5 {C 1,y,C m } belonging to the causal network, and Process Nodes Ph 5 {Ph 1,y, Ph n } representing the phasetype distribution. The Causal Network is modelled as a Bayesian belief network represented by the joint probability distribution of C({C 1,y, C m })

7 A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) Cause 1 Cause 2 Cause 3 Causal network Cause 4 Cause 5 Process model Phase 1 Phase 2 Phase 3 Event Fig. 2. The conditional phase-type model. PðCÞ ¼P PðC i jpaðc i ÞÞ ð9þ i where C i are the nodes in the network and pa(c i ) is the parent set of C i. The probability density function of the time for which the process is active, f(t), is defined by FðtÞdt ¼ probðprocess terminates ðt; t þ dtþþ: ð10þ The distribution of the process nodes Ph in the phase-type model may be found using (11) where pa( process) are the causal nodes which are parents of the process and p, Q and q vary according to the values of the causal nodes in pa(process) f ðtjpaðprocessþþ ¼ p expfqtgq: ð11þ The structure of the Bayesian belief network may be partially known due to expert advice. In the case of geriatric medicine, the clinician or geriatrician may already have experience or knowledge on how some of the variables may be interrelated. When this prior knowledge of the BBN structure is available, the BBN can be fitted for the data set using block recursive structures in the CoCo computer package (Badsberg, 1992). Block recursive structures represent the problem domain as an arrangement of blocks of variables according to prior knowledge about their possible relations. Variables contained within a block may only be connected by undirected edges thus representing an association between the variables. Potential causal relationships are only permitted between variables in different blocks and the connection represented by a directed edge. The CoCo package fits and tests various BBNs, using maximum likelihood estimation or information criterion, until a final model is selected that best describes the data set and relationships between variables. If prior knowledge of the BBN structure is not known, the package uses the maximum likelihood approach to assess all possible candidate networks and select the most appropriate. The parameters of the Coxian phase-type distributions for each set of values in pa( process) are estimated using the maximum likelihood function and the Nelder-Mead simplex algorithm (Nelder and Mead, 1965). A sequential procedure is adopted, to fit the phase-type distribution, whereby increasing numbers of n phases are tried for each cohort of individuals, starting with

8 572 A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) Causal component Gender Age Admission method Barthel grade Process model (Length of stay) Phase 1 Phase 2 Phase 3 Destination Leave hospital Fig. 3. Clinics four block structure. n 5 1 (corresponding to the exponential distribution), until there is very little improvement to the fit from adding an additional phase. Clinics C-Ph model A C-Ph model is developed to represent the Clinics data using the nodes, age, gender, admission method, Barthel grade ( patient dependency level), and destination to represent the causal network and the continuous variable patient duration of stay, to define the process model. A block recursive structure is utilised for the C-Ph model whereby a number of the interrelated causal nodes are grouped into four blocks according to the order in which they were recorded. Figure 3 illustrates the structure where the first block contains the patient personal details; age, gender, admission method, the second contains Barthel grade, the third contains the destination of the patient on departure from hospital and the final block, the continuous variable, duration of stay in hospital. The dotted edges in Fig. 3 represent the boundaries of the various blocks in the causal network. The CoCo package employs the coherent-direct strategy (Badsberg, 1992) along with the information criterion method to judge the suitability of the models and selection among models. Various candidate C-Ph models were investigated to find the most suitable representation for the elderly patients duration of stay in the Clinics data set. The most suitable C-Ph model selected to represent the Clinics data set is displayed in Fig. 4. The likelihood function for the C-Ph model is one of the contributing factors taken into account when selecting an appropriate model. For this particular model, the BBN structure will consist of the form in (12) where the product of joint probabilities is as follows: Y Pðx j ¼ i j jpaðx j ÞÞ ¼ PðBarthel gradejadmission MethodÞ j PðDestinationjAdmission MethodÞ PðAdmission MethodjAge and GenderÞ PðAge and GenderÞPðAgeÞ: ð12þ

9 A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) Gender Casual component Age Admission method Barthel grade Process model Phase 1 Phase 2 Phase 3 Destination Leave hospital Fig. 4. Clinics C-Ph model. The BBN structure chosen to represent the Clinics data set contains three undirected edges in the first block indicating the following associations between variables; the gender of the patient is associated with the age and admission method of the patient into hospital. The age of the patient is also associated with the admission method. The second block of the causal network contains the Barthel grade of the patient, the patient dependency on others to do everyday activities. The third block also features a single variable, the destination of the patients on departure from hospital. The admission method of the patient, (emergency admission, emergency GP, transfer, planned admission) has a direct influence on the Barthel grade and the destination of the patient (dead/home/transfer). The variables that directly influence duration of stay in hospital are Barthel grade and patient destination, the final outcome of the patient upon leaving hospital. Therefore the duration of stay of a patient in hospital appears to be directly linked with the patient s Barthel grade, measured on admission to hospital, and the patient destination from hospital. Table 1 displays the joint conditional probability distribution of patient Barthel grade and destination. It is reasonable that these two variables would have a direct influence on patient duration of stay in hospital. For example, those patients with a low Barthel grade of heavily or very dependent make up the majority (75%) of transfer patients where patients become increasingly Table 1 The joint probability distribution of the nodes destination and Barthel grade (includes the median duration of stay (in days) for each cohort) Barthel grade Destination Death Home Transfer Heavily dependent 0.09 (17) 0.31 (15) 0.05 (30) Very dependent 0.08 (37) 0.24 (18) 0.04 (51) Slightly dependent 0.03 (44) 0.04 (19) 0.02 (29) Independent 0.02 (8) 0.06 (13) 0.01 (30)

10 574 A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) dependent on hospital staff thus contributing to the transfer cases to nursing homes. The median duration of stay for heavily dependent patients is 30 days, which is a lot less than that of the very dependent patients who have a median duration of stay of 51 days. This initially surprising relationship is reasonable as the patients admitted as heavily dependent have already reached a further stage in their illness before admission to hospital and are transferred at a shorter duration of stay than the very dependent patients. The very dependent patients spend an additional period of time in hospital for two main reasons. The additional time may lead to recovery and final discharge or it may reflect a further progression in the patient s illness to heavily dependent and transfer to nursing-home care. Alternatively patients who are independent are more likely to return home quite quickly (median length of stay of 13 days). The duration of stay of the patients may be modelled using phase-type distributions based on the patient Barthel grade and destination of the patient. The different combinations of Barthel grade and destination provides 12 different groupings or cohorts of patients. The resulting phasetype distributions for patient duration of stay are shown in Table 2. The results show the duration of stay either as a single-phase distribution (in six of the cases) or a two-phase distribution in three of the cases with three cells in the table remaining empty. These empty cells represent very small cell sizes, cells with less than five observations, for cases that do not occur very often, that is patients admitted with Barthel scores higher than heavily dependent, (that is, very dependent, slightly dependent and independent ) who die in hospital. In fact, there were no patients who were considered completely independent on admission to hospital who later died in hospital. This would be expected as those patients who are more dependent on hospital staff and resources would be expected to be those who are more likely to die in hospital. The causal network may therefore be used to select the most likely situation, that is, Barthel grade and destination, for a patient based on the other variables in the network such as patient admission method and gender. The patient duration of stay may then be modelled using the estimates of the parameters of the phase-type distribution representing the cohorts of patients, as shown in Table 2. Table 2 Probability density function of patient duration of stay in hospital Probability density function of length of stay Destination Barthel grade 5 Heavily dependent Barthel grade 5 Very dependent Death f(t) e 0.04t e t * Home f(t) e B0.047t e t f (t) e Transfer f(t) e t f (t) e t t Destination Barthel grade 5 Slightly dependent Barthel grade 5 Independent Death * * Home f(t) e 0.041t e t f (t) e Transfer f(t) e t f (t) e *Not enough observations to estimate the parameters of the phase-type distribution t t

11 Conclusion A. H. Marshall and S. I. McClean/Intl. Trans. in Op. Res. 10 (2003) The conditional phase-type distribution is described in detail and used to represent the duration of stay of patients in hospital by conditioning the Coxian phase-type distribution on a Bayesian belief network. The benefit of using such a model is that BBNs allow for the representation of the variables as a graphical structure, providing users with a better understanding of the system and facilitating the inclusion of prior specialist knowledge about variable dependencies and causal relationships. The phase-type distributions permit the representation of continuous variables which are not constrained by the conditions of normality. Thus the combination of these two approaches to form the conditional phase-type distribution provides a way for modelling the duration of stay of patients in hospital. This paper has used the Clinics data set to demonstrate the development of the C-Ph model for patient length of stay. The C-Ph model may be used to estimate the patient length of stay in advance, so that clinicians may have prior knowledge of demands for resources. For instance, advance knowledge of the length of stay of current patients in hospital will assist the hospital managers in predicting the forthcoming workload for staff and the resource implications and scheduling issues involved with such a load. The resulting model in Fig. 4 produced various different models for patient length of stay according to Barthel grade and destination. The destination of the patient may be predicted using the causal network and used along with the patient Barthel grade to estimate the corresponding patient length of stay. Thus, clinicians could use the patient information available on admission to hospital to help predict the patient destination from hospital and upon assessing the patient s likely destination, the model will enable the prediction of the patient s anticipated length of stay. The model would therefore provide approximate predictions for patient length of stay according to the category of patient Barthel grade and destination. However there are areas for improvement within this modelling approach. The current C-Ph model assesses the patient variables measured on admission to hospital and models length of stay. The model could be improved by the inclusion of additional clinical variables that concern the condition of the patients which will most probably change with time as the patients continue their stay in hospital. For example the patient may undergo tests whose results may influence their stay in hospital. Additionally, the Barthel grade of a patient may change dramatically during their stay in hospital. If the model included such patient information which was able to change dynamically as time progresses, the accuracy in predictions could be improved while also assisting in the further understanding of the interrelationships between clinical variables. Another aspect worth investigating is the implementation of the C-Ph model for assessing the flow of patients throughout the hospital using discrete event simulation. Acknowledgements The authors wish to thank Professor Peter Millard for kindly providing valuable comments, feedback and the data for this study.

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