Computational and Mathematical Methods in Medicine

Volume 2018, Article ID 6367243, 13 pages

https://doi.org/10.1155/2018/6367243

## Understanding Dynamic Status Change of Hospital Stay and Cost Accumulation via Combining Continuous and Finitely Jumped Processes

^{1}School of Finance, Zhejiang University of Finance and Economics, China^{2}School of Data Sciences, Zhejiang University of Finance and Economics, China

Correspondence should be addressed to Xiaoqi Zhang; ude.olaffub@hziqoaix

Received 1 March 2018; Accepted 23 April 2018; Published 10 June 2018

Academic Editor: Kazuhisa Nishizawa

Copyright © 2018 Yanqiao Zheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Coxian phase-type models and the joint models of longitudinal and event time have been extensively used in the studies of medical outcome data. Coxian phase-type models have the finite-jump property while the joint models usually assume a continuous variation. The gap between continuity and discreteness makes the two models rarely used together. In this paper, a partition-based approach is proposed to jointly model the charge accumulation process and the time to discharge. The key construction of our new approach is a set of partition cells with their boundaries determined by a family of differential equations. Using the cells, our new approach makes it possible to incorporate finite jumps induced by a Coxian phase-type model into the charge accumulation process, therefore taking advantage of both the Coxian phase-type models and joint models. As a benefit, a couple of measures of the “cost” of staying in each medical stage (identified with phases of a Coxian phase-type model) are derived, which cannot be approached without considering the joint models and the Coxian phase-type models together. A two-step procedure is provided to generate consistent estimation of model parameters, which is applied to a subsample drawn from a well-known medical cost database.

#### 1. Introduction

Rising expenditures and constraints on health care budgets have prompted the development of a variety of methods for the analyses of hospital charge and length of stay (LOS) as discussed in Gold [1], Lipscomb et al. [2], and Lin et al. [3]. Correctly fitting the charge and LOS data is a critical step in optimizing the allocation of healthcare resources. But due to the protection of private information, the detailed information regarding the treatment process that patient experience in hospital is not available from many well-known medical outcome databases, like the New York State’s Statewide Planning and Research Cooperative System. The missing longitudinal information of the treatment process makes it more challenging to generate good fitting; meanwhile it becomes demanding to have a dynamic model, through which effective inference can be made against the hidden treatment process. To that goal, a bunch of stochastic-process-based models have been well developed and applied to analyze the medical datasets.

The continuous-time Phase-Type (PH) model has been widely used in the study of hospital charge and LOS data. Many authors focus in particular on a special subclass of PH model/distribution, namely, the Coxian phase-type (CPH) model/distribution Tang [4]; Faddy et al. [5]; Marshall et al. [6–8]; Fackrell [9]. Unlike other popular theoretical distributions widely used in inpatient data, such as log-normal and gamma distribution, the CPH model/distribution not only provides a theoretical distribution that can be used to fit the empirical data, but also gives us a sketch of the treatment dynamics that patient experience in hospital. In fact, from CPH models, we can track the pathways that patient went through in different medical stages (characterized by the discrete set of phases in the PH model) during a hospital stay. The pathway information makes it possible to cluster patients and facilitate the use of healthcare process improvement technologies, such as Lean Thinking or Six Sigma McClean et al. [10, 11].

The other popular approach to study hospital charge and LOS is through dynamically modelling the charge accumulation process and the determination of the time to discharge, which belongs to a more general class of joint models of the longitudinal measurements and time to event, Ibrahim et al. [12]; Tsiatis and Davidian [13]; Henderson et al. [14]; Kim et al. [15]; Sousa [16]; Lawrence Gould et al. [17]. In medical cost studies, the charge accumulation is a monotonic nondecreasing process; the joint model used in this case is reduced to a class of random growth with random stopping time (RGRST) models.

Like CPH models, the RGRST models do also capture the treatment dynamics that patient experience in hospital. But in contrast to tracking the pathways of patient moving through different medical stages, the RGRST models focus more on describing how patient and/or doctor makes the discharge decision in reaction to the change of actual charge level and the length of time that patient has stayed in hospital. Therefore, the story of RGRST models is more about the behavioural patterns of patient/doctor behind the treatment dynamics, while the story of CPH models is more on the medical side.

It is natural in this paper to think of the possibility of combining CPH models and RGRST models together in order to extract more information regarding the discharge decision-making on different medical stages. However, there is a natural gap between the two models. The CPH model is a finitely jumped stochastic process in essence, while the charge accumulation in the RGRST model is continuous. It is not trivial to combine a jump process with a continuous process. To deal with that difficulty, we propose a partition-based approach with each partition cell determined by solving a boundary differential equation. These boundary differential equations are subtly designed to merge the continuous charge into discrete “phases” involved in a Coxian phase-type model. In sum, the main contributions of this paper are as follows:

(i) We show that there is a natural way to convert a special subclass of RGRST models to CPH models.

(ii) We propose an algorithm to estimate the transition matrix of the CPH model converted from a given RGRST model and the parameters involved in that RGRST model.

(iii) Based on the correspondence between RGRST models and CPH models, we derive a variety of different measures of the “cost” of staying in a medical stage at each time. That “cost” information is important for the purpose of insurance payment and healthcare process improvement.

McClean et al. [11] tried a different way to incorporate the charge accumulation process into a CPH model. But in their work only the case that the charge accumulation process adopts a piece-wise linear form was discussed. It turns out that the piece-wise linear assumption is quite restrictive while crucial to their main result. Without it, the matrix technique in McClean et al. [11] is no longer applicable to achieve the th order moments of total charge for , while our differential-equation-based approach does still work. In fact, we believe our method extends the work of McClean et al. [11] in the following two aspects.

Instead of being piece-wise linear, we consider a much more general situation in which the charge accumulation process can take arbitrary forms as long as a conditional expectation function of that process satisfies a general regularity condition. In particular, within our framework, it is possible to consider the potential influence of the current charge level on the future charge accumulation which is neglected by the piece-wise linear assumption.

In addition to the moments of total charge, it is derivable from our model of the joint distribution of the total charge and LOS, and the joint distribution of the costs and time being spent on every stage by every fixed time . Therefore, our model provides more detailed information of the treatment that the patient experiences in hospital.

Although the motivation of our work is the analysis of the charge accumulation and the determination of hospital length of stay, it turns out that the proposed method is useful for many other problems where the relation among the time to event and a hidden continuous process as well as a jump process is in interest. For example, in the field of investment risk management, it is always important to detect how the default probability of the corporate bond issued by a firm is affected by the growth stage and profitability (say measured by the flow of revenue) of that firm. In this case, our model can definitely provide some insights if we identify the default as the event in interest and consider the revenue flow as determined by a continuous process similar to the charge accumulation and the transition among different growth stages of the firm as described by a CPH process. In addition to problems of the survival-type, it is also natural to extend our work to the case of competing risks, of which every stage in our model can be identified with a type of risk. Although in competing risk models, the CPH transition matrix is no longer sufficient, it turns out that the partition-based technique introduced below is extendible to derive the joint distributions of a wide class of the competing risk models, the details of which will be discussed in a related paper by the authors.

This paper is organized as follows. In Section 2, after a short review of the CPH models and RGRST models, we present the correspondence between them and briefly introduce the estimation algorithm. In Section 3, we conduct numerical studies to show the validity and usefulness of our model. A couple of interesting findings toward the medical outcome database, the New York State’s Statewide Planning and Research Cooperative System 2013, are discussed. Section 4 concludes the paper.

#### 2. Model

In this section, a new model (denoted as CPH-RGRST model) is constructed that connects the CPH models to RGRST models in the sense that

a CPH-RGRST model is a RGRST model;

charges in a CPH-RGRST model can be classified into a number of stages such that every stage is identified with a phase in a given CPH model in the sense that, at every time , the probability of staying in a stage is exactly given by the probability in the th phase of the CPH model.

In particular, the marginal distribution of LOS induced by a CPH-RGRST model is a CPH distribution. We shall state the detailed construction of the CPH-RGRST models after a brief review of the definition and some basic properties of RGRST models and CPH models.

##### 2.1. the Joint Model (RGRST) versus the CPH Model

A RGRST model can be formally defined as follows as discussed in Gardiner et al. [18, 19] and Polverejan et al. [20]:

where the process represents the actual charge level at each time. The random variable indicates the LOS, and is the indicator function. ( for short) is the event process representing whether or not to stay in hospital for longer time at each time point . is a nonnegative process characterizing the potential increment rate of charge per unit time provided that patient decides to stay, and is a nonnegative random variable representing the charge at the initial time. We shall denote by the potential charge accumulation process in distinguishing the actual charge process .

As shown in the supplementary materials (available here) note that a RGRST model can be completely specified by the initial probability density function (pdf), , induced by the initial charge and the following two conditional expectation functions:

And using (2), the joint probability density function (pdf) of the LOS () and the total charge () at the discharge time can be expressed as follows:

where the function in variable is the time-dependent pdf induced by . The detailed derivation of (3) can be found in the supplementary materials. Expression (3) is useful in the estimation algorithm stated in the next section as it is the key component of the likelihood function.

To associate RGRST models with the CPH models, the hospital length of stay, represented as the random variable in (1), should induce a CPH distribution generated from a CPH model, which is a finite-state continuous-time stationary Markovian process with only one absorbing state/phase (we shall use the term “phase”, by convention, in place of “state”). A CPH model is determined by an initial probability mass vector with and , and the transition intensity matrix

where and the entry of represents the transition intensity of a patient from phase to phase at every time ; formally:

As suggested in McClean et al. [10], a phase in a CPH model can be identified with a treatment stage during hospital stay, such as diagnosis, acute care, assessment, rehabilitation, and long-stay care. The transition of patients among these stages characterizes the treatment progress.

##### 2.2. Correspondence between CPH and RGRST Models

The main result of this section is that there does exist a correspondence between CPH and RGRST models. The correspondence is built through converting the continuous variable, charge, in a RGRST model to finite many discrete states by partitioning the product space, (representing charge and time, respectively), into a number of cells such that each cell corresponds to a phase in a CPH model, while the evolution of the probability of staying in those cells is exactly determined by the given CPH model. More precisely, we have the following theorem.

Theorem 1. *Fix a RGRST process represented as a triple with being the pdf of initial charge and , as defined in (2). Suppose functions , , and are smooth and , satisfyfor some constant . Then, for any fixed positive integer , an -dimensional vector with , and an -dim vector , there exists an partition of the space denoted as such that the following time-dependent probability mass function defined on the tuple :is generated by a CPH model with the initial mass and its transition matrix is given as in (4) with for .*

The proof of Theorem 1 is presented in the Appendix. From the proof, it is clear that the connection between RGRST models and CPH models is equivalent to a constraint put on the conditional probability function in (2) of the underlying RGRST model by the condition (6). In fact, the functional form of is completely determined by (6) and the function , which gives a first-order partial differential equation (PDE) of . This equation turns out to be solvable and has a unique solution for a given boundary condition. Therefore, using the characteristic method, Evans [21], we can solve (6) and express the function as follows:

where evaluated at is constantly which means that all patients have to stay in hospital for a positive time before discharge; the form of the boundary and the value of the function on (denoted as ) are constructed by iteratively solving the Initial Value Problem (IVP) (A.11) in the proof; the details of the iteration are presented in Algorithm 1 of Corollary 2. is the first time when the solution trajectory () of IVP (A.11) (starting from ) touches the boundary curve . Equation (8) is crucial to determining the parametric form of the joint pdf (3) and the likelihood function used for estimation.