Using Compartmental Models to Predict Hospital Bed. Occupancy

Transcription

1 Using Compartmental Models to Predict Hospital Bed Occupancy Mark Mackay and Michael D. Lee Department of Psychology University of Adelaide Running title: Predicting Bed Occupancy Address for correspondence: Mark Mackay Department of Psychology University of Adelaide SA 5005 AUSTRALIA Telephone: Facsimile: Electronic Mail: mark.mackay@dhs.sa.gov.au

2 Predicting Bed Occupancy 2 Summary Objectives: Hospital beds are an expensive resource. The modelling of hospital bed occupancy should lead to better decision-making concerning the use of this expensive resource. The authors use cross-validation to compare two approaches to developing compartmental models that can be used to describe and predict hospital bed occupancy. Methods: Compartmental models have been used to describe hospital bed occupancy by other authors (e.g. Harrison and Millard, 1991). Data describing census days and the year was modelled using compartmental models. The absolute error provides a simple statistic that can be meaningful interpreted in this applied context and was used to compare the fit of the various models to training and test year data. Results: A compartmental model can meaningfully represent acute care data. The absolute error of the model based developed using the entire training year data set was lower than the absolute errors of all census models. Conclusions: Model creation should be based upon the consideration of as many points of data as required in order to capture the variation within the data. Census models are unlikely to capture the variation in the data. The prediction of patient flow based upon compartmental flow models will be improved if based upon the consideration of more data. The use of more robust models should result in better decision-making. Keywords: occupancy, hospital beds, model, prediction

3 Predicting Bed Occupancy 3 Introduction In recent years the Australian public health sector has seen increasing pressure to do more with the same or reduced levels of expenditure. At the same time, the number of beds in public acute care hospitals has been declining, while the level of demand and patient expectation is increasing. Consequently, the provision of public health care has become a political risk, often with debate not progressing beyond the number of hospital beds provided by the different political parties. The percentage of the gross domestic product expended on the health sector in Australia has increased since While there was a slight reduction in the level of expenditure as a percentage of GDP during the early to mid 1980s, growth has continued during the 1990s [1]. This phenomenon is not unique to Australia, but has occurred across countries with which Australia is often compared in relation to health care, such as the United States of America, the United Kingdom and New Zealand. During the same time, the percentage of expenditure that relates to provision of public health care services in Australia has grown. Similar growth has occurred in the USA, but not in New Zealand or the United Kingdom. It is widely acknowledged in journals and numerous reviews [2 3] that, particularly in view of the ageing society, which is likely to result in increased demand for services, workforce issues and rising costs, the current level of health care provision is not sustainable. One powerful way to optimise resources is to build and use models of key aspects of the health care system. The ability to infer the underlying process that generated observed data is the goal of most behavioural research [4] and the goal of modelling the use of acute care hospital beds is no different. Developing a formal and quantitative model allows interpretation, understanding, and insight into how and why the distribution changes; the ability to generalise where data are not available (e.g., other hospitals); and the ability to make predictions where data cannot be available

4 Predicting Bed Occupancy 4 (e.g., into the future). Ultimately, the predictive capabilities of models enables them to be applied to the issue of resource allocation, including the determination of the number of hospital beds required for service provision for a given hospital or community. Compartmental flow models have been previously shown to be of use in describing geriatric patient bed occupancy [5 8]. These models are intuitively plausible, easily interpreted and can be used to examine the flow of patients through the system. The application of this approach to the acute care system, however, has been limited [9]. Against this background, the purpose of this paper is twofold. First, to reinforce that the compartmental flow models can be used to model acute care data. Secondly, to consider how flow models can be developed to provide the best results for the predictive modelling of hospital bed occupancy. Modelling Bed Occupancy The modeling of hospital beds and patient length of stay, which are intertwined, is not new. For example, work in this area has been undertaken by Yates [10], Pendergast and Vogel [11], and Sorensen [12]. Some of this work, such as that by Sorensen, develops models reliant upon the Average Length Of Stay (ALOS), which could be argued to be a slightly more advanced approach than what is often undertaken by health care managers and clinicians. It has been recognised, however, that the use of the ALOS for modeling hospital bed occupancy is flawed [13 14]. There are theoretical and practical reasons that using the ALOS is inappropriate for use in the development of models. First, the length of stay profile typically has a highly skewed distribution and that is not well summarised by its mean value. Secondly, the length of stay distribution is complex, often consisting of mixtures of patient types (i.e., medical and surgical, planned and unplanned admissions, young and elderly) and mixtures of outcomes (i.e., some patients die, some are discharged home, some to alternative care services such as nursing homes).

5 Predicting Bed Occupancy 5 While it might be argued that the introduction of casemix categories could reduce some of the complexity, recent work indicates that the problems associated with the average length of stay are still not overcome [15]. Furthermore, the ALOS does not take into account the time of day when a hospital is most busy. That is not to say, however, that some of the work previously undertaken has not yielded interesting findings, such as the focus on discharge destination by Sorensen [12]. Compartmental Modelling Compartmental models were introduced as a means of looking resource implications concerning hospital beds by Harrison, McClean and Millard [5 8,16]. These models recognise that patients do not flow through the hospital in a uniform manner, and so describe the flow of patients through notional time-related compartments. The model work was developed using data relating to the hospitalization of geriatric patients in the South of England. Mackay and Millard [9,14] extended the model to the acute care sector using data relating to the hospitalization of patients in acute care settings in Australia. Compartmental models describe the flow of something, such as patients, through a system, where the system is comprised of a finite number of homogeneous subsystems known as compartments. According to Godfrey [17], compartmental models have been widely applied as modelling solutions in the areas of biomedicine, pharmacokinetics and ecology. Harrison and Millard [5] drew an analogy to the decay of drugs and the decline of patient stay over time when initially developing this form of bed modelling. Although the compartment models may consist of many compartments, work to date has focused on two or three compartment models to describe the patient stay profile within the hospital [9,16], with additional compartments being added to incorporate the community [18]. Figure describes the flow of patients in a generic compartmental model of hospital bed occupancy.

6 Predicting Bed Occupancy 6 [Figure 1 about here.] The modelling work of Harrison was incorporated into software known as the Bed Occupancy Management and Planning System (BOMPS) 1. The software provides two mechanisms for creating the bed occupancy profiles: a daily census or an annual average day. Most of the work undertaken has focussed on the use of the daily census, where the parameters that describe the various rates of flow are derived using the occupancy profile data of a single day. This practice raises issues concerning the ability of these models to generalize and make predictions. For example, if the census method is used based upon data from a Monday in February, it is reasonable to question whether the obtained bed occupancy predictions distribution will generalise to data for other days of the week, or other months of the year. It is possible that the annual average day approach may be better, because it considers the variation in patterns of bed occupancy across different days. The purpose of this paper is to consider which methodology is likely to provide the best results for the predictive modelling of hospital bed use under a compartmental flow model. Evaluating Models Whenever a model makes predictions about data, there is a variety of performance measures that can be generated to describe the degree of fit between the two. For example, the least square and absolute error values are two common measures used for this purpose [19]. The BOMPS software provides a range of output including the correlation between the model and data, the least squares value and also a visual presentation of the fit. Optimization of such performance measures may lead to a model that best fits the data. However, it may also result in overfitting of the data, leading to a model that describes the current data well, but is less useful in predicting the future. Cross-validation is a recognised approach that can be used to determine

7 Predicting Bed Occupancy 7 whether models generalize to another data set [19]. There are two approaches that can be employed. One approach is to with-hold some of the data from within the period being examined and analyze the fit of the model to these data. The second approach is to use data from a different period, such as a future or prior period, and determine the fit of the model to these data. In both instances, the data are related, but the period of time differs. We suggest that the latter approach provides a better method of assessing generalisability for forecasting purposes, since the variation from year to year is likely to be greater in a hospital environment than if assessment is based upon the same period. It also reflects the nature of the applied problem, where the emphasis is on forecasting future demand, rather than generalizing forwards and backwards in time from a limited sample. Accordingly, in this paper we use cross validation of a training set to a test year to assess the predictive capability of flow models generated from census and average data. Methods De-identified data from a large teaching hospital within South Australia were used as the basis for this research. The data related to patients treated within a medical division or service, and therefore excluded the majority of patients who had been admitted for surgical procedures. Same-day elective or planned admission patient data were excluded from the analysis, because the beds for these patients were managed differently. Often the beds were chairs, or the area was only staffed Monday to Friday during normal business days, or both, and thus these beds were not generally available to all patients. The data included the date and time of patient admission and discharge. A subset of the data was used to create a profile of the busiest time of day at the hospital based upon bed occupancy at various times of the day using the admission and

8 Predicting Bed Occupancy 8 discharge data. As a consequence of this analysis, instead of using midnight census data for the remainder of the analysis, a midday day bed census profile was created for each day of and financial years. The profiles provided a count of how many patients were in bed at midday for a given date and how many days patients had been in bed (i.e., days since admission). The data was used as the training data and the data was used as the test data for cross validation. Formally, these data can be represented by a set of counts d train it, counting the number of patients with length of stay j days on the t-th day of the training year, and d test it, counting the number of patients with length of stay j days on the t-th day of the test year. We considered compartment models based on ten different data sets generated from the training data. Nine of these were simply a single day in the training year, and so followed the census approach. Seven of the census days were chosen at random, while the other two were chosen because they were the training days showing maximum and minimum occupancy across the year. The ninth data set was the arithmetic average of all days in the training year. Formally, each of these data sets takes the form of a vector of counts, y =(y 1,...,y n ), where n is the maximum observed length of stay. Where the c-th day in the training year is chosen as the census day y t = d train ct, while the annual average uses y t = i d train it. For all data sets we used two-compartment models, which predicts a length of stay count at t days by ŷ t = Ae bt + Ce dt, where A, b, C, and d are non-negative free parameters. The maximum likelihood values of these parameters, A, b, C, and d were found assuming that the counts followed a Poisson distribution. This is a common statistical assumption in modeling count data [20], and previous research involving the counts that define length of stay distributions in hospitals has used this assumption successfully [15,21]. The likelihood function is

9 Predicting Bed Occupancy 9 accordingly given by p (y A, b, C, d) = t yŷt t e yt Γ(ŷ t +1), where Γ ( ) is the Gamma function Γ (z) = 0 u z 1 exp ( u) dt. Results Consistent with previous findings, the occupancy profile based upon time since admission was found to be highly skewed (M =13.03, SD =28.59, and Skewness = 3.23). To evaluate the census day and annual average approaches, the goodness-of-fit achieved by the maximum likelihood models was measured for each census day model against the one day in 1998 from which it was generated, and for the annual average model against the average day for To measure goodness-of-fit, we relied on the absolute error because it provides a simple statistic that can be meaningfully interpreted in an applied context. Denoting the maximum likelihood compartment model counts as ŷt = A e b t + C e d t allows this error measure, relating to the single (census or average) day data set, to be expressed as E single = t y t ŷ t. The ability of the models to fit the data for all of the training days was measured by calculating E train = d train it ŷ t. i t Of course, in making this comparison, it should be noted that the annual average model has the advantage of having used all of the training year data in the first place to determine its parameter values. For this reason, the evaluation of the models against the test year data, which was not used to determine any parameter values, provides a true

10 Predicting Bed Occupancy 10 comparison of the predictive abilities of the census day and annual average approaches. These predictive errors, for all models, are given by: E test = d test it ŷ t. i t Figure 2 presents the results of these goodness-of-fit analyses using the E single, E train and E test measures. It can be seen that all of the models fit the individual days on which they were trained well. This is supported by the correlations between the model and training data, which ranged from to As expected, the predictions of the census day models for the remainder of training year are much worse than the average annual model. Most importantly, however, Figure 2 shows that the model generated from the annual average data continues to outperform the census models when evaluated against the test year data. [Figure 2 about here.] Discussion Hamel and Prahalad [22] have argued that implementation failures are really past foresight problems. The creation of credible models that can be used to describe and forecast hospital bed needs can help improve decision-making relating to hospital beds and reduce foresight problems that result in poor outcomes. Such modelling is important for three reasons. First, the models can be used to assist in decision-making around current and future resource utilization, such as bed management and workforce planning, ideally leading to better outcomes. These outcomes may not relate solely to the number of beds opened or closed, but may be more diverse, such as influencing the number of training places made available for health professionals to meet future forecast demands.

11 Predicting Bed Occupancy 11 Secondly, the creation of a bed model also provides decision-makers with the ability to pretest decisions relating to how current configurations might be changed in the future. McClean and Millard [7] have previously emphasized this as an important reason that such work should be undertaken. It enables decision-makers to experiment with various options prior to implementation, thus reducing the likelihood of mistakes occurring when the decision is implemented. Thirdly, the models can provide the foundation for more complex modelling work, assuming that they have a sound theoretical basis and capture the events being modelled in a useful manner [23]. The modelling of hospital or institutional (e.g., nursing home) beds is important, because this is where much of the service provision activity occurs in relation to admitted patients. However, linking such modelling to other activities that are involved in the total service provision may also be informative. For example, linking pharmacy, theatre, pathology, allied health and hotel services may provide a more complete picture of the relationship between different components of the acute care system. Changes implemented in one part of the system, be it an increase in the number of beds opened or a reduction in available theatre time, are likely to influence other components of the systems. Sometimes such outcomes may be counter intuitive. The one-day bed census methodology developed by Harrison and Millard [5] and validated and extended by Harrison [16] and others [6 8] also fitted our data well. The models we tested all performed well in terms absolute error and correlation statistics when measured against the training data. The results of our work suggest, however, that a well-fitted model based upon a single day census derived model may not lead to the best predictions when compared to one based upon the annual average census model. The results presented in Figure 2 indicate that in terms of absolute error, the annual average model was consistently better than the other census models when

12 Predicting Bed Occupancy 12 measured against both the training and test year data. This finding is consistent with the common modeling findings [19,24] that the training error of a model does not necessarily predict the test error well. The reason for the superior prediction performance of the annual average model over the census models is likely to stem from the fact that the model is based upon the consideration of many more data: Approximately 23,600 for the year compared to a maximum of 94 for any of the census days. The effect of increased number of data is to smooth the seasonal and daily patterns of fluctuation in demand and supply. This has the effect of reducing uncertainty about whether the model has captured the behaviour across the period, in terms of the flows of patients and bed use, that would otherwise exist with picking a day at random upon which to base the model. Conclusion Millard, Harrison and McClean have previously demonstrated that the ability to create compartmental models that fit hospital bed occupancy profiles well exists. This work confirms that such models can be created for a medical acute care service. However, we believe model creation should be based upon the consideration of as many points as necessary to capture the variation within the data. Our results suggest that a single day census model is unlikely to do this. Rather, an annual average model appears to provide better performance in terms of capturing the variation within the training data and predicting future events. The contribution of this paper is to show the potential improvement in prediction of patient flow using compartment models that are learned from more data. The value of establishing models that are capable of generalization is that it enables the question of policy and strategic decision-making to be explored in a more robust manner.

13 Predicting Bed Occupancy 13 Acknowledgments The authors thank Professor Peter Millard for his valuable comments regarding an earlier draft of this paper.

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17 Predicting Bed Occupancy 17 List of Figures 1 A diagrammatic representation of the flow of patients through the hospital system. The compartments are notional, in the sense that the patients may not actually change location within the physical hospital Error values for the ten different two-compartment flow models against single-day, training year, and test year data

18 FIGURES 18 Compartments Patients Enter System Compartment 1 (e.g., Short Stay Patients) Patients Leave System Flow To Next Compartment Compartment 2 (e.g., Long Stay Patients) Patients Leave System Flow To Next Compartment Compartment n Patients Leave System Figure 1: A diagrammatic representation of the flow of patients through the hospital system. The compartments are notional, in the sense that the patients may not actually change location within the physical hospital.

19 FIGURES Apr 99 Absolute Error Dec Dec Jan Feb Oct May Jul Sep 98 All Days 0 Single Day Training Year Test Year Evaluation Data Figure 2: Error values for the ten different two-compartment flow models against singleday, training year, and test year data.