Characterising Hospital Admission Patterns and Length of Stay in the Emergency Department at Mater Dei Hospital Malta

1. Introduction

By forecasting daily resource requirements for admissions, healthcare planners can develop a plan to ensure the efficient and effective quality of service at a minimal cost [8] to ensure the ideal use of resources [11]. Complex strategies are often required to solve problems of admission scheduling and resource requirements to efficiently and effectively manage the healthcare system. Healthcare planners frequently experience dilemmas of ensuring equitable allocation of hospital resources when faced with long waiting lists and overcrowded emergency departments having patients waiting for admission. Thus, by finding an efficient solution to this problem, it is possible to help healthcare resource managers, hospital staff, and policymakers make the hospital more efficient. The aim of this project included developing a mathematical model, which may be used to model LOS and admissions patterns through PTS trees [8,10,11,12,28,29,30]. Developing this model will help healthcare professionals create policies that ensure the optimal allocation of the limited resources available. This model would then predict patients' LOS and the number of admissions for independent data.

1.1. Background and Previous Research

There has been tremendous interest in estimating the hospital length of stay from a long time [31,32] and the factors affecting it [17,18] and recent studies [33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85]. There are some recent reviews of machine learning and statistical methods for the hospital length of stay estimation. [43,48,51,62,68]. Keegan [18] argued in favour of evidence showing that bed occupancy rate is a reliable key performance indicator for hospitals' capability to provide good quality care to patients. Cooke et al. [3] suggested that if a border is set and the bed occupancy rate is below it, then the waiting times in the Emergency Department will be reduced exceptionally. Jones [17] discussed several factors which affect hospital occupancy demand, including temperature (admissions may increase or decrease the possibility of certain conditions), injuries and infections (these do not happen during certain periods and have no significant patterns), clinical practice changes, weather and environmental factors such as viruses and the rising need of end-of-life care.

Many models have been developed in the past, some of which include queuing models (Worthington D. J. [24], Gorunescu et al. [9]). In contrast, others use computer simulation or population ratio-based models (Kuzdrall et al. [19]). Fackrell [4] suggested that a realistic approach to modelling a care system uses compartmental models mainly implemented using phase-type Distribution (PHD). However, these models are unsuitable for admission scheduling and capacity planning since they mostly model patient flow within the hospital and do not consider re-admissions or community care. Garg et al. [20] proposed an alternative model developed using Markov chain modelling where the resource necessities could be forecasted using patient pathways in the future. In this study, Markov Chains were used to construct a policy that satisfies any future resource availability for the care system. However, this study is limited since it assumes a fixed number of daily admissions, which is unrealistic and cannot be used for many practical situations. In a further study by Garg et al. [11], a model was introduced where scheduling admissions, allocating resources and forecasting requirements could be achieved. This model improves the previously mentioned model [20] since it may be used for fixed and variable daily admissions. The model presented in the study [11] considers two time-dependent covariates: the current calendar year and the patient's current age. The covariates for each patient are updated daily to have a more realistic model. This model is mainly based on predictable values and can therefore forecast the number of expected patients in each phase and the daily cost of care. The authors of a further study (Garg et al. [8]) propose using covariates, such as age, gender, time of admission and diagnosis, to carry out better hospital capacity planning since these characteristics affect a patient's LOS. In this study, a PTS tree is used for smaller patient groups with respect to the LOS distribution using the characteristics as a base. In this paper, Garg et al. [8] propose an adaptable and flexible approach to intelligent healthcare planning and patient organisation, considering patient heterogeneity, essential variability and system complexity.

1.1.1. The seasonal Effect on Patient Admissions

Moreover, patients' admission could also be affected by several seasonal effects. Fullerton & Crawford [6] states that patient admission increases substantially during winter, mainly in general medicine and orthopaedics. This study also found fewer admissions during Christmas than the rest of the year. Green, Fullerton and Crawford ([6], [13]) agree that fewer procedures are booked for the weekend, and no elective patients are admitted to keeping bed space for weekend demand. Fullerton & Crawford [6] believe that the seasonal effect is predictable. Therefore, chaos within hospitals could be avoided if healthcare planners had to plot the admission rates better and better utilise the resources of primary care institutions.

1.1.2. Models Used to Estimate the Patient's Length of Stay

Estimating the LOS of patients helps resource planners analyse and estimate the hospital's bed occupancy and thus forecast the required resources. The two most popular methods that provided almost accurate approximations in previous studies when estimating patient LOS include the Gaussian Mixture Model and the C-PHD. Using phase-type distributions in various stochastic models allows algorithmically tractable solutions to be found. General PHDs are said to be over-parametrised (Fackrell [4], Marshall A, McClean S. [21]). C-PHD are a subclass of general PHDs which requires less complex parameter estimation (Fackrell [4]). The literature provides different methods to approximate the parameters [4]. These methods included:

Maximum likelihood estimation (Asmussen et al. [2], Olsson [23], Faddy & McClean [7], Faddy [5] and Hampel [14]).

Moment matching (Johnson [16])

Least squares

Splitting the main part and the tail part of the distribution (defined on positive numbers with a PHD) and approximating them separately (Horvath and Telek [1]).

2. Materials and Methods

In this study, the patient flow within the hospital system is to be modelled. In several studies ([20], Garg et al. [11]), patient flow is categorised through the rate of transition of a patient between states. It is assumed that there are n hospital phases (acute, treatment, rehabilitation, long stay) and m community phases (dependent, convalescent, recovered). Patients move sequentially from one hospital phase to the next and similarly from one community phase to the next. A patient may be re-admitted into the first hospital phase from any community phase and may be discharged from any hospital phase to the first phase or die at any point in the process. Furthermore, the set-up consists of n hospital phases and one absorbing state within this study. This may be represented as a discrete-time Markov chain having n+1 states. In this model, patients are admitted in the first state. They could leave the system at any other state (through death or by being discharged). Figure 1 describes the possible patient flow through the system, where: HP_i represents being in hospital phase i, λ_i represents the transmission rate between HP_i to HPi+1, and μ_i represents the transmission rate between HP_i and the absorbing state, death or community phase. Therefore, as mentioned above, this study models patient flow within the healthcare system using C-PHD and PTS. These two distributions are further described below.

2.1. Coxian Phase-Type Distribution

C-PHD is a special type of PHD. It is defined as an n-state continuous Markov process with a single absorbing state which may be reached from any phase instead of only being reached from state n. Admission into the system may only be from the first state, allowing sequential movement through the states [10]. This study uses C-PHDs to model patient flow, as shown in Figure 1. The initial state distribution, p, is defined as [10]:

$$p=\left[\begin{array}{c}1 0 \cdots 0 0\end{array}\right]$$

(1)

the vector q denotes the absorption probabilities and is defined as:

$$q={\left[\begin{array}{c}{\mu }_1 {\mu }_2 \cdots {\mu }_{n-1} {\mu }_n\end{array}\right]}^T$$

(2)

and the transition matrix Q is defined as [10]:

$$Q =\left(\begin{array}{cccccc}-\left({\lambda }_1+{\mu }_1\right)& {\lambda }_1& 0& \cdots & 0& 0\\ 0& -\left({\lambda }_2+{\mu }_2\right)& {\lambda }_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \cdots & 0& 0\\ 0& 0& 0& -\left({\lambda }_{n-1}+{\mu }_{n-1}\right)& {\lambda }_{n-1}& 0\\ 0& 0& 0& \cdots & 0& -{\mu }_n\end{array}\right)$$

(3)

From these definitions, the log-likelihood function is obtained [10], where N is the total number of patients in the healthcare system and t_i is the LOS of patient i.

$$L={{\displaystyle \sum }}_{i=1}^N\left(\log \left(\mathit{p}\exp \left(\mathit{Q}{t}_i\right)\mathit{q}\right)\right)$$

(4)

Within our study, the above function is used to create PTS trees.

2.2. Phase-Type Survival Tree

Survival trees are a regression tool used to perform survival analysis [10]. It is an effective and efficient method of collecting survival data and understanding its connection with covariates, results from treatments and LOS data. A PHD distinctly models each node in a PTS tree. PTS trees have been used to model many applications in medical research, including forecasting bed requirements. In Garg et al. [10] study, PTS trees are implemented to cluster patients' lengths of stay.

PTS trees are created by repetitively partitioning the data into subsections depending on covariates through splitting and selection conditions aiming to maximise within-node similarity or between-node splits [12]. Splitting maximises node similarity based on improving log-likelihood functions [12].

The weighted-Average information criterion (WIC) is a weighted average of Akaike's information criteria (AIC) and the Bayesian information criteria (BIC) with a small sample size correction [8]. As the splitting criteria based on the WIC combines the strengths of both the AIC and the BIC, it works well with small and large sample sizes and also in the case when the sample size is not known [25]. The following formula is used to calculate the WIC [8]:

$$WIC\left(d\right)=-2\left(Loglikelihood\right)+d+\left(\frac{d\left(\left(log\left(N\right)-1\right)log\left(N\right)\right){(N-\left(d-1\right))}^2+2N\left(N+\left(d+1\right)\right))}{\left(2N+\left(log\left(N\right)\left(N-\left(d+1\right)\right)\right)\right)\left(N-\left(d+1\right)\right)}\right)$$

(5)

Since C-PHD is being used, each node is modelled separately. If a covariate X has k values and the node splits into k partitions, then the WIC for the split can be calculated using equation 5. After splitting the node by the covariate, the WIC gain can be calculated in equation 6, where WIC₀ is the WIC before splitting [8].

$$WI{C}_{tot}\left(d_{tot}\right)={\displaystyle \sum }_{i-1}^k(WI{C}_{K_i}\left(d_{X_i}\right)$$

(6[M1] )

$$G_X=\left(WI{C}_0\left(d_0\right)\right)\left(WI{C}_{tot}\left(d_0\right)\right)=\left(WI{C}_0\left(d_0\right)\right)-({\displaystyle \sum }_{i-1}^k\left(WI{C}_{K_i}\left(d_{X_i}\right)\right)$$

(7)

This study uses split and selection criteria to maximise node homogeneity as recommended as the most suitable criterion by [30]. The node that minimises the WIC is selected to recursively divide the node into child nodes by starting at the root node. If a node with a negative gain occurs, then the node is set as a terminal node, and no more splitting occurs. This is the stopping criteria.

3. Implementation

The EMphy package is used in which all algorithms and models are implemented in MATLAB using a PHD fitting program [22]. This package, developed by Asmussen et al. [2], uses the expectation minimisation algorithm to calculate the maximum likelihood parameter estimations.

3.1. The Dataset and Some Data Analysis

The results obtained from this study are based on datasets provided by Mater Dei Hospital, Malta and Free Metro, Online. Mater Dei Hospital provided data for patients discharged during 2011 and 2012. The patient data provided by Mater Dei was confidential and did not include any personal information. The covariates present in the data provided by Mater Dei Hospital included gender, age, locality of residence, source admission and discharge location, admissions and discharge wards and admission and discharge dates. Admission type is a crucial distribution of admission data represented in Table 1. We divided admission into three groups and included fields as No. of admissions, total Days spent during LOS and Average days spent during LOS.

3.1.1. Admissions Data

Between 2011 and 2012, 66601 and 68903 patients were admitted to Mater Dei Hospital. Table 2 depicts the number of admissions occurring each day of the week for each month throughout both years. From Table 2, the maximum number of admissions (2423 patients) occurred on Mondays during January, whilst the least number of admissions (849 patients) occurred on Sundays in August. From Table 2 and Figure 2, it could also be seen that the maximum number of admissions occurred during October when 12208 patients were admitted to the hospital. The minimum number of admissions occurred in December when 10001 patients were admitted to the hospital. From this data, it was observed that January, October and November are the months most patients were admitted to the hospital. Additionally, it may be seen from Figure 3 that the number of admissions is reduced substantially during the weekend. Figure 4 confirms that fewer elective and day cases are admitted during the weekend, mainly Saturday and Sunday, compared to the rest of the week. It may then be seen that the number of admissions on a Monday is much higher than the rest of the week in all three types of admissions. This agrees with previously mentioned studies by Fullerton and Crawford [6] and Green [13]. These authors agree that fewer procedures are booked during weekends to keep bed space for weekend demand.

3.1.2. Covariants

The covariates Gender, Age, District and SourceAdm were grouped into three groups, four groups, seven districts and five groups, respectively. The Gender covariate groups are; Male, Female and Unclassified and the Age covariate groups are; Age 0 (NewBorn), Ages 1 to 30 (Under30), Ages 31 to 70 (Under70) and Ages 70 to 105 (Over71). Admissions of patients over 70 have an average LOS of approximately 5.91, longer than younger patients (excluding newborns) average LOS 6.8.

The Locality of Residence covariate groups may be seen in table 3. These groupings were carried out to have better performance when running. Residents of Gozo and Comino had the minimum average LOS of 2.92 days, while the North of Malta had the longest average of 3.89 days. The SourceAdm covariate (the source of admission) groups are; Private Residence Home/ Usual Residence, Home for the Elderly (including St Vincent de Paule Residence and Zammit Clapp Hospital), Other (including Gozo Hospital, Labour Ward, Nursery, Public Hospitals (Government Institutes including Boffa Hospital & Mount Carmel Hospital) and Private and Foreign Hospitals), Police Custody and Unknown. Table 3 gives the number of admissions, total, and average LOS for each covariate group.

3.2. Phase-Type Survival Trees

The following PTS trees were generated using the WIC-based spitting criteria for the emergency data provided by Mater Dei Hospital. This approach was used to analyse the length of stay and admission patterns.

3.2.1. Length of Stay Analysis

Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25 and Table 26 represents the steps used to generate the LOS PTS tree shown in Figure 5, representing the WIC-based splitting criteria from the LOS data against the covariates related to the patient's characteristics.

First, we fit the complete data (represented as a root node (1) in Figure 5 and Table 4) to Coxian Phase Type Distribution (C-PTD) and calculate its WIC. Then, for each covariate, we split the LOS data and fit each group individually to C-PTD, total the C-PTD and compare it WIC of the root node (i.e., before the split) to calculate the gain in C-PTD. We select the covariate providing the maximum positive gain in C-PTD to split the data. Table 4 shows that covariate “Age“ offers the maximum positive gain in C-PTD. Therefore, we select this split to grow the tree. New nodes are shown in Figure 5 as 2 (newborn), 7 (under 30), 10 (under 70) and 20 (over 70).

Step 2: we split the LOS data for each remaining covariate (Gender, District and SourceAd), fit each group individually to C-PTD and total the C-PTD and compare the WIC of the node before the split to calculate the gain in C-PTD. We select the covariate providing the maximum positive gain in C-PTD to split the data. Here, in Table 5, Table 6, Table 7 and Table 8, for nodes 2 (newborn) and 7 (under 30), covariate “Gender“, while for nodes 10 (under 70) and 20 (over 70) covariate “District” offer the maximum positive gain in C-PTD. Therefore, we select this split to grow the tree, and new nodes are shown in Figure 5 as 3, 6, 8, 9, 11, 12, 13, 14, 15, 16, 19, 21, 24, 27, 28, 29, 30, and 33.

Step 3: Nodes 3, 6, 8, 9, 11, 12, 13, 14, 15, 16, 19, 21, 24, 27, 28, 29, 30, and 33 are splitted by remaining covariates, and fit each group individually to C-PTD and total the C-PTD and compare it WIC of the root node (i.e., before the split) to calculate the gain in C-PTD. We select the covariate providing the maximum positive gain in C-PTD to split the data. Here, in Table 5, Table 6, Table 7 and Table 8 for node 2 (newborn) and 7 (under 30), we can see covariate “Gender“ offers the maximum positive gain in C-PTD, while for node 10 (under 70) and 20 (over 70) it is covariate District. Therefore, we select this split to grow the tree and new nodes are shown in Figure 5 as 3, 6, 8, 9, 11, 12, 13, 14, 15, 16, 19, 21, 24, 27, 28, 29, 30, and 33.

Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25 and Table 26 show that only nodes 3 and 16 provided significant WIC improvement with a split by sourceAd, and nodes 21, 24 and 30 provided significant WIC improvement with a split by gender. All remaining nodes did not provide any significant WIC improvement by any split and therefore considered terminal nodes. In Figure 5, there are 23 terminal nodes representing the data's significant clusters. The survival plots for all the terminal nodes are shown in Figure 6, which outlines the model's goodness of fit. The total gain in WIC obtained is 5782.13, where the WIC of the root node is 355134.92, and the terminal nodes' WIC sum up to 349352.79. It is possible to identify the relationship between the patient's characteristics and their LOS by analysing Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25 and Table 26 and Figure 5. From the table, the most significant split is by the covariate age, with a WIC gain of 6511.88. The mean LOS value may significantly differ between the age groups for each split. Figure 5 shows that the most significant split for newborns (node 2) and patients under 30 (node 7) occurred for the gender covariate. In contrast, the most significant split for patients under 70 (node 10) and patients over 71 further occurred for the district covariate. Figure 6 plots the survival function related to Personal Characteristics for each terminal node for the Length of Stay analysis to show the goodness of fit.

3.2.2. Admissions Analysis:

Table 27, Table 28, Table 29, Table 30, Table 31, Table 32, Table 33, Table 34, Table 35, Table 36, Table 37, Table 38, Table 39, Table 40, Table 41, Table 42, Table 43, Table 44, Table 45, Table 46, Table 47, Table 48 and Table 49 represent the steps used to generate the admissions PTS tree shown in Figure 7, representing the WIC-based splitting criteria from the admissions data against the covariates related to the patient's characteristics. Similar to the length of stay analysis above, here also, first, we fit the complete data (represented as a root node (1) in Figure 7 and Table 27) to Coxian Phase Type Distribution (C-PTD) and calculate its WIC. Then, for each covariate, we split the admissions data and fit each group individually to C-PTD, total the C-PTD and compare it WIC of the root node (i.e., before the split) to calculate the gain in C-PTD. We select the covariate providing the maximum positive gain in C-PTD to split the data. Table 27 shows that covariate “Source of admissions (sourceAdm)“ offers the maximum positive gain in C-PTD. Therefore, we select this split to grow the tree. New nodes are shown in Figure 7 as 2 (Home/Private Residence), 3 (Home for the elderly), 4 (other), 5 (Police Custody) and 6 (Unknown).

Step 2: we split the admissions data for each remaining covariate (Gender, Age and District), fit each group individually to C-PTD and total the C-PTD and compare the WIC of the node before the split to calculate the gain in C-PTD. We select the covariate providing the maximum positive gain in C-PTD to split the data. Here, in Table 28, Table 29, Table 30, Table 31 and Table 32 for nodes 2 (Home/Private Residence), 4 (Other admission sources) and 5 (Police Custody) covariate “District“ and nodes 3 (Home for the Elderly) and 6 (unknown admission source) covariate “Age” offer the maximum positive gain in C-PTD. Therefore, we select this split to grow the tree, and new nodes are shown in Figure 7 as 7, 12, 17, 22, 27, 32, 37, 42, 43, 49, 52, 57, 62, 67, 70, 75, 80, 81, 84, 87, 90, 92, 93, 94, and 95.

Table 33, Table 34, Table 35, Table 36, Table 37, Table 38, Table 39, Table 40, Table 41, Table 42, Table 43, Table 44, Table 45, Table 46, Table 47, Table 48, Table 49, Table 50, Table 51, Table 52, Table 53, Table 54, Table 55, Table 56 and Table 57 show that only nodes 7, 12, 17, 22, 27, 32, 37, 52, 57, 62, 67, 70, 75, 81, 84, 87 and 90 provided significant WIC improvement with a split by Age, and nodes 49, 67 and 35 provided significant WIC improvement with a split by gender and only node 43 provided significant WIC improvement with a split by District. All remaining nodes did not provide any significant WIC improvement by any split and therefore considered terminal nodes. Figure 7 shows a graphical representation of the WIC-based splitting criteria from the admissions data against the covariates, directly affecting patients' characteristics. The survival tree consists of 70 terminal nodes representing the data's significant clusters. Table 27, Table 28, Table 29, Table 30, Table 31, Table 32, Table 33, Table 34, Table 35, Table 36, Table 37, Table 38, Table 39, Table 40, Table 41, Table 42, Table 43, Table 44, Table 45, Table 46, Table 47, Table 48, Table 49, Table 50, Table 51, Table 52, Table 53, Table 54, Table 55, Table 56 and Table 57 show the results used to generate the survival tree.

Similarly to the previously generated survival tree, the average WIC is taken for the covariates Age, Gender, District and sources of Admissions. The average WIC is calculated because the root node takes the data for the whole period (721 days), while each covariate takes the whole period per subgroup. For example, considering age in each subgroup (newborn, under 30, under 70 and over 71) calculates the WIC over the same 721 days. Therefore the age would have four times the data of the root node. Therefore, for covariate Age, the average WIC is calculated by dividing the WIC per subgroup by 4. Similarly, the covariates Gender, District and Source Admissions were divided by 2, 7 and 5, respectively. This was repeated as the tree was further split. The total gain in WIC obtained is 2378.89, where the WIC of the root node is 2561.45, and the terminal nodes' WIC sum up to 182.56. It is possible to analyse the relationship between the patient's characteristics and admissions using the results from Table 27, Table 28, Table 29, Table 30, Table 31, Table 32, Table 33, Table 34, Table 35, Table 36, Table 37, Table 38, Table 39, Table 40, Table 41, Table 42, Table 43, Table 44, Table 45, Table 46, Table 47, Table 48, Table 49, Table 50, Table 51, Table 52, Table 53, Table 54, Table 55, Table 56 and Table 57 and Figure 7. It may be seen that the most significant split occurs for the source of admissions covariate with a WIC gain of 4348.46. The next level shows that the most significant splits occurred for the district covariate for three groups (private residence, other and police custody) and the age covariate for two groups (elderly homes and unknown).

4. Result & Evaluation

A model's Goodness-of-fit (GOF) describes how well-observed data fits into a model. By deriving the GOF, it is possible to evaluate the effect covariates have on the model fit. One of the most popular methods used as a GOF statistic is log-likelihood. Usually, the log-likelihood function is calculated by approximating the chi-square distributions and determining significance levels. The smaller result from the log-likelihood function provides a better-fit model. If the result is 0, then the model is a perfect fit [15]. A model's GOF may be assessed using GOF statistics or GOF indices. However, GOF statistics provide problems due to the small expected probabilities of obtaining accurate p-values. Due to this, most researchers prefer to make use of GOF indices. The two popular GOF indices include AIC and BIC.

Authors Wu and Sepulveda [25] showed that the WIC provides strength and stability over the models tested (including AIC and BIC etc.). The results also showed that for a small sample size, WIC performs as well as AIC. On the other hand, for a large sample size, WIC results are as well as BIC results; however, WIC exceeds the results from other criteria. This highlights the strength of WIC.

Therefore, from the literature reviewed above, it could be concluded that using WIC and log-likelihood to generate the models provides strong and stable results and, thus, models.

4.1. Predictions:

This section tests the accuracy of the predicted mean LOS values for the LOS analysis models generated. Figure 5 and Figure 7 and their respective tables 11 and 12 (In the appendix) display the significant clusters based on patients' LOS and their relationship with temperature and patient characteristics, respectively. A total of 23 clusters are generated for the relationship with patient characteristics. The mean number of admissions for all the data with the same clusters is calculated from the 2013 data. From the respective PTS tree generation tables in Appendix A, the mean LOS of each terminal node corresponding to the group is taken and recorded. The diff between these values gives the Forecasting Error result.

4.1.1. Personal Characteristics Model

Table 58 tests the accuracy between the actual and predicted data for patient characteristics, i.e. using the covariates gender, age, district and source of admissions. The table shows that the highest percentage error of over 50% is the cluster for female patients from Gozo or Comino over the age of 71 (percentage error = 53.5%). This is followed by male patients from Gozo or Comino over the age of 71 who also have quite a high percentage error of 33.36%. In another cluster, patients under 70 from an unspecified locality have a percentage error above 20% (percentage error = 21.56%). It may also be seen that apart from these three groups, all other clusters have a low percentage error of below 16%.

4.2. Admissions Analysis

This section tests the accuracy of the predicted mean number of admissions for the admissions analysis models generated. Figure 7 and respective tables 14 (In the appendix) display the significant clusters based on the number of patients admitted and their relationship with temperature and patient characteristics, respectively. Seventy clusters are generated for the relationship with the patient characteristics. All clusters are tested for the admissions analysis against patient characteristics 10 clusters are chosen to be tested for accuracy.

The mean number of admissions is calculated by counting the number of admissions and the number of records with the same clusters as those taken to be tested and dividing the number of admissions by the number of records. From the respective PTS tree generation tables in Appendix A, the respective terminal nodes are found, and the mean number of admissions of the cluster is recorded. The difference between these values gives the Forecasting Error result.

4.2.1. Personal Characteristics Model

Table 59 tests the accuracy between the actual and predicted data for patient characteristics using the covariates gender, age, district and source of admissions. Randomly ten records were selected from the 70 significant clusters, two from each level 2 subgroup in Figure 7. It may be seen (Table 59) that the highest percentage error is for the group of patients under 30 admitted from their private homes and residing in the Northern Harbour area; the percentage error value is 44.74%. The group of patients under 70 admitted from an elderly home and residing in the North had the second highest percentage error of 27.78%. Three groups had a percentage error of 0%, while three other groups had a low percentage error of under 10%. Two randomly selected clusters had no patients admitted for 2013, and thus these groups could not be tested for accuracy.

More ever, our models predict the mean and actual LOS and the number of admissions while comparing results to those of independent data. The independent data is for patients admitted in 2013 as emergency cases, provided by Mater Dei Hospital. It is important to note that this data for 2013 was not used to create the model specified above.

Table 60 calculates the Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Deviation (MAD) and the bias for all the models. MSE takes the average squared error values, the difference between the actual and predicted values. The RMSE is the square root of the mean square error, representing the standard deviation of differences between the actual and predicted value. This error test does not show whether there was an increase or decrease. MAD calculates the average absolute value of the forecasted error; it can show which forecasts deviate most. The bias calculated the forecast error average.

The error results for all the tests are close to 0, having an average forecast error of - -0.69 for the personal characteristics model. The average forecast error obtained for the patient characteristics model was -0.82, respectively.

From these results, it could be concluded that the models created can help healthcare professionals to strategically plan future resources on accurate forecasts of demands predicted by patients' characteristics. More accurate results could be obtained by splitting the covariate groups differently or considering factors like the type of emergency diagnosis. Types of cases such as surgical or chest pain may cause a patient to be admitted to the hospital, thus making patients' LOS longer than those with minor injuries such as fractures or sprains. Analysing these factors can provide more accurate results on a patient's LOS and the number of beds and resources available.

5. Conclusions

This study used a C-PHD approach on a 2-year data set (2011/2012) provided by Mater Dei Hospital and Free Metro to generate PTS trees for admissions and LOS on patient characteristics groupings for emergent patients. The PTS trees reveal the factors which significantly affect the admission rate and LOS. The average admission rate and LOS were predicted using the models and compared to the actual average admission rate and LOS for 2013 (an independent data set). The difference between the predicted and actual results was evaluated using accuracy measures. These measurement results showed that the most accurate admission model created was related to patient characteristics, while the LOS models created both showed promising results.

Further improvement to the LOS results obtained in the predictions may be achieved by extending the model to use covariates such as diagnosis, type of admission (emergency/elective), and type of procedure (e.g. BUPA classification of complex major, major+, major, intermediate and minor). Admissions results may be improved by extending the model to use day or month of admissions to accommodate daily and monthly patterns. It is also possible that admission results will be improved by considering seasonal and weekend effects. These recommendations would be carried out by running the EMpht program and reconstructing a tree using the additional covariates. Once a model that provides accurate results is created, healthcare professionals can use the model for more accurate forecasting of demand and forward planning.