For a sustainable Healthcare expenditure policy, it is important to have accurate forecasts to avoid any sort of uncertainty about the rise in Healthcare expenditures. The Auto Regressive Integrated Moving Average (ARIMA) model and the Artificial Neural Networks (ANNs) models have got attention and have been the preferred model to forecast Healthcare expenditures for decades. However, it has been found that each of the two models has weaknesses in capturing linear and nonlinear relationships in data. It is with this background that, we proposed a hybrid model, which differs in combining the advantages of Auto Regressive Integrated Moving Average and Artificial Neural Networks. The hybrid model was tested on sets of Rwanda total Healthcare expenditure actual data, and the empirical results of the forecasted values in 2027 are 552,881.33 million Rwandan francs. Our results showed the effectiveness of the hybrid model which has a higher prediction accuracy as compared to the existing models. For policymakers to better plan for future resources and estimate the amount of the budget allocated to Healthcare, there is a need of accurate estimation of Healthcare expenditure as it impact government policy and planning.
Research Article
Towards hybrid artificial neural network model in predicting long term total healthcare expenditure in Rwanda
https://doi.org/10.21203/rs.3.rs-1752167/v1
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Box–Jenkins methodology
Artificial Neural Networks
Time Series forecasting
ARIMA-ANN hybrid model
Worldwide, both private and public expenditures on Healthcare have been increasing through the years among developed and developing countries (Lorenzoni & Belloni, 2014). This raises global concerns about Healthcare and the economy, and therefore the fiscal preparations as well as policy interventions are needed to ensure future fiscal sustainability (Medicare Center, 2011). A Healthcare system is mainly composed of a couple of Healthcare expenditures, which is explained as any type of spending to prevent and treat illnesses and diseases with the aim of improving the Health of people. Healthcare spending includes the three major categories of medical, hospital, and pharmaceutical expenses.
Since Healthcare expenditure has been growing over years, the increase in Healthcare expenditure is explained by factors that vary according to the investigation period of time (Di Matteo, 2010). In the short-term, Healthcare spending development is mainly influenced by the Healthcare sector government budget. In the medium-term, the Healthcare expenditure development is influenced by the technology advancement. While in the long-run, the main factor to influence the increase in Healthcare expenditure is chronic disease prevalence. There are also other factors, such as the elderly population, economic growth, as well as the development of treatment practice. Besides, other factors like population life expectancy, population size, and population structure greatly affect Healthcare spending (Astolfi & Lorenzoni, 2012). Finally, changes in consumption behaviors have been leading to high Healthcare spending by the population. The level of personal preferences as well as the social environment through different Healthcare programs such as disease prevention and Healthcare promotion have a magnitude impact on the level of consumption for Healthcare services and hence result in greater Healthcare expenditure.
According to the World Health Organization, 2017, the total Healthcare expenditure in Rwanda was 324,385 million Rwandan francs in 2017 where 48% of the total Healthcare expenditure was paid by the government and 52% was paid by households. Total Healthcare expenditure in Rwanda is mainly financed by the public sector, and both private payment providers and insurers. In Rwanda, public employees are covered by their employers while a variety of Healthcare insurance policies is provided by private companies. The increase in Healthcare expenses has to follow the economic growth of the country (OECD, 2019), otherwise, the progressive increase in Healthcare expenditure puts a heavy burden on resources which in turn fails to satisfy other needs of the population. There is an essential need in controlling the Healthcare expenditure related policies and for a sustainable Healthcare expenditure policy, there is a necessity of accurate Healthcare expenditure forecasts to avoid any sort of uncertainty. This brought a thirst to different researchers to develop different models. The main problem is that some models used in forecasting Healthcare expenditure have weaknesses. A new approach is therefore needed to forecast Healthcare expenditure. The aim is to develop a more sophisticated model which is efficient in estimating the Healthcare expenditure with higher accuracy than other models.
Previous studies, among others (Nikolaos, 2005), (Zheng, et al., 2020), (M. Ramezanian, 2019) tried to model Healthcare expenditure using ARIMA and (Nida Shahid, 2019) have estimated the Healthcare expenditure using ANN model. It has been found that each of the two models has weaknesses when linear and nonlinear behaviors are in the data, thus each of the ARIMA and ANN is no longer appropriate to model the series because the ARIMA model only performs better in capturing linear behavior and ANN is good in capturing non-linear behavior in data but ANN has limitations in over-training, which occurs when the capacity of the ANN for training is too great because it is allowed too many training iterations. But also ANN model is not effective for extrapolation, which is sometimes very important because the existing data used to train an ANN do not necessarily cover the entire range (Khashei M, 2009), (Khashei M B. M., 2011) and (Chungen Yina, 2003). A closer look at the literature on the prediction of Healthcare expenditure reveals several gaps and shortcomings. The Rwandan context, for example, (R. Muremyi, 2020) used the MARS model to predict out-of-pocket Healthcare expenditures in Rwanda using machine learning techniques where their findings show that the tests of the accuracy of the models were 50.16 % for MARS model, 74 % for decision tree model, 87 % for Tree Net model, 83 % for random forest model and gradient boosting 81 %. To the best of our nowledge, there is n previous research using the RIMA-ANN Hybrid approachto forecast Rwanda Healthcare expenditure yet it i crucial to predict total Healthcare expenditure of Rwanda beacause the results could inform policymakers on how to plan for future resources and allocate the necessary budget to Healthcare expenditure (Astolfi & Lorenzoni, 2012).
Data
The current study used secondary data from the Rwanda National Healthcare Account (NHA) produced by the Rwanda Ministry of Healthcare (MOH) in partnership with the World Health Organization (WHO). The data give detailed information on Rwanda’s Healthcare expenditure by both public and private sides. Since the result of the this study depend on the precision and accuracy of the data, this study used a quarterly time series data from 2006 to 2018 with the following variables: Total Healthcare expenditure current price, Government Healthcare expenditure and households Healthcare expenditure.
Models
In the analysis of historical data, time series forecasting is a very famous technique for forecasting a country’s Healthcare expenditure however, Healthcare expenditure data contains a mixture of linear and non-linear characteristics. Therefore, due to its complexity, a single model cannot capture all the patterns in Healthcare expenditure data. As result, a variety of methods including linear and non-linear time series methods are used in the prediction (Jianqing Fan, 2003).
Autoregressive Integrated Moving Average (ARIMA) models
ARIMA model is among the famous methods in modeling non-stationary data. The ARIMA model, rt is expressed as a function of lagged values and stochastic error components while this is not possible in linear regression models. The general and standard form of the ARIMA model is ARIMA (p, d, q). An ARIMA is presented as:
If\({w}_{t}={\varDelta }^{d}{r}_{t}={(1-B)}^{d}{r}_{t} \left(1\right)\)
Then\({w}_{t}={\phi }_{1}{w}_{t-1}+{\phi }_{2}{w}_{t-2}+\dots {\phi }_{p}{w}_{t-p}+{\epsilon }_{t}-{\theta }_{1}{\epsilon }_{t-1}-{\theta }_{2}{\epsilon }_{t-2}-\dots -{\theta }_{q}{\epsilon }_{t-q} \left(2\right)\)
The seasonal autoregressive integrated moving average model is presented as (p, d, q) (P, D, Q), where p represents the number of auto-regressive terms, q denotes the number of moving average terms and d denotes the number of times a series must be differenced to induce stationarity. And P is the number of seasonal autoregressive components, Q: number of seasonal moving-average terms, and D is the number of seasonal differences required for the series to be stationary (Box & Jenkins, 1994), (Brockwell & Davis, 1996).
Identification
When a positive d is set to transform a nonstationary series, the order p and q) of the corresponding intermediate terms must be identified to represent the dominant features of the information. Therefore, the filtering graph procedures (ACF and PACF schemes) were used.
Estimation
when estimating θ1...θp and φ1...φq of AR and MA processes respectively, the estimation is based on the PACF and ACF, respectively. If the series is an Auto-regressive process, the coefficients are estimated by the least-squares method. If there is a MA or ARMA process in this array, then non-linear estimation techniques, such as maximum likelihood, can be used to estimate parameters using numerical optimization algorithms.
Diagnostic Checking
after the model fits perfectly with the forecasting procedures mentioned above, the step to follow is to carry out the diagnostic checking for the model. The importance of diagnostic inspection coefficients refers to statistical tests for residual behavior and the order of the model. If the predicted model data represents the process of emergence, the residuals behave as white-noise
This means that the residues have not to be auto-correlated. Auto-correlation of the residue is examined by Lung-Box’s Statistics Q (1978, 1979) and is defined as:
$${Q}_{LB}=n\left(n+2\right)\sum _{s=1}^{m}\frac{{\widehat{\rho }}_{s}^{2}}{n-s} \left(3\right)$$
Where,
n: denotes the number of observations
m: denotes the number of auto-correlation coefficients and\(m=\sqrt{n}\)
\({\rho }_{s};\) denotes the sampled residual auto-correlation.
To test whether the residuals are normally distributed, the standard residual graph along with the Shapiro-Wilk test is used. When comparing the descriptive efficiency of alternative models that differ in both number of parameters and sample size, Akaike, Hannan-Quinn and Schwartz are used as the evaluation metrics. The lower the value for these three metrics, the better the model is. If partial auto-correlations and auto-correlations are both of low value, the model is then preferred as adequate for future forecasting.
Forecasting
The predictive method for future phase range values is derived from the previous phases, and it refers to the most appropriate model derived from the previous stages. The estimation of the Auto-Regressive Integrated Moving average model is evaluated using the Mean Squared Error (MSE) as the optimum metric. Other indicators commonly used to measure predictive accuracy are the Root Mean Square Error (RMSE), the Mean Absolute Error (MAE), and the Theil (U-Theil) inequality coefficient. These indicators are listed as follows
$$MSE=\frac{1}{T}{\sum _{t=1}^{T}({\widehat{y}}_{t}-{y}_{t})}^{2} \left(4\right)$$
$$MAE=\frac{1}{T}\sum _{t=1}^{T}|{\widehat{y}}_{t}-{y}_{t}\left| \right(5)$$
$$RMSE=\sqrt{\frac{1}{T}{\sum _{t=1}^{T}({\widehat{y}}_{t}-{y}_{t})}^{2}} \left(6\right)$$
Theil Inequality Coefficient is as follows:
$$U=\frac{\sqrt{\frac{1}{T}{\sum _{t=1}^{T}({\widehat{y}}_{t}-{y}_{t})}^{2}}}{\sqrt{\frac{1}{T}{\sum _{t=1}^{T}({\widehat{y}}_{t})}^{2}}+\sqrt{\frac{1}{T}{\sum _{t=1}^{T}({y}_{t})}^{2}}} 0\le U\le 1 \left(7\right)$$
Where,
\({y}_{t}\) : is the actual value of the endogenous variable y in time t.
\({\widehat{y}}_{t}\) : is the revised value of the endogenous variable y in time t.
T: is the number of observations in the simulation (of a sample).
When Theil Inequality Coefficient U = 0, the estimated values are equal to the actual values of the series \({y}_{t}\)=\({\widehat{y}}_{t}\) for all t. This represents a case where the actual and predicted values perfectly fit. On the other hand, if U = 1, the forecasting is not correct for the sample being investigated.
Artificial Neural Network (ANN) models
NNs are an architecture with interconnected neurons which that aim of mimicking the functionality of the human brain(McCulloch & Pitts, 1943). The ANNs structure's main challenge is to identify the optimum number of layers and nodes in the layer in the time series prediction. This is determined through experiment because there is no theoretical basis for identifying the parameters. Simple architecture for ANN model of \(p*q*1\) is
$${y}_{t}={\varnothing }_{0}+\sum _{j=1}^{q}{\varnothing }_{j}g+\left[{\theta }_{0j}+\sum _{i=1}^{p}{\theta }_{ij}{y}_{t-1}\right]+{\epsilon }_{t} \left(8\right)$$
Here,\(j\left(j\text{0,1},2\dots q\right),ij(i\text{0,1},2\dots p;j\text{0,1},2\dots q)\)
ARIMA-ANN hybrid model (Zhang’s hybrid model)
When linear and nonlinear behaviors are in the data under study, each of the ARIMA and ANN is no longer appropriate to model the series. Thus, the model that can capture both linear and non-linear patterns are preferred. Zhang stipulated that it is rational to develop a hybrid model that combines both ARIMA and ANN for modeling linear and non-linear behavior in the historical data (Zhang G., 2003.). According to Zhang, there are:
Additive hybrid model
Multiplicative hybrid model
Where, yt is the observation at time t and Lt, Nt denote linear and nonlinear components respectively at time t. Auto-regression is appropriate for linear component and the forecasted value is Lt at time t and \({e}_{t}={y}_{t}+\widehat{{L}_{t}}\) is the residual at time t.
Zhang confirmed that, ANN is appropriate in modelling the residuals from ARIMA which only have non-linear relationship (Zhang G., 2003.). Using \(\rho\) input nodes, the ANN for residuals is represented as follow:
\(\) \({e}_{t}f{e}_{t-1},{e}_{t-2},\dots ,{e}_{t-\rho }+{\epsilon }_{t} \left(9\right)\)
Where \(f\) denotes a non-linear function by the ANN and \(\epsilon\)t represents the white-noise. By ( \(\widehat{{N}_{t}}\))
Representing the ANN, then the hybrid estimate at time t is:
\(\widehat{{y}_{t}}=\widehat{{L}_{t}}+\widehat{{N}_{t}}\) For Additive hybrid model
\(\widehat{{y}_{t}}=\widehat{{L}_{t}}*\widehat{{N}_{t}}\) For Multiplicative hybrid model
The evaluation metrics namely Mean Absolute Percentage Error (MAPE) and Root Mean Squared Error (RMSE) are used to evaluate the hybrid models.
The analysis focused mainly on developing a model that can accurately predict the total Healthcare expenditure in Rwanda.
Data visualization and exploration
We first visualized the data using the Matplotlib function through its library called “pyplot” to see if there are any trend and seasonality in data. This is crucial and is important due to the fact that to estimate using historical data, we first need to ensure that the data are stationary, therefore, data visualization is one of the preliminary steps toward checking the stationarity of the time-series data.
We plotted the moving average and see if it varies with time. Here, at any time‘t’, the average/variance of the last year is considered. We can see that parameters are changing at some point and our rolling estimates captured the instability. But again, this is more of a visual technique. In addition to this, ADF which is the famous test for stationarity testing is used. The null hypothesis that states that the data has a unit root is rejected when p-value is less than 0.05. Even if the variation in standard deviation is small, the mean is continuously increasing with time indicating that the series are not stationary.
Model estimation and removing trend
For historical data to become stationary, the first step is to remove the trend and seasonality from the data. We used the transformation techniques to normalize the data and remove the rising trend. After the logarithmic transformation, the rolling statistic was computed. A rolling average was computed from taking input for the past 12 months and assigning a mean expenditure value at every point further ahead in the series.
The results from the transformation showed that the series has no longer a unit root. However, in some cases, the seasonality appears high in the data. In such situations, eliminating the trend does not help much. Instead, more attention is on the seasonality in data. Differencing is the famous method for this task. Differencing is used to eliminate the series dependence on time which includes trends and seasonality. Differencing helps in stabilizing series by eliminating the variation in the level of a series, reducing the magnitude of both seasonality and trend. Differencing is conducted by taking the difference between previous and present values.
Finding the best parameters for model and Fitting model
Before building the forecasting models, we first identified the optimal parameters for the forecasting models. We plot the ACF and PACF and a non-seasonal ARIMA model stated in the form of ARIMA (p, d, q).
To identify the best parameters, we first check, where ACF and PACF graph for the first time breaks off the root or decreases to zero. Looking at Figure 3, p and q values are close to 1. The ARIMA (1, 1, 1) is the tentative ARIMA model for Healthcare expenditure.
Figure 4 maps the actual and ARIMA (1,1,1) estimates. We noticed the small deviation of estimated values from actual values.
Predictions
The ARIMA (1, 1, 1) predictions show an increasing trend as indicated by Figure 5, where the total Healthcare expenditure in Rwanda will continue to increase in the long run.
ARIMA, Artificial Neural-Network and Hybrid Model
After the model training, we evaluated the performance of the model on both the training data and on test data. In this analysis, we split the series into train and test data sets with 60% of the observations being for training and the remaining 40% for testing the model.
Table 1 forecast accuracy for total Healthcare expenditure data
|
Measures of fit (Training set) |
Forecast accuracy (Testing set) |
|||||||
|
|
MD |
SD |
MAD |
MSE |
MAPE |
MAD |
MSE |
MAPE(%) |
|
ARIMA |
0.3023 |
0.35634 |
0.1149 |
0.0102 |
3.6023 |
0.1115 |
0.0159 |
3.713 |
|
ANN |
0.3198 |
0.4189 |
0.1178 |
0.0104 |
3.5042 |
0.110 |
0.0168 |
3.498 |
|
Hybrid |
0.309 |
0.3679 |
0.1029 |
0.0062 |
3.3438 |
0.0099 |
0.0075 |
3.543 |
After fitting the models, the performance was evaluated using three performance metrics namely MAD (Mean Absolute Deviation), MSE (Mean Squared Error), and MAPE (Mean Absolute Percentage Error). The results presented in Table 1, show that the same ANN architecture of 18 X 9 X 1 provided accurate predictions than both the ARIMA and ANN models. In relation to MAD, MSE and MAPE, the ARIMA-ANN Hybrid model improve the predictive capability compared to the remaining models.
While designing the ARIMA-ANN Hybrid model, the period from 2006 to 2018 was divided into two sections. After running the experiment for many times, it was found that the 60% data were used as the training data and the other 40% data were used as the test data. After fitting the three models: ARIMA, ANN and ARIMA-ANN Hybrid, the performance was evaluated using three performance metrics. We evaluated the performance of the three models on both the training data and on test data., and the ARIMA-ANN Hybrid model performed better in the prediction. We found that ARIMA-ANN Hybrid model improve the predictive capability compared to the remaining models.
The ARIMA-ANN Hybrid prediction model in this paper is runned and realized in Python3. Based on the National Healthcare Account quarterly time series data from 2006 to 2018, the Healthcare expenditure is predicted in the future ten years. Then, through the experimental operation, the prediction graph of the experimental output is obtained on Fig. 7.
Looking at Fig. 7, the forecasted and actual values produced by the three different models, we noticed that the predicted values for the ARIMA-ANN Hybrid model are closer to the actual values than the two other models.
All models employed in the analysis showed that in ten years ahead, there will be an increase in total Healthcare expenses. The forecasted values by ARIMA model will be 557,299.97 million Rwandan francs in 2027 quarter-four from 87,270.89 in 2017 quarter four. While using the ANNs model, the forecasted values are 555,090.65 million Rwandan francs in 2027 quarter-four from 86,924.92 million Rwandan francs in 2017 quarter four. Moreover, ARIMA-ANN Hybrid model forecasted values in 2027 is 552,881.33 in million Rwandan while they were 86,578.95 million Rwandan francs in 2017 quarter four.
It is worth discussing these interesting facts revealed by the results from both traditional and machine learning-based approaches to analyze, forecast, explain, and present the time series data of Healthcare expenditure in Rwanda. Comparing with other models, the best model has been selected as the final model. We provided also a method for prediction based on data, which is applicable and useful to government and insurance companies. First, we used the Box-Jenkins procedure to model total healthcare expenditure in Rwanda. The time series was found to be non-stationary in mean and variance. Transformation and differencing suitably rendered the series stationary. The analysis on Healthcare expenditure time series in Rwanda and the established model showed that the ARIMA model by the B-J method has good forecasting validity when modeling analysis is made for non-stationary time series. The results in this study showed the superiority of the model established by using the radial basis functions than other methods. Consequently, the results showed that the ARIMA-ANN Hybrid out-performed the ARIMA and neural network models. The forecasting error for the ARIMA model and ANN model is slightly higher than the forecasting errors for ARIMA-ANN Hybrid model. Therefore, the use of ARIMA-ANN Hybrid model to predict Healthcare expenditure is preferred. Overall, the Hybrid model is the one that obtained the most robust results with higher accuracy than other proposed models. The results of this study agree with the work of (Li Wang, 2013) who compared the traditional models like ARIMA and ANN architecture with the hybrid model. They concluded their study saying that the hybrid model showed the superiority in forecasting over the traditional models. Note also that, the predictive capability of the ANN model is higher than the ARIMA model in terms of predictive capability from the test data but the ARIMA-ANN Hybrid outperfom both ARIMA and ANN models. Figure 7 shows that ARIMA-ANN Hybrid is better when compared to ANN and ARIMA models for Healthcare expenditure forecasting. We noticed that the ARIMA model has a directional pattern, as shown in Fig. 7 while ANN models are slightly deviating from atual values. This work also agrees with the results of (Zhang G., 2012) that the hybrid approach performs better than ARIMA and ANNs models when applied separately and confirms the opinions reported in the literature that the ARIMA-ANN Hybrid based models are superior in predictive capability over the ARIMA and ANN models in time series forecasting.
This study compared the predictive capabilities of three models namely ARIMA, ANN and ARIMA-ANN Hybrid to predict total Healthcare expenditure. The results showed the effectiveness of the hybrid model, which has a higher prediction accuracy as compared to other models considered in this study. At last, the ten years’ forecasts showed that total Healthcare expenditure and household Healthcare expenditure will experience an increasing trend in the long run.
For policymakers to better plan for future resources and estimate the amount of the budget allocated to Healthcare, there is a need of accurate estimation of Healthcare expenditure as it impact government policy and planning.
Author contributions Isaac Hategekimana conceptualized, analyzed, structured the methodology, did resource acquisition, validation, formal analysis, investigation, original draft preparation, visualization and prepared the draft manuscript. Annie Uwimana, advised and supervised conceptualization, acquisition of the data, analysis, structuring the methodology, resource acquisition, original draft preparation, visualization and prepared the draft manuscript. Joseph Nzabanita, Venant Habarugira and Zhang Shuguang advised on writing and preparing the draft manuscript. All authors commented on previous versions of the manuscript. All authors have read and approved the final manuscript.
Funding No funding was received to assist with the preparation of this manuscript.
Data availability Data used are Rwanda National Healthcare Account (NHA) data produced by the Rwanda Ministry of Healthcare (MOH) in Partnership with the world Healthcare organization (WHO).
Code availability Python codes used is available on request and will be made public on publication of this paper
Conflict of interest All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.
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