Dynamic Evaluation and Convergence Analysis of the Efficiency of Rural Medical Health Centers in China


 Background: Rural medical health centers (RMHCs) are the foundation of the three-level primary medical and health service system in China. The efficiency of RMHCs is related to the rationality of the health care resource allocation for China’s 560 million rural population. Methods: This study analyzed the dynamic changes in the efficiency of RMHCs and its convergence using the non-guided SBM–DEA window model and convergence model, in Shanxi Province, China. Results: The results showed that the medical service efficiency is low. The average scores of the comprehensive technical efficiency and pure technical efficiency of the RMHCs in China from 2016 to 2018 were only 0.1541 and 0.1670, respectively, with nearly 63% of the values being lower than their average scores. The comprehensive technical efficiency and pure technical efficiency of the RMHCs in 2016–2018 exhibited a downtrend year on year. The convergence analysis results also showed that the current rural health clinic medical service efficiency had absolute convergence and conditional convergence. If appropriate policies are followed, the medical service efficiency of the RMHCs can be improved to reach its steady-state level.Conclusions: The medical service efficiency of RMHCs remains low and may gradually decline. The government should promote the efficiency of medical service in RMHCs to reach their own optimal level by formulating corresponding medical and health resource allocation policies.

diagnosed and treated in RMHCs, accounting for 20.13% of all outpatient visits in medical institutions at all levels across the country during 2018. Moreover, RMHCs take 40% of the responsibility of the basic public health services in China. The medical service efficiency score of RMHCs is related to the distribution of medical and health resources for the 560 million rural population of China [3] . Therefore, it is particularly important to evaluate the medical service efficiency of RMHCs and allocate medical and health resources reasonably to solve the existing problems in rural China.
Currently, some studies have evaluated the medical service efficiency of primary medical and health institutions. However, these studies mainly focused on county and town hospitals, with very few studies on RMHCs [4][5][6][7] . Therefore, this study takes RMHCs in Shanxi Province as the research object, and the data envelopment analysis (DEA) method is applied to measure their medical efficiency. The DEA method is a mature efficiency measurement method, which has been used by several scholars studying the medical service efficiency [8][9][10][11] . Moreover, we use the absolute and conditional convergences to analyze the convergence of RMHC efficiency. The convergence theory has been applied to the field of health care since 2000, and has been paid increasing attention by scholars since then. The convergence theory has been widely used in studying health service evaluation indices such as the total health cost, life expectancy, and mortality rate [12][13][14] . In this research, we apply the convergence analysis to answer whether the difference in the medical service efficiencies between RMHCs is gradually narrowing or expanding? Is the development of regions with a lower medical service efficiency lagging behind those with a higher medical service efficiency or surpassing the latter, which leads to the so-called convergence phenomenon?
In comparison with existing literature, the contributions of this paper are as follows: First, a dynamic evaluation of the medical service efficiency of China's RMHCs is carried out, and the true level and change trend in the current medical service efficiency of the RMHCs are obtained. Second, the convergence theory is applied to analyze the difference in the medical service efficiencies, so as to provide a basis for optimizing the dynamic management of rural medical and health resources in the future and promoting the rationality of health resource allocation.

Data sources
The data presented in this study were derived from the Shanxi Rural Health Institute's 2016-2018 Health Statistics Report, which involved 4834 RMHCs; patient information was not collected.

Input and output variables
Input and output variables are used in the DEA model, and the efficiency of the medical service is calculated using the input made by the hospitals and the output obtained from the input. Unlike county and town hospitals, RMHCs mainly provide rural residents with general diagnosis and treatment and referral medical services for common and frequently encountered diseases. Therefore, the input-output indicators of the RMHCs are quite different from those of county and town hospitals. We chose the number of rural doctors, technician training times for doctors, and drug expenditure as the input variables. The output variables were the number of patients and income from essential drugs.

Data envelopment analysis model
The DEA is a new field in operations research, management science, and mathematical economics. It is a quantitative analysis method to evaluate the relative effectiveness of comparable units of the same type by means of linear programming based on multiple input and output indices [15] . Conventional radial DEA models, such as the CCR and BCC models, only include the proportion of all inputs (outputs) reduced or increased. When evaluating an institution's efficiency, the influence of slack variables is not considered, which may result in the deviation of the efficiency measure. Tone [16] proposed the SBM-DEA model (slack-based measure, as a function of the furthest distance to the frontier), which solved the problem that the radial model did not include the slack variable in the measurement of the inefficiency. The SBM-DEA model in this study is as follows: The objective function: minρ = indicates that the RMHC is highly effective and located in the efficiency frontier, and each slack is 0. When ρ is close to 0, it indicates that the RMHC is inefficient.
In addition, when using the SBM-DEA model to calculate the efficiency value, the calculated efficiency value is not comparable between different years, because the leading edge of each RMHC differs from year to year. The DEA window analysis method, proposed by Charnes [17] , is preferred to solve this type of problem. Its

Convergence analysis
There are three types of convergence analysis: α convergence, absolute β convergence, and conditional β convergence [18] . Notably, the β convergence is a necessary but insufficient condition of the α convergence, because if there is an α convergence, the efficiency of the individual absolute gap in narrow, low efficiency of this type of situation only when individuals have higher than high level of efficiency of individual growth rate can be set up; however, individuals with β convergence do not necessarily have α convergence.
In this study, the α convergence refers to the process whereby the medical service efficiency of RMHCs changes over time. The expression is: where T represents the medical service efficiency of the ith RMHC, and n represents the total number of RMHCs. Based on the absolute β convergence, the lower the medical service efficiency of the RMHCs, the higher the growth rate of the medical service efficiency. Therefore, as time passes, inefficient RMHCs will catch up with the efficient ones and reach a stable convergence state at the same speed. The expression of the absolute β convergence is [19] : indicating that there is a "catch-up effect" among RMHCs. In other words, the RMHCs with a low efficiency score will eventually have a higher growth rate, and the medical service efficiency will eventually approach the same steady-state level. On the contrary, if ,0 is significant and positive, it indicates a widening gap in the medical service efficiency among RMHCs. The conditional β convergence allows the RMHCs to have different characteristics, so that the gap between efficient and inefficient RMHCs persists. The expression of the conditional β convergence is [20] : lnT , − , −1 = + , −1 + , ln T , is the logarithm of the medical service efficiency score in the period t of the ith RMHC, and β is the convergence coefficient. If , −1 is significant and β is negative, it indicates that the efficiency score of the ith RMHC will converge to its steady-state level over time. Table 1     The absolute β convergence test results showed that the regression coefficient of β is negative and significant at the 1% level, indicating that the efficiency exhibited absolute β convergence, which means that RMHCs with a low initial medical service efficiency had a higher growth rate, whereas RMHCs with a high initial medical service efficiency had a lower growth rate. As a result of this growth gap, the medical service efficiency gap among RMHCs will gradually narrow.

Results
In addition, the coefficient in the conditional β convergence model is negative and significant at the 1% level, indicating that the medical service efficiency of RMHCs approaches its steady-state level over time (see Table 2).  [21] . The reason is that their income source mainly includes three aspects: basic drugs system subsidy, basic public health services subsidy, and diagnosis and treatment income. The subsidies were the main source of income. However, as RMHCs are at the bottom of the rural medical system, the subsidies were deducted, with little left for rural doctors [22] . In addition, rural doctors not only undertake basic medical services for rural residents but also undertake considerable basic public health services. This makes them dissatisfied with the current situation [23] . On the other hand, the decrease in the drug expenditure is significantly related to the limited essential medicine varieties. In this situation, patients prefer to go to higher-level hospitals [24] .
From the efficiency evaluation results of the RMHCs, the average TE and PTE scores of RMHCs in the past three years were only 0.1541 and 0.1670, respectively.
In particular, in 2018, nearly 63% of the TE and PTE scores were below their average level. At the same time, the TE and PTE scores were still decreasing year on year.
Based on the previous analysis, it can be concluded that the decrease in the TE and PTE scores year on year is strongly related to the decline in the number of rural doctors and the insufficient input of drug resources. RMHCs should be paid more attention in this regard. Therefore, an effective way to improve the medical service efficiency of RMHCs is to enhance the dynamic management of rural doctors and drug procurement. First, the government should improve the income subsidy mechanism for rural doctors, increase the proportion of national financial support, and reduce the proportion of county-level subsidies to ensure the availability of subsidy funds [25] . Second, county, township, and village health service networks should be established to set up a service platform for sharing human resources for healthcare.
Third, the medical insurance agency should adjust the list of essential drugs dynamically to ensure their adequate supply.
From the efficiency of RMHCs' development trend, the absolute β convergence and conditional β convergence indicate that health equalization in rural area can be achieved; however, notably, the connotation for promoting the equalization of rural medical and health resources among regions is that rural residents have equal access to medical and health resources when they need to visit hospitals. Therefore, the equalization of regional medical and health resources pursued by the government is not an absolute β convergence among regions but a conditional β convergence based on certain factors. In other words, when allocating medical and health resources, the government should dynamically manage medical and health resources depending on the level of economic development, and population size and age structure of the rural residents in rural areas, to ensure that the RMHCs reach their own optimal level of medical service efficiency.

Conclusion
Overall, the medical service efficiency of RMHCs remains low and may gradually decline. The government should promote the efficiency of medical service in RMHCs to reach their own optimal level by formulating corresponding medical and health resource allocation policies.

Limitation
This paper has several limitations. First, the sample RMHCs were selected from a northern province in China; other provinces were not selected in our study. Second, because of data access permissions, none of the output indicators in the DEA model were medical quality indicators. In the future, scholars can include more provinces in China and choose some medical quality indicators in the DEA model.