A Study on the Efficiency and Productivity of County-Level Hospitals from the Perspective of Regional Contrast in China: Data Envelopment Analysis and the Malmquist Index

Background . County-level public hospitals play an important role in China's medical tertiary health care network. Since a new round of medical reforms occurred in 2009, county-level public hospitals have conducted continuous exploration and reforms. To analyze the efficiency and productivity of 36 county-level public hospitals based three provinces in China. Methods. We randomly selected 12 county-level hospitals from each 3 provinces based on economic levels and regional differences in China, finally, a total of 36 county-level hospitals were chosen, and a self-made questionnaire was used to investigate hospital operations for collecting data from 2011 to 2015. 2011-2015 is the twelfth five-year period of China's national economic and social development. Four input indicators and three output indicators were selected. Data envelopment analysis and the Malmquist index methods were used to measure the efficiency and productivity by the key indicators for each region. Results. On average, four input indicators in three regions have continued to grow from 2011 to 2015. The output in the three regions is directly proportional to the upward trend in inputs. On average, the three output indicators of hospitals in the eastern region are higher than the central and western regions. The technical efficiency of county-level public hospitals in the central, eastern, and western regions of China were on an upward trend, and the number of the technical efficiency, the pure technical efficiency, and the scale efficiency values reaching 1 in the three regions was more than half, respectively. The average of total factor productivity change for 2011-2015 in the central, eastern, and western regions was 1.016, 0.997, and 0.930, respectively. adverse consequences of the expansion of hospital scale. The decline in productivity in the western region was due to the decline in the technological efficiency and technical efficiency. The hospitals should strengthen hospital management and blind scale expansion. Financial subsidies in the western region have a significant role in promoting the development of hospitals. The internal management innovation of hospitals in the eastern region has a positive effect on the technical efficiency of hospitals. The medical reform measures in the central region have positively promoted the efficiency of hospitals. analysis of hospital productivity. This study combined the DEA and Malmquist Index methods to analyze the operating efficiency of county-level


Background
China is a vast country with a large population. Medical services in China have always been an important livelihood issue for people-government relations. How nearly a billion rural residents can enjoy equal, fair, and highly accessible primary health care services has always been a focus of the government. As early as the 1950s, the formation of a rural tertiary health care network provided strong support for China's primary medical services [1] . County-level public hospitals, as the leaders of China's three-tiered rural health care network, have a positive effect on the normal use of tertiary health care networks [2][3][4] . As medical institutions for the treatment of frequently occurring and common diseases, county-level public hospitals have been affected by their own efficiency in addition to objective reasons such as regional economic and social development [4][5] . If a hospital's operating efficiency is low, more government investment will fail to improve its efficacy and waste resources, exacerbate the shortage of resources, and affect patients' medical experiences. If a hospital focuses only on the extensional development of human and material resources and other factors, it will ignore the core technologies of its competitiveness, which is the primary consideration for enhancing its efficiency. This is not conducive to the hospital's sustainable development [6] . The importance of studying county-level hospitals is obvious. This is not only a review of past efficiency analyses, but also a summary of the reason for the increase or decrease of past efficiency. This study describes how to scientifically and effectively evaluate the efficiency of a hospital, and the appropriate scientific analysis method is the key.
At present, the most common evaluation methods used worldwide to evaluate hospital efficiency are data envelopment analysis (DEA) and stochastic frontier analysis (SFA) [7][8] . Although SFA can separate the contribution of random effects and changes in technical efficiency, SFA cannot handle multiple outputs or consider random factors in medical care [9] . Thus, it is less practically used in health care efficiency research [9][10] . Given that DEA can handle a variety of output and random factors in healthcare, we are more willing to use DEA for efficiency evaluation in the medical industry [9][10] .
We used the DEA method to evaluate hospital efficiency and the Malmquist Index for a cross-period efficiency analysis of hospital productivity. This study combined the DEA and Malmquist Index methods to analyze the operating efficiency of county-level public hospitals in the eastern, central, and western regions of China. Our application of DEA is based on the concept of technical efficiency and is directed at production technology at a distinct time. However, the production of decision-making units (DMUs) is generally a long-term and dynamic process [11] . Therefore, we must consider the impact of overall efficiency and technological efficiency on changes in productivity.
A review of the related literature found that China has a large number of research on the operating efficiency and productivity of hospitals as a whole, or on the research regarding allocation of resources to hospitals in the field of hospital efficiency and productivity [18][19][20] . However, hospital efficiency has yet to be assessed according to geographical factors. This paper analyzes the efficiency and productivity of hospitals in different areas based on regional variations in China.

Methods
This paper studies the efficiency of a sample of Chinese hospitals based on the DEA method. DEA is a non-parametric research efficiency method based on the concept of relativity developed by Charnes, Cooper, and Rhodes [11] . It allows users to directly apply input and output data to create a non-parametric DEA model for analysis [15][16] . By judging whether the production DMUs is located on the "production frontier" that production may gather, the production frontier is an extension of the production function to the multi-output situation in economics, and it is a surface composed of Pareto optimal solutions with minimum input or maximum output [17] .
DMUs are the subject of research using the DEA method, and may be an enterprise, a hospital, or the equivalent representative. If a DMU is at the front line of production, it means that it is in a dominant position with the current level of technology and scale.
When we use DEA to evaluate the efficiency of DMUs, we can get some relevant management information. With the information, we can not only analyze technical efficiency and scale efficiency but also the input redundancy, output shortage, and shadow analysis, which provide hospital managers and decision-makers evidence and support for the data. As part of management, reforms of research on management information are of great significance for clarifying the effectiveness and role of hospital reform.
Currently, frequently used models are Charnes, Cooper and Rhodes (CCR) models and Banker, Charnes and Cooper (BCC) models [11,18] [18][19] . Based on the data observed by each DMU, we determine whether the DMU is valid for DEA. It is essentially to determine whether the DMU is on the "production frontier" of the production set. That is to say, under certain technical conditions, we can get the maximum output set that each DMU input can form. The production frontier is the one in the economics where the production function is more productive [8,20].
In the CCR model, if the efficiency of DMU is 1, the DMU is on the effective frontier of production and is at an ideal scale. However, if the DEA value is not 1, the DMU is ineffective. In the BCC model, if the PTE is 1, then it will means that the hospital has achieved the optimal output under the fixed input of resources such as existing human resources and financial resources; on the contrary, if it is not 1, the resources invested will be excessive or insufficient, and it needs to improve based on actual conditions. If the SE is 1, the hospital will be in the stage of constant scale returns, indicating that if the hospital increases resources N times, the output will increase by N times; if SE is not, the income scale will be increasing or decreasing. If the TE is 1, it will means that the hospital is on the effective production frontier, and at the ideal scale, that is, in a "perfect" state [ There are three types of returns to scale in the decision-making unit: which are constant, diminishing, and increasing. The constant of returns to scale means that the proportion of increase in the output indicator is equal to the increase in the input indicator. The diminishing of returns to scale means that the proportion increase in the output indicator is less than the increase in the input indicator. The increasing of returns to scale means that the proportion of increase in output indicators is greater than the increase in input indicators.
The DEA model has two orientations: 1) input-oriented and 2)output-oriented.
The methodology used in this study is input-oriented. Because, in the area of health services, the input indicators are easier to control than the output indicators. When the output is constant, we can optimize use of resources by controlling the amount of input. [23][24] .
The Malmquist Index (MI) was used by Caves et al. to introduce the Malmquist consumption curve into production analysis to calculate the productivity index by estimating the ratio between the distance functions [25][26] . It has been widely used since Färe (1994) used it to measure the distance function [27] . Productivity reflects the relationship between output indicators and input indicators. We used the Malmquist Index to analyze the productivity of hospitals over time. Through a cross-period analysis and comparison of five-year data panels, the dynamic changes in hospital efficiency in each region were observed. As with the DEA model selection, we still choose CRS and input-oriented when using the Malmquist index.
The Malmquist Index is used to calculate the productivity level of an organization that analyzes multiple input and multiple output indicators. Total factor productivity change (TFPC) represents the change of productivity levels during t to t+1 period.

Caves et al.(1982) further decomposed the index into technological change (TC) and
technical efficiency change (TEC) [26] . Färe (1994) further decomposed efficiency change into pure technical efficiency change (PTEC) and scale efficiency changes (SEC) [27] . and . MI < 1 indicates productivity decline, MI = 1 indicates productivity unchanged, and MI > 1 indicates growth. All MI averages are geometric means. Its formula is expressed as: Where is the Malmquist productive index ; is the DMU's output of period t+1; is the DMU's input of period t+1; is the DMU's output of period t; is the DMU's input of period t; is a vector of distance function estimate for the n DMUs in period 1 relative to estimated technology in period 1; is a vector of distance function estimate for the n DMUs in period 2 relative to estimated technology in period 2; is a vector of distance function estimate for the n DMUs in period 2 relative to estimated technology in period 1; is a vector of distance function estimate for the n DMUs in period 1 relative to estimated technology in period 2 [28] .

Samples and sources of data
In early 2015, Jiangsu, Fujian, Anhui, and Qinghai Provinces in China were identified as China's comprehensive pilot programs for a deepening medical reform. According to regional differences, we have selected one province from eastern, central and western China, respectively, which were Fujian, Anhui and Qinghai Province.
When selecting county-level hospitals in each province, the following two factors were considered: Firstly, regional and economic factors, four hospitals in the eastern, central , and western regions of each province were selected, and finally, 12 hospitals from each province were selected. Secondly, the willingness of investigating hospitals to participate in our study were considered. Each of the 12 county-level public hospitals
Thus, four input indicators and three output indicators were selected. The four input indicators are: 1) the actual number of physicians, 2) the actual number of nurses, 3) the total expenditures, and 4) the fixed assets. The three output indicators are: 1) the number of outpatient and emergency visits, 2) the medical income, and 3) the actual occupancy of the total bed days.

Statistical analysis
Microsoft Excel 2007 software was used for data entry. 12 sample hospitals in each of the three regions were used as DMUs to perform a static analysis based on CCR model and BCC model calculation efficiency by using DEAP 2.1 software. The productivity analysis was based on the Malmquist Index.

Description of the three regions' input and output indicators
Regarding inputs, Table 1 shows that, on average, four input indicators in three regions have continued to grow from 2011 to 2015. From 2011 to 2015, the hospitals in the eastern region invested more than central and western regions, with the least in the western. Regarding output, Table 2 shows that the output in the three regions is directly proportional to the upward trend in inputs. On average, the three output indicators of hospitals in the eastern region are higher than the central and western regions, with the least in the western.

Distribution of the number of TE, PTE and SE values reaching 1 of countylevel public hospital
In Table 3, the numbers Table 3.

Returns to Scale of County-Level Public Hospitals in the Studied Regions
The proportion of hospitals with increasing returns to scale in the studied regions was relatively high, and the decreasing proportion of the central hospitals was the lowest.

County-Level Public Hospital Efficiency Means in the Regions
The

Analysis of the results based on the DEA-Malmquist Index model
From the overall average TFPC, the average TFPC from 2011-2015 for the five years in the central region was 1.016, which was a 1.6% increase. The average TFPC for the eastern and western regions from 2011-2015 was 0.997 and 0.930, down 0.3% and 7.0%, respectively. The change trend of TFPC in the central region was consistent with the overall change in TEC and TC (Figure 4). TEC was consistent with SE ( Figure 4). The changes in TFPC and TC in the eastern region were consistent ( Figure 5). TEC was consistent with changes in scale efficiency ( Figure 5). The extent of TFPC and TEC in the western region remained highly consistent ( Figure 6). TEC and SEC were consistent, but the former was much lower than the latter ( Figure 6).  Table 4 and Figures 4-6).
TEC in the three regions increased to varying degrees. The western region rose rapidly, compensation mechanisms, personnel distribution, and cost structures [5] . Although the TE of the hospitals in the three Chinese regions declined, overall it was trending upward.

Conclusion
The Not applicable.

Consent to publish
Not applicable.

Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.