Data sources and regional division
Data were derived from the Henan Statistical Yearbook (2017–2021) and Henan Annual Monitoring Report of Public Hospitals (2016–2020), which covered 18 cities in Henan Province. The 18 cities were Zhengzhou (ZZ), Kaifeng (KF), Luoyang (LY), Pingdingshan (PDS), Anyang (AY), Hebi (HB), Xinxiang (XX), Jiaozuo (JZ), Puyang (PY), Xuchang (XC), Luohe (LH), Sanmenxia (SMX), Nanyang (NY), Shangqiu (SQ), Xinyang (XY), Zhoukou (ZK), Zhumadian (ZMD) and Jiyuan (JY), respectively. A series of longitudinal data were used to evaluate the equity and efficiency of public MCH institution’ health resource allocation in Henan Province from 2016 to 2020. Based on the economic development level, the 18 cities were divided into four groups: Central Urban Agglomeration (CUA), North Henan (NH), West and Southwest Henan (WSH) and Huanghuai Region (HHR) (Fig. 1).
Indicators of equity and efficiency
PubMed, Embase and Web of science were searched between 1 January 2015 and 1 March 2022 for English-language literatures by using the keywords “hospital”, “efficiency”, “DEA”, “healthcare”, and “efficiency evaluation”. Based on a systematic review of relevant literature on hospital efficiency and the availability of data, the number of employees, health technicians, licensed (assistant) doctors, registered nurses, total assets, fixed assets, building business room area, the number of equipment above 10 thousand, the number of actual beds, and the number of MCH institutions, these 10 indicators are finally included in the input indicator set. The number of discharged patients, the number of outpatients, bed days occupied by discharged patients, average hospitalization days, the utilization rate of hospital beds, total income, business income, financial subsidy income, the number of open bed days and the actual bed days are included in the output indicator set. A total of three indicators are used in equity analysis: the number of health technicians, number of beds, and government financial subsidies to public MCH institutions.
Data envelopment analysis model
The DEA model is a linear pro-gramming approach used to measure the efficiency of a decision-making unit (DMU). A DEA model can be either input- or output-oriented. An input-oriented model focuses on optimizing input levels for a given level of output, while an output-oriented model is targeted toward achieving the largest possible proportional expansion in outputs at given input levels. The DEA model, an evaluation method that respects the personality of hospitals, can make a more effective comparison without information about relative prices, and is widely used in the evaluation of hospital operation efficiency17, 18.
Some articles7, 19focused on the application of DEA in the health care department, to improve the discriminative power of DEA, and have summarized the guidelines for input-output indicators: a) Input and output indicators should have the same direction principle, that is, the changing trend of output indicators should be consistent with input indicators. b) By using the inputs and outputs it is important not to mix absolute and relative data, otherwise, the results can be seriously distorted, as absolute values are consequently bigger for larger units. c) Most importantly, twice the number of inputs and outputs together should not exceed the amount of DMUs to ensure enough discrimination between units.
Statistical analysis
Gini coefficients and Lorenz curves
The Gini coefficients (G) and Lorenz curves are commonly tools to assess the equality of health resource allocation. For the Lorenz curve, the x-axis represents the cumulative percentage of population or geography, and the y-axis represents the cumulative percentage of health care resources (health technicians, beds and government financial subsidy). A 45° line indicates absolute equity. A larger distance from the 45° line indicates greater unfairness. The formula is given as follow:
$$G=\sum _{i=1}^{n}{W}_{i}{Y}_{i}+2\sum _{i=1}^{n}{W}_{i}\left(1-{V}_{i}\right)-1$$
Vi: the cumulative percentage of health resources in the total number of health resources from the lowest number to the highest number.
Yi: the proportion of the number of health resources (health technicians, beds and government financial subsidies) in the ith city in the total number of health resources.
Wi: the proportion of the resident population or geographical area of the ith city in the total population or geographical area of the province at the end of the year.
Theil index
Theil index (T) is the optimal tool to assess the inequity of health resource allocation. The range of T is from 0 to 1. Generally, the closer T is to 0, the better the fairness. The formula is given as follow:
$$T=\sum _{i=1}^{n}{P}_{i}log\frac{{P}_{i}}{{Y}_{i}}$$
Pi: the proportion of the resident population or geographical area of the ith city in the total population or geographical area of the Henan province at the end of the year.
Yi: the proportion of the number of health resources (health technicians, beds and government financial subsidies) in the ith city in the total number of health resources.
T can be further decomposed into the Tintra and Tinter. The formulas are given as follows:
$$\text{T}={T}_{intra}+{T}_{inter}$$
$${T}_{intra}=\sum _{\text{g}=1}^{k}{P}_{\text{g}}{T}_{\text{g}}$$
$${T}_{inter}=\sum _{\text{g}=1}^{k}{P}_{\text{g}}log\frac{{P}_{\text{g}}}{{Y}_{\text{g}}}$$
Tintra: the intraregional distribution of health resources (health technicians, beds and government financial subsidies) among the four regions.
Tinter: the interregional distribution of health resources (health technicians, beds and government financial subsidies) among the four regions.
Pg: the proportion of the population of the four regions to the total population of Henan Province
Yg: the proportion of the health resources (health technicians, beds and government financial subsidies) of the four regions to the total health resources of Henan Province.
Health Resources Density Index (HRDI)
The HRDI displays the influence of population and geo- graphical factors on the agglomeration of health resources while avoiding bias caused by a single population or geographical aspect. The formula is given as follow:
$$HRDI=\sqrt{{X}_{i}{Y}_{i}}$$
Xi: the number of health resources (health technicians, beds and government financial subsidies) per thousand people in the ith city/region
Yi: the number of health resources (health technicians, beds and government financial subsidies) per square kilometer in the ith city/region
Correlation analysis
Based on the principle of homogeneity of input and output indicators of the DEA model, our study used correlation analysis to test the homogeneity of all indicators in the set of input-output indicators. The uncorrelated and negatively correlated indicators in the input and output indicators set were excluded. Pearson’s correlation coefficient was used as the criterion of correlation. The closer the value of the Pearson correlation coefficient is to 1, the higher the correlation between the two indicators is, and the closer to 0, the lower the correlation between the two indicators is.
Cluster analysis
Cluster analysis, also known as point group analysis, is a kind of exploratory analysis. It can combine groups with similar attributes and properties into one class without giving specific classification criteria in advance from the sample data itself20. Compared with traditional classification methods, the results of cluster analysis are more detailed, comprehensive, and reasonable. Considering the diversity of input and output indicators in efficiency research in medical institutions and twice the number of inputs and outputs should not exceed the amount of DMUs. Therefore, based on the predetermined set of input and output indicators, this study uses ward cluster analysis to further classify.
Data envelopment analysis
In DEA analysis, an input-oriented model has been implemented. There are three reasons for using the input-oriented model: Firstly, it is easier to control the inputs in a hospital environment than the outputs. Secondly, the input-oriented approach quantifies the input reduction without changing the output quantities21. Thirdly, public institutions are non-profit entities seeking to provide better services, with less of a focus on financial profit. We used the constant return to scale (CRS) model expanded by Charnes and colleagues and return to scale (VRS) model developed by Banker and colleagues to calculate three types of efficiencies: technical efficiency (TE) provided by the CRS model, the pure technical efficiency (PTE) provided by the VRS model, and the scale efficiency (SE)22–24. There is a formula between these three types of efficiencies:
$$TE=\text{P}\text{T}\text{E}\times \text{S}\text{E}$$
TE refers to the ability of the current input to achieve the maximum output under certain production conditions, which can comprehensively measure the resource allocation ability and use efficiency of the DMU, and can represent the operational efficiency of the hospital to a certain extent. PTE refers to the change in production efficiency caused by factors such as management and technology in DMU. When PTE = 1, it means that DMU produces in an effective way. However, PTE<1, it indicates that production is not performed in an effective manner. SE refers to the change in production efficiency caused by scale factors in DMU. When SE = 1, it indicates that the DMU scale is valid. When SE<1, DMU may have two states: decreasing returns to scale (DRS) or increasing returns to scale (IRS). When it is DRS, it indicates that DMU investment is too high; when it is IRS, it indicates that DMU investment is insufficient.
Malmquist Productivity Index
The traditional DEA model can only evaluate the relative efficiency of DMU at a specific time level, and cannot infer whether the time factor will affect the efficiency change. Fare et al. based on Malmquist’s concept25–27, improved and proposed the malmquist productivity index (MPI) to measure the productivity changes of DMUs. total factor productivity (TFP) was used to measure its variation. Furthermore, TFP can be further decomposed into efficiency change (EC) and technological change (TC), and EC can be further decomposed into pure technical efficiency change (PEC) and scale efficiency change (SEC). TFP index and the other indexs are expressed using the following formula28:
$$TFP=\text{E}\text{C}\times \text{T}\text{C}=(\text{P}\text{E}\text{C}\times \text{S}\text{E}\text{C})\times \text{T}\text{C}$$
When TFP > 1, it indicates that efficiency is improved, TFP = 1 means efficiency remains unchanged, and TFP < 1 means efficiency decreases.
Tobit regression analysis
In order to explore the factors affecting the TE of DMUs, this study took the TE of DMUs as the dependent variable. Through the research and summary of relevant literatures29–32, the utilization rate of hospital beds, average hospitalization days, the number of outpatiens, total income, the actual bed days, health technicians, total assets, building business room area, financial subsidy income, and per capita Gross Domestic Product (GDP) in the region were selected as independent variables for regression analysis. The TE score of DMUs ranges from 0 to 1, most of the values are continuously distributed, and a small part of the data is quite concentrated on a certain value. Tobit regression model, which allows the value to be limited to no more than 1, is considered a better technique to solve this problem and is widely used to investigate the factors influencing efficiency scores33, 34. Therefore, Tobit regression was adopted in this study to further explore the factors affecting TE.
Microsoft Excel 2019 was used to calculate the G, T and draw the Lorenz curve. ArcGIS 10.8 was employed for mapping the distribution of health resources. Correlation analysis and cluster analysis involved using SPSS 26.0 and DEA analysis were performed with DEAP 2.1. Tobit Regression was performed with Stata 14.0 (Stata Corp, College Station, TX, USA). Two-sided P < 0.05 was considered statistically significant.