Service Eciency of Nursing Homes Based on DEA and Tobit Model: a Longitudinal Study in China

Background: Low utilization of care resources in China continues to be a challenging problem even though the rapid development of pension services. However, few studies have explored the utilization of care services with either the production of nursing homes at a national and longitudinal level. Methods: The present study analyzes the spatio-temporal distribution of TE and productivity of nursing homes by using the data from China Civil Affairs statistical yearbook (2012-2016). Analysis was conducted based on the Data Envelopment Analysis and Tobit model, which have been widely applied to integrate several quality measures as a comprehensive benchmark. Results: The average TE, PTE and SE of nursing homes were 0.909. 0.928 and 0.979 from 2012 to 2016, respectively. The TE and SE decreased from 2012 to 2014, but improved after 2014. TE is 0.98 in the Eastern region, 0.93 of that in the Central region and 0.91 of that in the Western region, with a decrease range of 2%, 7% and 9%, respectively. The average improvement range of the ve input indexes of the non-DEA effective nursing homes was 27.26%, 20.62%, 19.77%, 22.04%, and 38.84%, respectively. The inuencing factors of eciency value of nursing homes indicated that if there are more social workers, more patients in the nursing homes, and more employees who are aged 56 and above, the TE and productivity of nursing homes will be higher. Conclusion: We nd that TE of nursing homes in Western regions are all at the lowest, as the current facilities can neither provide equitable nor optimal access and utilization. The decline of the eciency and productivity before the healthcare reform implementation in 2014 was signicant, where the inuencing factors could be age and gender of employees, the number of social workers, and older people. Finally, suggestions are put for improving the eciency of allocation and utilization of care services.

In order to solve the problems brought about by the aging of the population, Chinese government has arranged a total investment of 9.17 billion yuan in the central budget, focused on supporting the construction of nursing homes for the elderly and facilities for elderly care with integrated medical care.
By the end of January 2019, there were 163,800 nursing homes and various types of facilities in the country, of which 29,700 were nursing homes, 46,600 were community service facilities for elderly, and 87,300 were mutual care facilities. A total of 7.46 million beds were provided for the various types of elderly care services, and among them, there are 3.928 million beds in nursing homes, while only account for 50% bed occupancy rate [6]. Total beds account for 3% of the country's older population. Generally, developed countries can reach 5% -7%. From the calculation, the number of beds required should be at least 12.45 million to accommodate China's elderly, who are in the need of long-term care (LTC) [7]. It has been further estimated that 300 000 registered long-term care workers nationwide and over 40 million older people with disabilities who need care [8], the overall shortage of beds for the elderly, with a large space to adjust, showing a very obvious contradiction between supply and demand.
Various contradictions between supply and demand still inevitably occur in China, such as the serious shortage of beds, the corresponding number of nursing staff, the low level of professionalism, smallinfrastructure scale of residential care, and the lack of risk management system and the service evaluation system for elderly care [9,10], plus insu cient inspections and ineffective enforcement of rules [11,12]. Therefore, how to effectively improve the service quality, service e ciency, and also the output e ciency of nursing homes in the process of increasing investment and rationally allocating service resources have become an urgent problem for both the government and academia.
With the deepening of the research on the service quality of nursing homes, foreign scholars began to pay more attention to the problem of inputs and outputs of the nursing homes at the same time. A sample survey was conducted in nursing homes in certain areas of Finland, Italy, Japan and Norway respectively, which investigated the relative e ciency and in uencing factors, based on explorations on application of Data Envelopment Analysis (DEA) method and regression model [13][14][15][16]. DEA was de ned as an advantageous non-parametric technique for evaluating performance in terms of relative e ciency in the presence of multiple inputs and outputs [17]. E ciency can be simply used as a tool to explain the relationship between the inputs and outputs. And even technical e ciency (TE) can be applied to nd out whether any waste can be eliminated without worsening any input or output [18].
Although most of researches in developed countries on the service e ciency of pension service institutions are relatively complete, their research approaches and results may not be applicable to China due to different levels of development. There exist amounts of shortcomings in the researches related to the e ciency of China's nursing homes. For example, some only focused on theoretical study for institutional planning of LTCFs, including the exploration of institution building and service standards, service contents, and service personnel [19], some mainly focused on individual nursing home in a certain province or city [20], while some others just proposed analytical methods, but lack quantitative studies, which cannot provide scienti c basis for promoting the overall healthy development of the nursing homes in China from a macro perspective [21,9]. In addition, most of the previous researches use the cross-sectional data to apply the single stage and DEA Tobit two-stage models in conducting a regional static research [22].
The purpose of this study is to make comprehensive evaluation of service e ciency and spatio-temporal variation of nursing homes in China, combined with DEA method and Tobit model to provide an evidence for decision-making by integrating and optimizing resource allocation, as well as to comprehensively improve the service e ciency of LTCFs.

Data
Data were obtained from the China Civil Affairs statistical yearbook [23], a national survey compiled by the Statistical Annual Report of the Social Service Industry and the reports of relevant departments, which is related to the social services in each year among all of the nursing homes in 31 provinces of China from 2012 to 2016. This project was deemed exempt from Institutional Review Board (IRB) review because the data are de-identi ed and publicly available.

Data Envelopment Analysis
DEA is an advantageous non-parametric technique for measuring the relative e ciency of homogeneous decision making units (DMUs) because of its advantages in dealing with multi-input and multi-output indicators [17]. DEA can handle multiple input and chromatography at the same time. Also, it doesn't need to set arti cial weight coe cient [24]. Furthermore, it has nothing to do with the dimension in selecting the suitable input and output items and it avoids calculating the standard cost of each service [25].
Therefore, the DEA model can be regarded as the better choice for the purpose and current situation of the research on the comprehensive e ciency of nursing homes. Selection of the model 1. Charnes-Cooper-Rhodes (CCR)and Banker-Charnes Cooper (BCC)model DEA calculates a maximal performance measure for each DMU with the sole requirement that each DMU lie on or below the frontier [26]. The traditional DEA model mainly included CCR model based on constant returns to scale and BCC model with variable returns to scale [17]. The redial CCR model [27]and BCC model [28] suffers from one shortcoming; they neglect the slacks in the evaluation of e ciencies. To overcome this shortcoming e ciency scores can be computed using the ''slack based'' non-radial and non-oriented DEA model [29]. This paper measures the overall TE, pure technical e ciency (PTE) and scale e ciency (SE) of nursing homes. The nal scores related to TE, PTE and SE were compared by Kruskal-Wallis test, given the sampling error.

Slack-based model (SBM) model
The Additive Model (AM) or SBM is based on input and output slacks [29]. In this study, a non-oriented and non-radial model known as SBM-DEA model has been used [30]. BCC model and SBM model all choose input-oriented. We used the input-oriented model in this demonstration. An input-oriented DEA model was used to compute TE scores of nursing care can be expressed by the following formula.
In the case of θ = 1, S − = 0, S + = 0, the nursing home is fully e cient, whereas θ < 1 means that a nursing home is ine cient. BCC model adds constraint conditions on the basis of CCR model: At this time, it means that the return on scale of DMU remains unchanged and reaches the maximum output scale. In addition, when this means that returns to scale are increasing. If the input of DMU is appropriately increased on the basis of the original input, the output will be increased by a higher proportion. Whereas it means diminishing returns to scale, and increasing input does not lead to a higher proportion of output. In addition, for the formula of SBM-DEA, please refer to the paper written by Mogha SK for details [31]. Due to TE values have truncated characteristics and e ciency values are relative, using general multiple regression models results in bias and parameter estimation instability [32].

Malmquist model
Productivity measures changes in a production unit's e ciency in transforming inputs into outputs from time t to time t+1 [33]. It was generally used in panel data. MPI was named after Malmquist [34] and was introduced by Caves, Christensen and Diewert to evaluate productivity changes among different production units [35]. MPI (also called total factor productivity changes (TFPC)) can be decomposed into technical e ciency changes (TEC) and technological changes (TC). TEC can also be decomposed into pure technical e ciency changes (PTEC) and scale e ciency changes (SEC) [36]. The formula shown as following: TFPC = TEC × TC= (PTEC × SEC) × TC

DEA classi cation
We adhere to all the assumptions made by the Troutt et al [37]. and the Seiford and Zhu [38] studies to classify the DMUs into Eastern, Central and Western regions. Assuming non-negative inputs and an output of unity for all the acceptable cases and using the DEA model proposed by Banker et al [39].

(Supplementary materials)
Variable selection

Input-output indicators
There is no consensus conclusion on the choice of input-output variables for the operation e ciency of old-age institutions [40]. Based on the theory of production factors in economics, input factors can be divided into capital, labor, and material inputs, and output factors can be divided into economic bene ts and social bene ts [41]. As LTCFs were labor-intensive industries. Scholars usually used xed assets as capital input index and various types of institutional staff as human input index [42,9]. As in terms of material inputs, the actual number of beds was a resource that was easily controlled by managers in material resources. A number of studies considered the number of beds, the number of institutions and the original price of xed assets as indicators of material resources [43,18]. In addition, the older care industry is a labor-intensive industry, the capital of the labor substitution is very small. Comprehensively, choosing human capital as input index is an appropriate index for e ciency comparison, like the main care resources including the number of social workers and the number of employees [44]. Therefore, the input indicators established in this article are the number of institutions, the number of employees at the end of the year, the original price of xed assets, the number of social workers, the number of beds at the end of the year (Supplementary Table 1).
Output indicators are important in evaluating the quality and quantity of long-term care services [45]. However, most researchers only consider the input of human capital or nancial capital, but fail to consider the service quality of pension service, which directly determine the nal effect of elderly care and the development prospects of the residential care [46,47]. The service quality can be re ected by the number of the disabled, the number of partially disabled the number of complete independents and in residential cares at the end of the year. Therefore, the output indexes should include operating income, the number of disabled, the number of partially disabled and the number of independents in residential cares at the end of the year, and also the number of rehabilitation and medical outpatients (Supplementary Table 1). Supplementary Table 2 presented summary statistics of input-output variables.

Results
Comparison of the e ciency of nursing homes in China from 2012 to 2016.     Supplementary Fig. 2, the ve e ciency values of LTCFs in combined total of 31 provinces and municipalities reached the lowest in 2014 and the highest in 2015. During this period, the results showed a trend that initially went upward and then eventually downward.  The regions were divided into three major classi cations, namely Eastern (developed), Central (generally developed), and Western (underdeveloped) regions, which refers to the regional division in the China Health Statistics Yearbook.

Supplementary
Supplementary  As shown in the Supplementary Fig. 3, the e ciency values of the three regions all showed a trend of an initial decline, and then an increase, as the agency e ciency value reached the lowest point in 2014. The Eastern region, in 2014, also showed a slight decline at rst, and then gradually increase. Based on input-oriented (assuming that the output is unchanged), the CCR model was used to calculate the projected value of non-effective LTCFs in all provinces and municipalities in terms of inputs, the number of adjustments (actual value-projected value), and adjustment ratio[(adjusted number / actual value) ×100% ]of 15 non-effective LTCFs in provinces and municipalities in terms of investment are calculated.
The results indicated that if the original output stay unchanged and the input is effectively optimized, it will result in the following: the number of beds in China's LTCFs can be reduced by 360038.9, the number of institutions can be reduced by 3464.9, and the number of social work can be reduced by 497, the total value of assets can be reduced by 87,0381.7 yuan, and the number of employees can be reduced by 3,5465.9. (Supplementary Table 5) The e ciency values calculated by SBM and BCC models were used as the Tobit in uencing factors in regression analysis The LTCFs with a greater number of women, social workers, and annual older people have a greater e ciency value. In addition, the more people within the age range of 35 and below, and 46 to 55 in the LTCFs, the lower e ciency value is. However, if the LTCFs with the greater number of people over the age of 56, the e ciency value will be higher. (Table 5) Table 5 Tobit [48,20], but lower than the average TE, PTE and SE of LTCFs in Xiamen [22], suggesting that the nursing homes in most of the areas in the country may have problems of insu cient utilization of constructions or lower TE. This might be associated with the unreasonable utilization of some LTCFs construction resources, as SE re ects the e ciency of production scale utilization.
As this study showed that there was a decreasing trend of nursing homes in effective TE from 2012 to 2014. The possible reasons behind for the decrease of the e ciency might be the blindly expanded the construction of nursing homes with less standardized management previously. However, China's Ministry of Civil Affairs effectively strengthen the policy support to build up pension service three years action plan (2014-2016). As for the standardized management, it has compiled local standards such as service standards for pension institutions, classi cation and evaluation of pension institutions, and formulated implementation rules for the establishment of pension institutions and measures for administration pension institutions to standardize the management system of pension institutions. Moreover, service reform was carried out by the national pension service comprehensive reform since 2014 [49]. A series of national policies laid a great foundation to promote economy and development motivation of pension institution by improving the service speci cation standards and national investment to expand the scale of the institution.
2. Spatial, temporal and regional distribution analysis of the operational effectiveness of nursing homes in China Nursing homes in Beijing, Guangdong, Tianjin and other economically developed areas have always been technically e cient from 2012 to 2016. Referring to the guidelines for the quality inspection of nursing homes nationwide, the policy of "one district one case" and "one hospital one policy" have been adopted to make up for some existing shortcomings [49,50]. The results showed that nursing homes in Shandong province changed from being technical ine ciency in 2014 to an effective TE in 2015. The existing reasons behind might be that in 2014, the "China's 12th ve-year plan for the development of undertakings for the elderly", and "Suggestions on accelerating the development of the elderly care service industry" were comprehensively implemented. Furthermore, the government has allocated services development special funds of 2.4 billion yuan to support the development of pension services industry of market-oriented polit area in Jilin, Shandong and other eight provinces [22]. All of these were done to effectively promote the nursing homes in Shandong in one year by means of a rapid rise from being technical ine cient into technical e cient.
TFPC of the sample nursing homes from 31 provinces declined by 2.14% during the 5-year period.
Although TEC has improved with an average of 0.1%, the average1.7% decrease of TC leads to the decrease of TFPC. Therefore, improving TC to boost TFPC is essential. The results obtained in the nursing homes of 31 provinces are not distinct. Torabipour et al [51] studied the productivity changes of hospitals in Ahvaz county from 2007 to 2010 and found that TC played a major role in productivity changes. Li explored the changes in e ciency and productivity of 45 THs in Anhui Province and determined that the e ciency of these hospitals exhibited a downward trend from 2009 to 2011; TEC and TC also decreased by 3.8% and 14.1%, respectively [52]. 4. The adjustment of projected value for non-DEA effective nursing homes in 31 provinces and municipalities of China in 2016 Five input indexes including the number of institutions, the number of employees at the end of the year, the original price of xed assets, the number of social workers, the number of beds at the end of the year have to be adjusted to hit projected value. It was noticeable that the number of beds at the end of the year in non-DEA effective nursing homes needs to be improved the most, which indicated that the construction scale of nursing homes in some provinces were blindly expanded, while the beds occupancy rate of private nursing homes was lower, with amount of idle beds. All of these caused wasting amount of expenditure and capital [9]. A survey showed that there were more than 30 private residential cares in Chengdu, with a total of more than 8,000 beds with only 4,500 older people [55]. It was presented that only about half of the beds in the elderly care institutions in Chengdu urban area were applied, and 32.5% of the urban institutions in Tianjin, Harbin, Jinan, Wuhan and other cities are in de cit for a whole year, with an overall low bed occupancy rate [56]. The number of social workers also needs to be improved by 22%, which is consistent with the current situation that China's integrated medical and nursing system is still far from perfect, the corresponding number of nursing staff is still small, and also the professional level is very low [57]. Social workers, as the mainstay resources to ensure the service quality, their competence can be applied to comprehensively evaluate nursing homes. The development of social workers needs the support of the government because of its relevant policies it can introduce. Also, it needs to attach importance to the prospects being trained in social work-related majors in colleges, universities and society, so as to ensure that highly quali ed social workers can play an important role in manpower for the elderly care service [58].
5. The analysis of factors affecting the e ciency of nursing homes, based on Tobit model The e ciency value of nursing home was in uenced by variable factors. The results presented that the age of the employees has an in uence on the e ciency value. The more employees who are aged 56 and above, the e ciency value gets to be higher, as employees in this age might get higher level of professional ability. However, those who are around 40 years old, do not have any sense of longevity with their jobs due to their lack of su cient skills [59]. They are expected to have more patience and dedication to elderly care, so that the elderly can obtain better service quality to improve the outputs and e ciency of nursing homes in the process [22]. A study showed that social workers play an important role in resource integration, emotional supplement, communication and coordination, spiritual support, social function and also in managing of nursing homes. This situation re ects the inadequacy of competent personnel, which resulted in the decline of the e ciency in the social work services [60]. The government should strengthen the support for the development of social workers by setting up policies, standards and norms to ensure that highly competent social workers can ll in the manpower needed by elderly service institutions. Rosko et al. (1995) conducted a research in Pennsylvania, which presented that the occupancy rate of institutions affected the service e ciency [61]. As the pension industry development information asymmetry, it is easy to nd the phenomenon of vacant beds and a bed at in different regions. The bed occupancy rate signi cantly affects the service e ciency, that is, for every 1% increase in occupancy rate, the comprehensive e ciency of pension institutions will increase by 0.871% [62].

Strengths And Limitations
This study combined DEA method and Tobit model to comprehensively evaluate the performance and in uencing factors of nursing homes in China, based on panel data. As it is representative of the overall country, which can improve the understanding of the complex e ciency issues in the different regions of China.
However, this study should be interpreted in light of several potential limitations. In this paper, there are restricted variables due to data limitations. Even though we compensate the limitations of previous studies to include outcome quality, further research should be conducted and include more variables of the quality of care. Since DEA is a data-oriented method, the e ciency values obtained are directly affected by the combination of inputs and outputs applied. However, further research should explore the standardized criteria and mechanism to appropriately selected to measure the e ciency of decisionmaking units, for example, qualitative studies could be applied to explore the reasons for the changes in TE and productivity.

Conclusions
This study provides an empirical representation of the e ciency and productivity changes of nursing homes in the process of advancing the marketization of pension services in China. As the improvement of the productivity and TE of nursing homes has not kept pace with the development of the scale of institutions, the adverse alteration in TC and PTEC should be emphasized. The TE, PTE, SE of nursing homes in Eastern regions were all the highest. Among these three e ciency values, TE of nursing homes is lowest due to the higher SE of nursing homes in China. The results showed that the in uencing factors of in nursing homes might include gender and age of employees, the number of social workers, the number of older people in a year. It was suggested that the country have to attach great importance to the cultivation and improvement of the professional level, educational background and experiences of the professionals for elderly care, so as to meet the diversi ed needs of different types of elderly groups.