A Regional Efficiency Assessment of Long-Term Care Services in Taiwan

Abstract

Taiwan is currently an aging society and will be a super-aging society in the near future. The purpose of this research is to use two models of data envelopment analysis (DEA)—the slacks-based measurement (SBM) model and the dynamic slacks-based measurement (DSBM) model—to analyze the efficiency of long-term care (LTC) in Taiwan. This analysis aims to explore the current situation of LTC in Taiwan and provide policy recommendations for LTC. The computation empirical result on the LTC efficiency score presents that the DSBM model exhibits higher efficiency than the SBM model after considering the carry-over variable in the former model. The result from the SBM model indicates that Taiwan’s outlying islands display the worst LTC efficiency, but this result does not appear in the DSBM model. Lastly, these two models both indicate that the number of elderly people being serviced in institutions exhibits higher efficiency and lower slack than those serviced in homes in 2017 and 2018. This paper concludes that the DEA approach is a viable method for examining the performance of the LTC services system as Taiwan approaches a super-aged society.

1. Introduction

The aging population is a global issue, as highlighted by the World Health Organization (WHO) for Sustainable Development Goal (SDG) 3—Good Health and Well-being, which unveils a Global Strategy and Action Plan for Ageing and Health for 2016–2020 [1]. The report by OECD (Organization for Economic Co-operation and Development) indicates that the countries with the oldest populations are Finland, Italy, and Japan, and the fastest-aging countries are Greece, Korea, Poland, Portugal, Slovenia, and Spain in the OECD [2]. Rudnicka et al. [3] point out that Brazil, China, and Saudi Arabia are the fastest-aging countries in non-OECD countries. The trajectory of the global aging population can be traced by evidence that no country with its population aged 65 and over had more than 11% in 1950. Until 2000, the country with the highest percentage of population aged 65 and over was 18%. According to a forecast, this percentage will rise to 38% in 2050 [4]. The old age refers to people over 65 years old. A society with an old age population over 7% of the total population is defined as aging; when the percentage is over 14%, the society is called aged. The World Health Organization [5] defines a super-aged society as one in which the percentage of old age population is over 20%. Taiwan became an aging society in 1993, an aged society in 2018, and will become a super-aged society in 2025. Data from the Directorate General of Budget, Accounting, and Statistics of the Executive Yuan of Taiwan in 2021 show that the population above the age of 65 exceeded the population under the age of 14 in 2017, which implies that fewer young people can provide care services to the ever-aging population. Hence, it is necessary to construct a long-term care (LTC) services system because Taiwan is nearing a super-aged society.

Because two-member families and nuclear families in Taiwan often work full-time, the demand for LTC for older adults is rising. The LTC should involve professional health care and medical services. Institutional care and home care are two common LTC modes in Taiwan, which is also the main discussion point in this study. These two kinds of healthcare modes can be found in Wang et al. [6] and Chen et al. [7], in which they were named as community-based and home-based health services. Different countries may have different LTC modes, such as nursing homes and residential care homes, like the LTC modes in Poland. Chruściel and Dobrowolska [8] study the correlation between social support and life quality in nursing homes and residential care homes. They conclude that the former prompts the life quality of older people, and the latter is a social nature institution that needs more social support than the former. Andersen [9] establishes the Andersen healthcare utilization model to explain the use of healthcare services in three types of determinants, including the predisposing factors, enabling factors, and need factors. Shih et al. [10] apply Andersen’s healthcare utilization model to explore Taiwan’s LTC need and suggest that the policy of the aging population should consider healthcare capacity and quality, especially for middle and high-dependent demanders. Some studies find that the institutional care needs in the Netherlands and Denmark have fallen quite dramatically [11,12,13]. The report [14] also indicates that the percentage of the population over 80 years in institutional care presents a strong decline in the Netherlands. Alders et al. [15] study the factors that change the demand for institutional long-term care. They find that the trend of demand for institutional care is reducing; hence, policies to encourage community living to stress home care are necessary.

Data envelopment analysis (DEA) is comprehensively applied to the performance estimation of businesses, non-government organizations, and government departments. The original thought behind this method comes from Farrell [16], who proposes an efficient production function as a foundation for measuring production efficiency. Charnes et al. [17] apply linear programming to establish a production efficiency frontier under the assumption of constant return to scale (CRS). Their method is named the CCR-DEA model, which takes the CRS-based production frontier and uses the radical approach to adjust the input and/or output factors of a decision-making unit (DMU) in order to reach an efficient production situation. Tone [18] proposes slacks-based measurement (SBM), which is a non-radical approach for a DMU to adjust input and/or output factors in order to reach production efficiency. His model is named the SBM model. The efficiency measurement previously only focused on a one-shot period, and each period for a DMU was independent until Färe et al. [19] emphasized the relation of each period and developed the dynamic DEA model that had a carry-over effect. Tone and Tsutsui [20] involve the slacks-based measurement approach in the dynamic DEA model and create the dynamic slacks-based measurement (DSBM) model. In past studies, we have seldom seen the LTC model involving the carry-over effect. The LTC issue can be referred to as SDG 3 with “Good Health and Well-being” so that the LTC is also relative to sustainability and suitable to introduce the carry-over effect in the LTC model. Since some studies about health care have connected to the SDGs, such as Chiu and Fong [21], Díaz Díaz et al. [22], and Shevelkova et al. [23], we utilize the SBM model without carry-over effect and the DSBM model with carry-over effect to analyze the LTC issue and compare the results from these two models.

Since Japan is a country with a super-aged society, there are many studies related to its LTC services, such as a study by Yamauchi [24], who applies the DEA method to investigate the efficiency of LTC services in Japan’s 47 administrative regions. The conclusion notes that labor costs and the number of LTC employees in Japan are the main sources of inefficiency and that they occur due to distribution inefficiency. This implies that the distribution of LTC employees in Japan presents inequality. The Japanese government should place more LTC employees in those administrative regions that lack LTC employees and push the development of the LTC service industry in its rural areas. Aside from Japan’s LTC service study, Zhang et al. [25] conducted an investigation into China’s LTC services. They took 32 LTC service institutions in Xiamen and applied the Tobit regression to find factors of waste or mismanagement. Employees in China’s LTC service industry can be divided into medical personnel, administrative staff, medical assistant personnel, and medical care workers. Among the four kinds of LTC employees, the number of medical personnel has the largest shortage within China’s LTC service institutions. This result implies that increasing the number of medical employees may usefully prompt the performance of its LTC service institutions. Based on studies in Japan and China, Taiwan is also a society with a rapidly aging population. Yeh [26] investigated the reform and challenge of Taiwan’s “The Ten-year Long-Term Care Plan 2.0”, unveiled in 2017, which calls for a reform of the LTC system in Taiwan. He thinks that the challenge of implementation of the plan will be the institutional and socio-cultural tensions. The former is major in the import of many foreign LTC workers, and the latter focuses on conflicting values between universal modern society and traditional Confucian ethics in Taiwan. Chen and Fu [27] also discuss Taiwan’s “The Ten-year Long-Term Care Plan 2.0”, but they focus on the LTC system design and service strategy. Lin et al. [28] studied the operating efficiency of senior care facilities and the care quality in Taiwan. They found that the senior care facilities with high care quality performed at low operating efficiency. The result implies that high-quality senior care facilities may not effectively utilize the input resources.

The LTC services in administrative regions and service institutions are critical for improving LTC service performance. Hu et al. [29] apply the carrying capacity theory to compute the target quantity of medical staff in Taiwan’s administrative regions. Their study touches not only on the number of LTC service employees but also on LTC efficiency in different administrative regions. The carrying capacity theory can help compute the potential increase in LTC service employees for enhancing regional LTC service performance. Ozbugday et al. [30] view LTC service as an industry to explore LTC industrial productivity changes. They use the Malmquist index to investigate the efficiency change in the LTC service industry in 17 OECD countries. One conclusion is an improvement in average performance during the research period, but an increasing old age population has brought pressure on governments’ fiscal expenditure. The LTC service dynamic literature can allow us to understand DMU performance and the changes and trends in DMU performance. Wu et al. [31] adopt panel data analysis on the LTC service performance of Taiwan’s administrative regions. They summarize that the area around a metropolitan city usually has a shortage of LTC service employees.

The DEA approach is suitable for LTC service studies, as confirmed by Cheng et al. [32], Chiu et al. [33], and Yucen and Min [34]. Following Taiwan’s Ministry of Health and Welfare announcement that Taiwan’s LTC services have been advancing year by year from 2017 to 2019 [35], this study aims to deeply probe Taiwan’s LTC services via the DEA approach. Aside from applying SBM and DSBM models on LTC service issues, we also confirm the effect of a carry-over variable in the DSBM model in which Tone and Tsutsui [20] point out that the efficiency score in this model is higher than that in the SBM model. We also executed a slack analysis to examine the potential capacity of Taiwan’s LTC services.

LTC services represent a long-term social welfare initiative, and it is more effective to use the carry-over concept to reflect improvements in efficiency over time. We believe that the carry-over effect plays a significant role in assessing LTC service efficiency; however, its limitation lies in potentially reducing the model’s discriminative power due to the increased number of variables. This aspect has seldom been addressed in the existing LTC services literature. This study aims to utilize two DEA models, namely, the SBM (static) model and the DSBM (dynamic) model, to assess LTC efficiency in Taiwan. The goal is to explore the current situation of LTC in Taiwan and provide relevant policy recommendations for its development. To assist readers,

Table 1. List of acronyms.

Acronym	Full Noun
DEA	Data Envelopment Analysis
SBM	Slacks-based Measurement
DSBM	Dynamic slacks-based Measurement
LTC	Long-term Care
WHO	World Health Organization
SDG	Sustainable Development Goal
OECD	Organization for Economic Co-operation and Development
CRS	Constant Return to Scale
DMU	Decision-making Unit
CV	Coefficients of Variance

lists all the specific acronyms used in this paper.

The remaining structure of this paper is as follows: Section 2 outlines the empirical model applied in this study; Section 3 presents the results of the numerical analysis; Section 4 provides a discussion; and the Section 5 concludes with policy recommendations.

2. Methodology

The SBM model is a refined approach within the DEA method, while the DSBM model extends the SBM model by incorporating a temporal dimension. Both the SBM and DSBM models provide valuable insights into efficiency assessment within the DEA framework. The former identifies current inefficiencies through slack analysis, while the latter examines how efficiency changes over time. The choice between these models depends on the context of LTC in Taiwan and the information available to guide implications regarding static efficiency performance and dynamic efficiency changes in the region.

In this section, we introduce the output-oriented SBM model and output-oriented DSBM model. The output-oriented SBM model can be presented as

$$\mathrm{Minimize}{\mathit{\rho }}_0={\displaystyle \frac{1}{1+\left(\frac{1}{L}\right){\sum }_{r=1}^L\frac{S_{r0}^{+}}{y_{r0}}}}$$

subject to Xλ = x_j0 + s_j0⁻,

Yλ = y_r0 − s_r0⁺,

0 ≤ λ ≤ 1, s_j0⁻ ≥ 0, s_r0⁺ ≥ 0.

(1)

The objective function in Model (1) is DMU 0’s efficiency score, which depends on its r^th output slack; i.e., s_r⁺, where r = 1, 2, …, L. It is obvious in the objective function that a large (small) output slack causes a low (high) efficiency score. In other words, s_r⁺ = 0 causes ρ₀ = 1, which means that the DMU performs the best efficiency. On the other hand, s_r⁺ > 0 makes ρ₀ < 1, meaning that the DMU is inefficient. When s_r⁺ → ∞, it means that ρ₀ → 0. The feasible interval for the efficiency score is from 0 to 1; i.e., ρ₀ ∈ [0, 1]. The variables X and Y, respectively, stand for input and output factors, in which X = (x_ji) ∈ R^M×N and Y = (y_ri) ∈ R^L×N. The former presents that the input matrix X is combined by the j^th input vector for the i^th DMU, where j = 1, 2, …, M and i = 1, 2, …, N, and X is in a real number space with M rows and N columns. The latter presents that the output matrix Y is combined by the r^th output vector for the i^th DMU, where r = 1, 2, …, L, and Y is in a real number space with L rows and N columns. The variable λ is defined as the weight vector that connects input and output factors.

Färe et al. [19] propose the Dynamic DEA model, which is based on a radical approach in order to handle the carry-over effect in the DEA model. Tone and Tsutsui [20] provide the Dynamic Slacks-based Measurement (DSBM) model that controls the carry-over effect by a non-radical approach. To achieve the maximum output target, we also apply the output-oriented DSBM model in this paper and present it as

$${\displaystyle \frac{1}{{\tau }_o^{*}}}=max{\displaystyle \frac{1}{T}}\sum _{t=1}^T \left[1+{\displaystyle \frac{1}{L}}\left(\sum _{r=1}^L {\displaystyle \frac{s_{rot}^{+}}{y_{rot}}}\right)\right]$$

X_tλ_t = x_j0t,

Y_tλ_t = y_r0 − s_r0t⁺,

Z_tλ_t = z^fix_0t,

Z_tλ_t = Z_tλ_t+1,

0 ≤ λ_t ≤ 1,

(2)

where τ^*₀ is an overall efficiency score for DMU 0. The variables X and Y in this model are the same as those in the SBM model, which, respectively, represent input and output factors. The variable s_r0t⁺ is DMU 0’s r^th output slack at t term period, where s_r0t⁺ ≥ 0 and t = 1, …, T. The variable Z_t, a specific component of the DSBM model, represents the matrix of the carry-over variable. The variable z^fix_0t denotes DMU 0’s carry-over variable with a fixed style at the t-term period, and the variable λ_t (λ_t+1) is defined as the weight vector that connects the input and output factors at the t (t + 1)-term period. The efficiency score at the t-term period for DMU 0 can be obtained by

$${\tau }_{ot}^{*}={\displaystyle \frac{1}{1+\frac{1}{L}\left({\sum }_{i=1}^L \frac{s_{iot}^{+*}}{y_{iot}}\right)}}$$

(3)

In order to examine the outcome differences between SBM and DSBM models, we employ the Mann–Whitney U Test. In detail, the Mann–Whitney U Test can examine the fitness of two populations [36]. Brockett and Golany [37] present a pioneer study that applies the Mann–Whitney U Test to rank efficiency scores from the DEA model. Based on their idea, there are two populations: A with n₁ samples and B with n₂ samples. We then use the formula to evaluate the fitness of two populations as U₁ = n₁n₂ + [n₁(n₁ + 1)]/2 − W₁ and U₂ = n₁n₂ + [n₂(n₂ + 1)]/2 − W₂, where W₁ is the rank sum for n₁ samples, and W₂ is the rank sum for n₂ samples. We use U₁ (U₂) when the sum of efficiency scores in the n₁ (n₂) sample is smaller than the number of the n₂ (n₁) sample.

Given n₁ ≥ 10 and n₂ ≥ 10, we can use the statistics as follows to compute the mean value E(U₁) = E(U₂) = (n₁n₂)/2 and variance V(U₁) = V(U₂) = [(n₁n₂)(n₁ + n₂ + 1)]/12. We next apply the Z-test to test H₀: the efficiency score in sample A ≥ the efficiency score in sample B, or H₁: the efficiency score in sample A < the efficiency score in sample B. Given the significance level α, when Z < −Z_α, H₀ is rejected; if Z ≥ −Z_α, then H₀ is not rejected.

3. Empirical Analysis

This section presents the data analysis and the empirical results.

3.1. Data Source

The research objectives are 22 administrative regions in Taiwan. The data source is the website of the Ministry of Health and Welfare of Taiwan, and the data period spans from 2017 to 2019 [35]. Since the government’s data publication is not continuous, the research period in this study is also limited. The input factors include (i) the expenditure of LTC services by the government (x₁) and (ii) the labor numbers for LTC services, including medical personnel, administrative staff, medical assistant personnel, and medical care workers (x₂). The output factors include (i) the number of old age serviced in care institutions (y₁) and (ii) the number of old age serviced in homes (y₂). Different from the SMB model, there is a carry-over variable from Wu et al. [31] in the DSBM model. In our paper, we let variable z be the carry-over variable and denote it as the number of population over 65 years. We take 2017 as a base year to remove the influence of inflation in all monetary data.

3.2. Descriptive Statistics

Table 2. Descriptive statistics for all variables.

Statistics	Old Age in Institution (y1)	Old Age in Home (y2)	Government Expenditure on LTC Services (x1)	Number of LTC Service Employees (x2)	Old Age at 65 Years Old and Above (z)
Unit	Person	Person	NTD (Thousand)	Person	Person
Avg.	2077	6495	981,965	1735	156,192
Max	9029	35,065	3,385,322	6828	578,511
Min	25	9	7634	35	1361
Std Dev	2180	7879	884,079	1689	155,149
Coefficient of variation	1.050	1.213	0.900	0.973	0.993

presents the results of descriptive statistics on all variables. The government’s input resources in Taiwan’s 22 administrative regions for LTC services show a small difference, such that the coefficients of variance (CV) of expenditure and LTC labor are individually 0.900 and 0.973. Compared to them, the coefficients of variables for the number of elderly people being serviced in LTC institutions and those being serviced in homes, as well as the number of people above 65, have relatively high values of 1.050, 1.213, and 0.993, which imply that the biggest difference is in the number of those being serviced in homes, while the second biggest difference is in the number of those being serviced in LTC institutions. The government’s resources should, thus, not be uniform in each administrative region of Taiwan, but rather, their input in Taiwan’s administrative regions should be based upon their needs.

We apply the SBM model and DSBM model to investigate the performance of LTC services in 22 administrative regions of Taiwan.

3.3. Empirical Results from the SBM Model

Table 3. Results of the SBM model for Taiwan’s LTC efficiency.

Statistics\Year	2017	2018	2019
Avg.	0.784	0.717	0.776
Max	1	1	1
Min	0.234	0.069	0.088
Std Dev	0.210	0.217	0.226

shows the result of the SBM model for Taiwan’s LTC efficiency. The average LTC efficiency scores from 2017 to 2019 form a V-shape, and the minimum LTC efficiency score for Taiwan’s 22 administrative regions in 2018 was 0.069 for Lienchiang County.

Table 4. The Mann–Whitney U Test on Taiwan’s LTC efficiency score in the SBM model.

Year/Year	2017	2018
2017	-	-
2018	1.188	-
2019	−0.012	−1.117

shows no significant difference in average LTC efficiency scores between 2017, 2018, and 2019. However, a worrisome phenomenon is that the standard deviations of LTC efficiency scores for Taiwan’s 22 administrative regions from 2017 to 2019 have become larger. This implies that the difference in LTC efficiency scores in the 22 administrative regions is expanding yearly. Based on the average LTC efficiency scores and the standard deviations of LTC efficiency scores in Table 3, Taiwan’s LTC efficiency did not show continuous improvement from 2017 to 2019. This does not support the saying in the white book by Taiwan’s Ministry of Health and Welfare that the performance of long-term care services in Taiwan has improved year by year from 2017 to 2019 (2020) [35].

The computation results of LTC efficiency score for Taiwan’s 22 administrative regions are in

Table 5. SBM efficiency score for Taiwan’s administrative regions.

DMU	2017		2018		2019
	Score	Ranking	Score	Ranking	Score	Ranking
New Taipei City	0.718	13	0.924	4	0.992	6
Taipei City	0.610	19	0.631	15	1	1
Taoyuan City	0.589	20	0.461	20	0.677	17
Taichung City	0.684	17	1	1	1	1
Tainan City	0.841	11	0.694	12	0.947	8
Kaohsiung City	0.714	14	0.599	19	0.997	5
Six Municipalities	0.693	-	0.718	-	0.936	-
Yilan County	1	1	1	1	0.900	9
Keelung City	1	1	1	1	1	1
Hsinchu City	0.654	18	0.608	18	0.632	20
Chiayi City	0.807	12	0.888	5	0.991	7
Hsinchu County	0.703	16	0.449	21	0.713	14
Miaoli County	0.712	15	0.614	17	0.679	16
Changhua County	0.894	9	0.650	14	0.731	12
Nantou County	1	1	0.834	7	0.677	18
Yunlin County	0.951	7	0.814	8	0.745	11
Chiayi County	0.872	10	0.692	13	0.722	13
Pingtung County	0.989	6	0.627	16	0.809	10
Taitung County	0.910	8	0.742	11	0.659	19
Hualien County	0.991	5	0.880	6	0.683	15
Thirteen Cities and Counties	0.883	-	0.754	-	0.765	-
Penghu County	1	1	0.797	9	1	1
Kinmen County	0.234	22	0.790	10	0.419	21
Lienchiang County	0.377	21	0.069	22	0.088	22
Three Outlying Islands	0.537	-	0.552	-	0.502	-

in which we further divide the administrative regions into three groups: a group of six municipalities, including New Taipei City, Taipei City, Taoyuan City, Taichung City, Tainan City, and Kaohsiung City; a group of three outlying islands, including Penghu County, Kinmen County, and Lienchiang County; and a group of the other 13 cities and counties. Table 5 shows that the outlying islands group always has the worst LTC efficiency; the group of 13 cities and counties had the best performance in 2017 and 2018, and the group of six municipalities had the best performance in 2019. Furthermore, the efficiency scores of the group of six municipalities have improved from 2017 to 2019.

3.4. Empirical Results from the DSBM Model

Table 6. Result of the DSBM model for Taiwan’s LTC efficiency.

Statistics	Overall Efficiency Score	2017	2018	2019
Avg.	0.914	0.874	0.943	0.980
Max	1	1	1	1
Min	0.521	0.275	0.632	0.786
Std Dev	0.122	0.193	0.109	0.051

presents the results of the DSBM model relative to Taiwan’s LTC efficiency. The average LTC efficiency scores from 2017 to 2019 exhibit an improvement trend, and the minimum score is 0.275 of Kinmen County in 2017.

Table 7. The Mann–Whitney U test on Taiwan’s LTC efficiency score in the DSBM model.

Year/Year	2017	2018
2017	-	-
2018	−1.067	-
2019	−0.650	−1.686 *

Note: * Significance at 10% level.

shows a significantly negative relationship in average LTC efficiency scores between 2018 and 2019. It implies that some administrative regions exhibited low (high) LTC efficiency in 2018, but they presented high (low) LTC efficiency in 2019. In addition, one good phenomenon is that the standard deviations of LTC efficiency scores from 2017 to 2019 have become smaller. This illustrates that the difference in LTC efficiency in Taiwan’s 22 administrative regions is growing smaller. Based on the average LTC efficiency scores and standard deviations of LTC efficiency scores from 2017 to 2019 in Table 6, Taiwan’s LTC efficiency has had a continuous improvement from 2017 to 2019. This result from the DSBM model supports the saying in the white book by Taiwan’s Ministry of Health and Welfare that long-term care services in Taiwan are improving year by year from 2017 to 2019 [35].

Table 8. The DSBM efficiency score for Taiwan’s administrative regions.

DMU	Overall Efficiency Score	2017 Efficiency Score	2018 Efficiency Score	2019 Efficiency Score	Ranking of Overall Efficiency
New Taipei City	1	1	1	1	1
Taipei City	1	1	1	1	1
Taoyuan City	0.856	0.665	0.999	1	17
Taichung City	0.918	0.790	1.000	1	16
Tainan City	0.853	0.880	0.750	0.955	18
Kaohsiung City	0.760	0.732	0.632	1	20
6 Municipalities	0.898	0.844	0.897	0.993	-
Yilan County	1	1	1	1	1
Keelung City	1	1	1	1	1
Hsinchu City	1	1	1	1	1
Chiayi City	0.937	0.832	1	1	13
Hsinchu County	0.777	0.844	0.712	0.786	19
Miaoli County	0.925	0.805	1	1	15
Changhua County	0.935	1	0.892	0.918	14
Nantou County	1	1	1	1	1
Yunlin County	0.996	0.988	1	1	9
Chiayi County	0.975	0.930	1	1	10
Pingtung County	0.958	1	0.883	1	12
Taitung County	1	1	1	1	1
Hualien County	0.969	1	1	0.911	11
13 Cities and Counties	0.959	0.954	0.960	0.970	-
Penghu County	1	1	1	1	1
Kinmen County	0.521	0.275	0.887	0.999	22
Lienchiang County	0.734	0.481	1	0.994	21
3 Outlying Islands	0.752	0.585	0.962	0.998	-

lists the computation results of LTC efficiency scores for six municipalities, 13 cities and counties, and three outlying islands in the DSBM model. We find that the group of three outlying islands has the best performance on yearly efficiency scores in 2018 and 2019 but the worst overall LTC efficiency among the three regional groups. This is caused by the group of three outlying islands having an extremely low yearly efficiency score in 2017 at 0.585, and then their yearly efficiency scores soared to 0.962 and 0.998 in 2018 and 2019, respectively. This implies that LTC efficiency in the group of three outlying islands is improving; in other words, the old age population in the group of three outlying islands can enjoy LTC services because of LTC efficiency improvement. The result of the old age population obtaining LTC services also appears in the groups of six municipalities and 13 cities and counties, in which their year efficiency scores from 2017 to 2019 have been higher and higher.

From micro-level observations, Table 8 shows that DMUs with the best LTC efficiency score include New Taipei City and Taipei City in the group of six municipalities, and the percentage is 33%; Yilan County, Keelung City, Hsinchu City, Nantou County, and Taitung County have the best LTC efficiency score in the group of 13 cities and counties, and the percentage is 38%; only Penghu County has the best LTC efficiency score in the group of three outlying islands, and the percentage is also 33%. According to the percentages, the group of 13 cities and counties performed better than the others on LTC services.

3.5. A Comparison Between the SBM and DSBM Models

In this section, we perform comparisons from efficiency score analysis and slack variable analysis between the SBM and DSBM models.

3.5.1. Comparisons Between Efficiency Scores and Standard Deviations of Efficiency Scores

We compared the results of SBM and DSBM in Table 3 and Table 6 and found that the average LTC efficiency scores and the standard deviations of LTC efficiency scores in the DSBM model continuously improved from 2017 to 2019 when the average LTC efficiency scores were higher and higher. However, the standard deviations of LTC efficiency scores are smaller and smaller, but this result does not appear in the SBM model. Hence, the carry-over variable in the DSBM model has different implications from the SBM model without the carry-over variable.

Table 8 from the DSBM model shows that the outlying islands group performed the best on year efficiency scores in 2018 and 2019. This is very different from the SBM model (Table 5), in which the outlying islands group has always had the worst performance on LTC efficiency. This may be caused by the number of elderly people aged 65 and above being the carry-over variable in the DSBM model, but there is no carry-over variable in the SBM model. When we only focus on a comparison between the group of six municipalities and the group of 13 cities and counties, the same results in Table 5 on the SBM model and in Table 8 on the DSBM model show that the group efficiency scores of 13 cities and counties are better than those of six municipalities for 2017 and 2018, but the result of group efficiency scores reverses in 2019. This finding tells us that the carry-over variable applied in this model may change the performance of a DMU with the poorest performance, such as the group of three outlying islands.

Table 9. Comparison between the leader group taking the DSBM model as a standard.

	2017					2018					2019
	DMU	DSBM		SBM		DMU	DSBM		SBM		DMU	DSBM		SBM
		Score	Ranking	Score	Ranking		Score	Ranking	Score	Ranking		Score	Ranking	Score	Ranking
Leader Group	Taipei City	1	1	0.610	19	New Taipei City	1	1	0.924	4	New Taipei City	1	1	0.992	6
	Yilan County	1	1	1	1	Taipei City	1	1	0.631	15	Taipei City	1	1	1	1
	Changhua County	1	1	0.894	9	Taichung City	1	1	1	1	Taoyuan City	1	1	0.677	17
	Nantou County	1	1	1	1	Yilan County	1	1	1	1	Taichung City	1	1	1	1
	Pingtung County	1	1	0.989	6	Miaoli County	1	1	0.614	17	Kaohsiung City	1	1	0.997	5
	Taitung County	1	1	0.910	8	Nantou County	1	1	0.834	7	Yilan County	1	1	0.900	9
	Hualien County	1	1	0.991	5	Yunlin County	1	1	0.814	8	Nantou County	1	1	0.677	18
	Penghu County	1	1	1	1	Taitung County	1	1	0.742	11	Chiayi County	1	1	0.722	13
	Keelung City	1	1	1	1	Penghu County	1	1	0.797	9	Pingtung County	1	1	0.809	10
	Hsinchu City	1	1	0.654	18	Keelung City	1	1	1	1	Taitung County	1	1	0.659	19
	-	-	-	-	-	Hsinchu City	1	1	0.608	18	Penghu County	1	1	1	1
	-	-	-	-	-	Lienchiang County	1	1	0.069	22	Keelung City	1	1	1	1
	-	-	-	-	-	-	-	-	-	-	Chiayi City	1	1	0.991	7

arranges DMUs with an efficiency score of unity in the DSBM model into the leader group, and then we compare the DMUs’ efficiency results in the DSBM and SBM models by taking the DSBM model as a standard. We find that the number of DMUs with the best efficiency score in the DSBM model is increasing. In addition, we find a large difference in estimation results by the DSBM and SBM models in the leader group. For example, the DSBM efficiency score for Taipei City in 2017 ranks number one by the DSBM model, but it is ranked 19th by the SBM model. This result implies that the DSBM and SBM models have large inconsistencies in the ranking of top DMUs. It also implies that the carry-over variable has a critical influence on the ranking of top DMUs.

Note that the discrimination in the DEA model loses its discriminative power when there are more variables in the DEA model [38]. This result can be confirmed by Table 4, Table 7, and Table 8 in this paper, in which the variables in the DSBM model are more than those in the SBM model, and the number of effective DMUs in the DSBM model is also more than those in the SBM model.

3.5.2. Analysis of Output Slack Variables

The slack variable ratios of LTC services in institutions or those in homes are defined as the percentage of the slacks in either place to the optimal numbers of LTC services in either place. For example, the slack of LTC services in institutions is 20, which is the gap between the actual number of LTC services in institutions and the optimal number of LTC services in institutions. If the optimal number of LTC services in institutions is 100, then the slack variable ratio of LTC services in institutions is 20%. A low (high) ratio of a slack variable means that the actual number is close to (far from) the optimal number. A low (high) slack ratio implies a high (low) efficiency in which the actual number is close to (far from) the optimal one. Hence, slack analysis and efficiency analysis are the same thing in the DEA approach. The former focuses on the gap between the actual number and the optimal number, and the latter emphasizes the magnitude of the efficiency score between zero and one.

Figure 1 and Figure 2 illustrate the pattern of slack ratios of the number of old age being serviced in institutions or those being serviced in homes in the SBM and DSBM models, respectively. We find that the number of elderly being serviced in homes has a larger potential capacity than those who have been serviced in institutions in 2017 and 2018, with no difference between the SBM and DSBM models. In 2019, the potential capacities for the number of elderly serviced in institutions and homes are similar. Going into more detail, the potential capacity for the number of elderly serviced in homes is falling. This result implies that the number of old age serviced in their homes sharply increased in 2019, which caused their potential capacity to drop sharply. It may be a warning sign that LTC services in homes are approaching low capacity, which is similar to LTC services in institutions. We suggest that the capacity of LTC services in homes should be higher than that in institutions since LTC services in homes are the first defense line for elderly people with LTC needs, while LTC services in institutions are the final defense line for those with LTC needs.

4. Discussion

Based on the empirical results of the SBM and DSBM models, we find that these two models yield different outcomes. The DSBM model demonstrates weaker discrimination than the SBM model, and they show different patterns of LTC efficiency scores as well as varying potential capacities for LTC services. These differences primarily arise from the carry-over effect in the DSBM model, which is not present in the SBM model. Therefore, the carry-over variable plays a critical role in estimating LTC in Taiwan. The significance of the carry-over effect in long-term health and medical research has been highlighted in studies by Duncan-Jones [39] and Chiu et al. [33].

According to the white paper from Taiwan’s Ministry of Health and Welfare, LTC performance has improved year by year from 2017 to 2019 [35]. The empirical results from the SBM model do not support this trend, while the results from the DSBM model do. The former model does not include a carry-over variable, whereas the latter incorporates the population aged 65 and above as a carry-over variable, which critically influences the results of both models. The SBM model identifies current inefficiencies, while the DSBM model examines how efficiency changes over time. Therefore, health and medical events require long-term and dynamic observation [40].

The literature on Taiwan’s regional studies typically divides the 22 administrative regions into three groups: six municipalities; 13 cities and counties; and three outlying islands [41,42,43]. Following this classification, our research results from the SBM model indicate that the group of three outlying islands exhibits the worst LTC performance. In contrast, the results from the DSBM model show that no group dominates the others. Furthermore, when comparing the DMUs with the highest rankings in LTC efficiency in both the SBM and DSBM models, we find significant inconsistencies in their top rankings. This discrepancy is likely due to the influence of the carry-over variable in the DSBM model. Nevertheless, both the SBM and DSBM models provide the same information: in 2019, the efficiency of the number of elderly individuals being serviced in institutions and homes was high, with low slack. This implies that the efficiency scores for the number of elderly individuals serviced in institutions and homes decreased in 2019.

5. Conclusions and Policy Recommendation

Taiwan became an aging society in 1993 and is expected to be a super-aging society by 2025, thus showing huge potential LTC demand. The Taiwan government currently announces statistical data on LTC services to illustrate LTC performance, but these data cannot provide the potential capacity for LTC services. This study employs the SBM model and DSBM model with the number of elderly aged 65 and above as a carry-over variable to estimate LTC performance and the potential capacity of LTC services in Taiwan.

The SBM model indicates that the group of three outlying islands has the lowest LTC efficiency, while the DSBM model shows that no group dominates the others. This suggests that the three regional groups in Taiwan exhibit a short-term efficiency gap, but their LTC performances are relatively similar. The government should focus on the DMUs with low rankings in care efficiency according to the SBM model and help them improve their short-term care efficiency. The carry-over variable is a critical factor in LTC analysis, indicating that it is essential for long-term assessments.

When comparing the two care modes in Taiwan—services provided in institutions and those provided at home—both the SBM and DSBM models show that the efficiency of elderly individuals serviced in institutions and at home was high, with low slack in 2019. This implies that there was significant slack in the home care mode in 2019. Therefore, the government should encourage home-based services, not only to reduce slack but also to improve the efficiency of home care.

The contributions of this paper are as follows. (i) We use the DEA approach to confirm the descriptive statistics in the white book published by Taiwan’s Ministry of Health and Welfare. (ii) The results of different DEA models confirm that the LTC performance on the number of elderly being serviced in institutions and in homes presented high efficiencies and low slack in 2019. Because the data were collected from yearbooks, this paper focuses solely on the capacity of healthcare for older adults. In the future, we can examine the reasons for the need for LTC to improve its quality, particularly for elders suffering from dementia, who gradually lose cognitive function and face a decline in quality of life. Research on LTC quality is a valuable direction for the country as it confronts the challenges of an aging society.