Performance measurement of nonhomogeneous Hong Kong hospitals using directional distance functions

Abstract

Cook et al. (Oper Res 61(3):666–676, 2013) propose a DEA-based model for the performance evaluation of non-homogeneous decision making units (DMUs) based on constant returns to scale (CRS), extended by Li et al. (Health Care Manag Sci 22(2):215–228, 2019) to variable returns to scale (VRS). This paper locates these models into more general DDF models to deal with nonhomogeneous DMUs and applies these to Hong Kong hospitals. The production process of each hospital is divided into subunits which have the same inputs and outputs and hospital performance is measured using the subunits. The paper provides CRS and VRS versions of DDF models and compares them with Cook et al. (Oper Res 61(3):666–676, 2013) and Li et al. (Health Care Manag Sci 22(2):215–228, 2019). A kernel-based method is used to estimate the distributions as well as a DEA-based efficiency analysis adapted by Simar and Zelenyuk to test the distributions. Both DDF CRS and VRS versions produce results similar to Cook et al. (Oper Res 61(3):666–676, 2013) and Li et al. (Health Care Manag Sci 22(2):215–228, 2019) respectively. However, the statistical tests find differences for the different technologies assumed as would be expected. For hospital managers, the more generalised DDF models expand their range of options in terms of directional improvements and priorities as well as dealing with non-homogeneity.

Supplementary Information

The online version contains supplementary material available at 10.1007/s10729-022-09625-0.

Highlights

• Generalised DDF models for dealing with nonhomogeneous DMUs

• Identifying directions and priorities for performance improvement in a hospital setting

• Comparing the DDF models for nonhomogeneous DMUs with Cook et al. (2013) and Li et al. (2019)

Introduction

Hospitals are supposed to be alike, but in fact may not be similar due to the different processes involved. For instance, most tertiary hospitals provide emergency services rather than community services, and most primary care facilities provide community services rather than emergency services. Data Envelopment analysis (DEA) is a non-parametric approach to gauge the relative efficiencies of decision making units (DMUs) [1], which is widely used in healthcare performance measurement. However, in conventional DEA, DMUs exhibit homogeneity, with the same type of inputs and outputs (in differing amounts). Golany and Roll [2] believe that when using DEA to measure performance of DMUs, DMUs must satisfy homogeneity, that is, they have the same goal or task, the same external environment, and the same input and output variables and dimensions. Zhang et al. [3] set rigorous exclusion criteria to screen homogeneous hospitals for DEA evaluation, which demonstrated that non-homogeneity was a factor contributing to hospital efficiency.

Non-homogeneity among DMUs arises from differences in the external environment and internal conditions. Dyson et al. [4] suggest that non-homogeneity among DMUs stems from their environment and economies of scale. Some examples of non-homogeneity include Castelli et al. [5] subunits with interdependent DMUs which did not have the same inputs and outputs; Haas and Murphy [6] DMUs with different business processes, input / output sets, and environments; Saen et al. [7] DMUs where the inputs and outputs were not completely identical because of functional discrepancies. Non-homogeneity also occurs as noted by Thompson et al. [8] where outputs not produced are replaced by zeros and obviously, there will be no inputs involved for these outputs.

Various methods have been proposed to address non-homogeneity. Cook et al. [9] and Imanirad et al. [10] proposed corresponding solutions for different outputs and inputs, by grouping DMUs, decomposing their inputs/outputs, and evaluating their efficiency with the same inputs/outputs. Li et al. [11] proposed an approach for processing heterogeneous DMUs with different input configurations. Du et al. [12] constructed a non-homogeneous parallel network structure model considering DMUs with different outputs. Barat et al. [13] developed a network DEA-based method to address the problem of non-homogeneity in settings where subunits operate in a mixed network structure. Li et al. [14] and Lin et al. [15] extend the work of Cook et al. [9] from a constant returns-to-scale (CRS) version to a variable returns-to-scale (VRS) to measure the efficiencies of non-homogeneous hospitals. In another study with different inputs and different outputs, Wu et al. [16] split the inputs and assign them to each output, and similarly split outputs and assign them to each input, and then present a DEA model that places DMUs into homogeneous subgroups, calculates each subgroup’s overall efficiency in, and applies it to evaluate the environmental efficiency among industries in China. In an application to a university in China, Ding et al. [17] split each DMU into subgroups with homogeneous inputs and outputs, in a two-stage network structure for faculty research processes and student research processes. Chen and Wang [18] define the inclusion relationship of the production structure of nonhomogeneous DMUs and constructed the DEA cross-efficiency model for them by integrating the perspectives of self-evaluation and peer-evaluation.

However, in a non-homogenous environment, hospital managers need not only to select the direction of performance improvement, but also to determine the priority of improvement. Various studies have used nonradial models to address this. Färe and Lovell [19] specify an LP (linear programming) model to measure non-radial Pareto-Koopman efficiency. The LP of directional distance function (DDF) is first set out by Shepherd [20] and introduced by Luenberger [21, 22] as a shortage function, which considers that inefficiency can be measured by the distance from the observed DMU to the production possibility frontier (PPF). Färe et al. [23] developed an input distance function, though the model assumed that all inputs scaling pro-rate were similar to the Farrell efficiency measure. Chung and Färe [24], Chambers et al. [25] made important contributions to the theoretical development and defined the basic properties of the directional distance function. The directional distance function has been increasingly used for efficiency evaluation over the last twenty years [26]. It has the advantage over existing approaches of being a more general model that can provide both radial and non-radial measures and easily accommodates input and output radial DEA models. It can also accommodate both desirable and non-desirable inputs and outputs.

To our knowledge, there is no literature to extend DDF models to the non-homogeneity problem in DEA. This study develops DDF-based models for performance evaluation of nonhomogeneous DMUs applying these to Hong Kong hospitals. The CRS and VRS versions of the DDF-based models are compared with Cook et al. [9] and Li et al. [14] respectively. The remainder of this paper proceeds as follows. Section 2 reviews the DEA models with non-homogeneous DMUs proposed by Cook et al. [9] and Li et al. [14], and presents the CRS and VRS versions of our proposed DDF models for nonhomogeneous DMUs. In Sect. 3, a data set of Hong Kong hospitals illustrates our proposed models to conduct performance evaluation and provide efficiency improvement suggestions, as well as comparing these with Cook et al. [9] and Li et al. [14] models. The last section discusses the main findings, implications and possible future research directions.

Methodology

For the grouping of non-homogeneous DMUs, it is necessary to divide all DMUs into subgroups with the same kinds of inputs and outputs. Both Cook et al. [9] and Li et al. [14] assume that all DMUs have the same inputs, and only discuss the case of different outputs. This paper also assumes the DMUs employ co-inputs to produce nonhomogeneous outputs.

Assume there are n DMUs divided into p mutually exclusive subgroups, where all DMUs within a subgroup contain the same kinds of inputs and outputs while other subgroups contain different kinds of outputs. Table 1 illustrates with eight DMUs, each of which has the same kinds of inputs and at most three kinds of outputs.

Table 1.

Example of DMUs grouping

		Input (I)		Output (O)
Group	DMU	I1	I2	O1	O2	O3
p = 1	1	√	√	√
	2	√	√	√
p = 2	3	√	√		√	√
	4	√	√		√	√
	5	√	√		√	√
p = 3	6	√	√	√	√	√
	7	√	√	√	√	√
	8	√	√	√	√	√

A tick (√) indicates that a DMU has this input or output. These eight DMUs have the same inputs, but produce different outputs. If we classified the DMUs into several subgroups with the same inputs and outputs to make up a production possibility set (PPS), the PPS would contain too few DMUs to calculate the DEA efficiencies. For example, DMUs with the same kinds of inputs and outputs can be classified as a subgroup, denoted by L. In Table 1, the eight DMUs are divided into three subgroups, i.e., L₁ = {DMU = 1, 2}, L₂ = {DMU = 3, 4, 5}, L₃ = {DMU = 6, 7, 8}. There are only two or three DMUs in each group.

Figure 1 depicts a block diagram and steps of performance measurement for non-homogeneous DMU that is decomposed into three stages.

• Stage 1: Each output is considered as a subunit, and the outputs and inputs with the same DMUs are merged into a maximum subunit. The inputs are then split for each maximum subunit. In this stage, we need to know the proportions or ratios to split the inputs for each output.

• Stage 2: Each maximum subunit and inputs make up a production possibility set (PPS), and the efficiency values of DMUs for each maximum subunit under the same PPS are calculated;

• Stage 3: The DMU efficiency is aggregated by weighting the efficiency values of corresponding maximum subunits.

Fig. 1.

The diagram of performance measurement of nonhomogeneous DMUs

In Fig. 1, a subset with the inputs and the same output is referred to as a subunit, denoted by R. Then the subunits are R₁ = {DMU = 1, 2, 6, 7, 8}, R₂ = {DMU = 3, 4, 5, 6, 7, 8}, R₃ = {DMU = 3, 4, 5, 6, 7, 8}. The subunits containing the same elements are combined to constitute a maximal subunit, see Cook et al. [9] or Algorithm A1 of Appendix in the electronic supplemental material of this paper. In this example, R₂ and R₃ are merged, i.e. R₂’ = {R₂, R₃}, then the combined maximum subunit is unique, proved in reference [9] p.669. We use p for the index of the subgroups, k for the index of maximal subunits. Thus, R₁ = {p = 1, 3}, R₂’ = {p = 2, 3}, L₁ = {k = 1}, L₂ = {k = 2’}, L₃ = {k = 1, 2’}. For convenience, the notations used in this paper are summarised in Table 2.

Table 2.

Definitions of notations

Variable	Definition	Variable	Definition
i	Index of inputs	r	Index of outputs
j	Index of DMUs	o	Index of DMU under evaluation
p	Index of subgroups	k	Index of subunit
po	The p-th subgroup that contains observed DMU o	ko	The k-th subunit of observed DMU o
Lp	The subgroup p	Rk	The subunit k
αikp	The proportion of the i-th input assign to subunit k of subgroup p	ωko	The weight of the subunit k to observed DMU o
vi	The multiplier assigned to the i-th input	ur	The multiplier assigned to the r-th output
${\epsilon }_k^o$	The unrestricted variable of observed DMU o	${\lambda }_j^k$	The intensity variable of DMU j in subunit k
eko	The efficiency of subunit k of observed DMU o in conventional DEA	βko	The inefficiency of subunit k of observed DMU o in DDF
eo	The overall efficiency of observed DMU o	βo	The overall inefficiency of observed DMU o
aikp	The lower limit of αikp and aikp > 0	bikp	The upper limit of αikp and bikp > 0

In discussing efficiency, it is important to define the production possibility set and its efficient frontier, and then to identify the method of measuring the efficiency of inefficient DMUs. Suppose the technology reference set Ω = {(x, y): x can produce y}. There are j = 1, …, J DMUs (i.e., firms or plant) in the dataset, each DMU uses input X = ($x_1$,$x_2$,...,$x_{\mathrm{m}}$)∈${\mathbb{R}}_{+}^{\mathrm{m}}$ to jointly produce desirable outputs Y = ($y_1$,$y_2$,...,$y_s$) ∈ ${\mathbb{R}}_{+}^{\mathrm{s}}$.

In the context of nonhomogeneous DMU, Cook et al. [9] and Li et al. [14] developed DEA multiplier models under the CRS and VRS technologies as presented in Eqs. (1) and (2), respectively. For the model description, the initials of the author's surname are used for short, namely the Eq. (1) is CHIRZ and the Eq. (2) is LLM.

$$\begin{array}{cc}[\mathrm{CHIRZ}]& e_o=\max {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}[u_k{y}_{\mathit{ko}}/({\sum }_i{\alpha }_{ik{p}^o}{v}_i{x}_{\mathit{io}})]\\ & s.t.{\sum }_{r\in R_k}{u_{r}y}_{\mathit{rj}}/({\sum }_i{\alpha }_{\mathit{ikp}}{v}_i{x}_{\mathit{ij}})\le 1,j\in p,k\in L_p,\forall p\\ & {\sum }_{k\in L_p}{\alpha }_{\mathit{ikp}}=1,\forall i,p\\ & {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}=1\\ & {\alpha }_{\mathit{ikp}}\in [a_{\mathit{ikp}},b_{\mathit{ikp}}]\\ & v_i,u_k\ge 0,\forall i,k\end{array}$$ 1

$$\begin{array}{cc}[\mathrm{LLM}]& e_o=\max {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}\left[{(u}_k,y_{\mathit{ko}},+,{\epsilon }_k^o\right)/({\sum }_i{\alpha }_{ik{p}^o}{v}_i{x}_{\mathit{io}})]\\ & s.t.{\sum }_{r\in R_k}({u_{r}y}_{\mathit{rj}}+{\epsilon }_k^o)/({\sum }_i{\alpha }_{\mathit{ikp}}{v}_i{x}_{\mathit{ij}})\le 1,j\in p,k\in L_p,\forall p\\ & {\sum }_{k\in L_p}{\alpha }_{\mathit{ikp}}=1,\forall i,p\\ & {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}=1\\ & {\alpha }_{\mathit{ikp}}\in [a_{\mathit{ikp}},b_{\mathit{ikp}}]\\ & v_i,u_r\ge 0,\forall i,k\\ & {\epsilon }_k^o,\mathrm{free}\end{array}$$ 2

Both CHIRZ and LLM are the conventional multiplier DEA-based models. The directional distance function aims to increase (desirable) outputs and decrease inputs in directions specified as follows:

$$\overrightarrow{D}\left(x,,,y,;,g\right)=\sup \{\beta :(x-\beta g_{\mathrm{x}},y+\beta g_{\mathrm{y}})\in \mathrm{\Omega }\}$$ 3

where the nonzero vector g = (-g_x, g_y) defines the directions in which inputs and outputs are projected.

When $\overrightarrow{D}\left(x,,,y,;,g\right)$=0, DMU j is technically efficient and no additional improvements in outputs and inputs are feasible.$\overrightarrow{D}\left(x,,,y,;,g\right)$>0 indicates technical inefficiency. There are various options for choosing the direction (g_x, g_y), as pointed out by Färe et al. [27]. Zofio et al. [28] choose a directional vector towards profit maximizing benchmarks, while Lee [29] modify the direction towards marginal profit maximization. Atkinson and Tsionas [30] use Bayesian methods with the first-order conditions for cost minimization and profit maximization to estimate the directional distance function. For the objective function of DDF, Chen [31] summarizes six types from the energy economics literature. Wang et al. [26] comprehensively review the selecting techniques of direction for the directional distance function. Equations (1) and (2) show both CHIRZ and LLM consume ${\alpha }_{\mathit{ikp}}{x}_{\mathit{ij}}$ produce outputs. We choose the direction as ($g_x$, $g_y$) = (${\alpha }_{ik{p}^o}{x}_o$, $y_o$) to compare our results with CHIRZ and LLM.

The directional distance function model for non-homogeneous DMUs under constant returns to scale (CRS) technology is presented as Eq. (4), and variable returns to scale (VRS) technology as Eq. (5) [32] (“o” for the observed DMU).

$$\begin{array}{cc}[\mathrm{DDF}-\mathrm{CRS}]& {\beta }_o=\max {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}{\beta }_{\mathit{ko}}\\ & s.t.{\sum }_i{\lambda }_j^k{{\alpha }_{\mathit{ikp}}x}_{\mathit{ij}}\le \left(1,-,{\beta }_{\mathit{ko}}\right){\alpha }_{ik{p}^o}{x}_{\mathit{io}},j\in p,k\in L_p,\forall i,p\\ & {\sum }_{r\in R_k}{\lambda }_j^k{y}_{\mathit{rj}}\ge \left(1,+,{\beta }_{\mathit{ko}}\right)y_{\mathit{ro}},j\in p,k\in L_p,\forall p\\ & {\lambda }_j^k\ge 0,j\in p,k\in L_p,\forall p\\ & {\sum }_{k\in L_p}{\alpha }_{\mathit{ikp}}=1,\forall i,p\\ & {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}=1\\ & {\alpha }_{\mathit{ikp}}\in [a_{\mathit{ikp}},b_{\mathit{ikp}}]\end{array}$$ 4

$$\begin{array}{cc}[\mathrm{DDF}-\mathrm{VRS}]& {\beta }_o=\max {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}{\beta }_{\mathit{ko}}\\ & s.t.{\sum }_i{\lambda }_j^k{{\alpha }_{\mathit{ikp}}x}_{\mathit{ij}}\le (1-{\beta }_{\mathit{ko}}){\alpha }_{ik{p}^o}{x}_{\mathit{io}},j\in p,k\in L_p,\forall i,p\\ & {\sum }_{r\in R_k}{\lambda }_j^k{y}_{\mathit{kj}}\ge \left(1,+,{\beta }_{\mathit{ko}}\right)y_{\mathit{ro}},j\in p,k\in L_p,\forall p\\ & {\lambda }_j^k\ge 0,j\in p,k\in L_p,\forall p\\ & {\sum }_{j\in p}{\lambda }_j^k=1,k\in L_p,\forall p\\ & {\sum }_{k\in L_p}{\alpha }_{\mathit{ikp}}=1,\forall i,p\\ & {\sum }_{k\in L_p}{\omega }_{\mathit{ko}}=1\\ & {\alpha }_{\mathit{ikp}}\in [a_{\mathit{ikp}},b_{\mathit{ikp}}]\end{array}$$ 5

For ease of description, we call Eqs. (4) and (5) nonhomogeneous DDF (NH-DDF). To distinguish ${\beta }_o$ under the different technologies, we call ${\beta }_o$ under the CRS technology ${\beta }_{\mathit{CRS}}$ and ${\beta }_o$ under the VRS technology ${\beta }_{\mathit{VRS}}$. Note that the proportions α_ikp are not the same in different sets for each DMU_j and the constraint variables a_ikp and b_ikp are used to impose upper and lower limits on the size of the α_ikp variables.

Assuming β_i denotes the inefficiency of a subunit, and 1-β_i denotes the efficiency of a subunit, 1-β_o denotes the efficiency of a DMU under evaluation, and ω_i denotes the weight to aggregate the efficiencies of the subunits. Since the β of the DDF models represents the inefficiency, there are two approaches to aggregate the inefficiencies of the subunits into DMUs’ efficiencies. One is aggregating the 1-β_i to be the efficiency of a DMU namely $1-{\beta }_o={\sum }_{i=1}^n{\omega }_i(1-{\beta }_i)$, the other is aggregating the inefficiency of subunits and then subtract from 1 namely $1-{\beta }_o=1-{\sum }_{i=1}^n({\omega }_i{\beta }_i)$. However, the two approaches are equivalent.

Proof

$1-{\beta }_o={\sum }_{i=1}^n{\omega }_i(1-{\beta }_i)$=${\sum }_{i=1}^n({\omega }_i-{\omega }_i{\beta }_i)$=${\sum }_{i=1}^n{\omega }_i-{\sum }_{i=1}^n{\omega }_i{\beta }_i$=$1-{\sum }_{i=1}^n({\omega }_i{\beta }_i)$ since ${\sum }_{i=1}^n{\omega }_i$=1.

Empirical analysis

Data and variables

A data set was obtained from the Hospital Authority Annual Report of Hong Kong for the fiscal year 2012–2013.1 We chose this data set because it contains one output that all hospitals provide (total patient days) but the other outputs vary in provision across the hospitals in the data set. Thus it is easily seen that these are nonhomogeneous hospitals for the outpatient activities. The outputs show considerable non-homogeneity in outpatients which provides a good test for our DDF models. Note that the available data does not include undesirable outputs which would be a valuable extension to our models.

Li et al. [14] also employed the data set, but they used total length of stay (LOS) rather than total inpatient days as the output. Numerous studies have used total inpatient days as a measure of hospital output rather than the number of cases treated because this is more medically homogeneous (more complex patients require longer hospital stays) [33–35]. O'Neill et al. used either total inpatient days or inpatient days disaggregated by medical and surgical as outputs categories for a list of studies [36]. The inpatient days are a patient starting when formally admitted to a hospital with a doctor’s order, whereas the length of stay (LOS) is used to judge hospital efficiency and to predict whether the hospital is making money or losing money on different diagnoses, and is defined as whether a patient is considered admitted to the hospital at midnight. LOS is primarily used for a given diagnosis-related group (DRG) [37] or case-based group [38], the smaller the better.

Hong Kong's hospitals display non-homogeneity in producing different outputs. There are 37 hospitals, all of which consume the same two kinds of inputs, i.e. number of full-time equivalent (FTE) staff (I₁) and number of beds (I₂), and produce at most six outputs, i.e. total inpatient days (O₁), total Accident & Emergency attendances (O₂), total Specialist Outpatient attendances (O₃), family medicine specialist clinic attendances (O₄), total allied health outpatient attendance (O₅), and general outpatient attendances (O₆). All DMUs have total inpatient days (O₁), which means they all provide inpatient services. Thirty-five hospitals (95%) have allied health outpatient attendances (O₅). Descriptive statistics of the inputs and outputs are summarized in Table 3, including the number in each output. The detailed information is reported in Table A1 of the Appendix. In column 7 of Table 3, CV represents the extent of variability in relation to the mean of the population. Although the CV of total accident & emergency attendances (O₂) is the lowest of these variables, the maximum of that is over 20 times the minimum. The same phenomenon occurred in other variables, indicating large differences in the operational scale in these hospitals, which provides the justification for the use of VRS technology.

Table 3.

Descriptive statistics of raw data

	No. of samples	min	max	Mean	Std. Dev	CV
Inputs
Number of FTEs (I1)	37	57	5,870	1,635	1,719	1.05
Number of beds (I2)	37	26	1,843	734	563	.77
Outputs
Total inpatient days (O1)	37	3,370	568,259	197,249	158,290	.80
Total accident & emergency attendances (O2)	16	10,975	228,871	140,832	48,784	.35
Total specialist outpatient attendances (O3)	32	21	690,407	214,502	231,603	1.08
Family medicine specialist clinic attendances (O4)	16	251	58,190	17,369	19,889	1.15
Total allied health outpatient attendances (O5)	35	125	223,020	64,007	70,462	1.10
General outpatient attendances (O6)	19	26,117	766,062	296,495	199,749	.67

As described in Algorithm A1 of the Appendix, the outputs can be divided into six maximal subunits, with the DMUs classified into eight subgroups as reported in Table 4. Each subunit has a vector for the inputs and the same outputs of that subunit. R_k is defined as the subunit k composed of DMU subgroup sets. Therefore, in this example, the six subunits and subgroup sets can be expressed as:

• R₁ = {p = 1, 2, 3, 4, 5, 6, 7, 8}; R₂ = {p = 2, 3, 8}; R₃ = {p = 2, 3, 4, 6, 7, 8};

• R₄ = {p = 2, 7, 8}; R₅ = {p = 1, 2, 3, 4, 6, 7, 8}; R₆ = {p = 2, 3, 4, 7}.

Table 4.

Classification of subgroups and subunits

Subunit	Inputs (I1, I2)	O1	O2	O3	O4	O5	O6
Subgroup
1	√	√				√
2	√	√	√	√	√	√	√
3	√	√	√	√		√	√
4	√	√		√		√	√
5	√	√
6	√	√		√		√
7	√	√		√	√	√	√
8	√	√	√	√	√	√

For example, subunit 1 comprises subgroups 1, 2, 3, 4, 5, 6, 7, and 8; subunit 2 comprises subgroups 2, 3, and 8. The rest is deduced by analogy.

In terms of the DMU subgroups consisting of subunits, the 37 DMUs can be divided into eight subgroups, with each DMU composed of inputs and subunits. Lp is defined as the subgroup p composed of subunit sets. Therefore, in this example, the eight subgroups composed of subunit sets can be expressed as:

• L₁ = {k = 1, 5}; L₂ = {k = 1, 2, 3, 4, 5, 6}; L₃ = {k = 1, 2, 3, 5, 6}; L₄ = {k = 1, 3, 5, 6};

• L₅ = {k = 1}; L₆ = {k = 1, 3, 5}; L₇ = {k = 1, 3, 4, 5, 6}; L₈ = {k = 1, 2, 3, 4, 5}.

Each subgroup has different subunits, e.g., p = 1 consists of subunits 1 and 5, and p = 6 consists of subunits 1, 3 and 5. Hospitals classified in the same subgroup have similar characteristics. For example, hospitals in subgroup p = 1 are sanatorium and rehabilitation hospitals, those in subgroup p = 2 are large acute general hospitals, those in subgroup p = 4 are community-based health care facilities, those in p = 6 are more specialist hospitals that focus on particular medical fields such as psychiatry, ophthalmology and so on.

Results and analysis

The proportion of each input assigned to each subunit is used to allocate input for each subunit. Both CHIRZ and LLM provide limits on input proportions. CHIRZ employs assurance regions (AR) while LLM uses an interval number [a, b] to limit the input splitting proportions of each subunit. For simplicity, we directly use Tables A2 and A3 of the Appendix to split input 1 (number of FTEs) and input 2 (number of beds) into each subunit, respectively.

Following the methodology in Sect. 3, models (4) and (5) are used to calculate DDF efficiencies under CRS and VRS respectively. The performance of each DMU is the aggregation of the inefficiency scores of subunits by taking a weighted average. The aggregation weights (ω) of subunits to DMUs are provided in Table A4 of Appendix.

The results of DDF-CRS (4) and DDF-VRS (5) are shown in Tables 5 and 6, respectively. Columns 3–8 report the inefficiencies (β) of subunits, columns 9–10 the overall inefficiencies (β) and overall efficiencies (1-β) of DMUs, and the last column shows the rankings of DMUs.

Table 5.

Efficiency scores of subunits and DMUs under the CRS technology

Subgroup	DMU	Subunit1	Subunit2	Subunit3	Subunit4	Subunit5	Subunit6	βCRS	1-βCRS	Rank
p = 1	1	.355				.994		.591	.409	25
	25	.471				.989		.684	.316	29
	29	.351				.970		.599	.401	26
p = 2	2	.020	.181	.074	.597	.283	.268	.381	.619	12
	3	0	0	.211	.585	.436	.179	.345	.655	8
	11	.119	.294	.704	.448	.200	.423	.500	.500	18
	16	0	.122	.115	.805	.690	.273	.476	.524	16
	18	0	0	.026	.969	0	0	.148	.852	2
	19	.066	.040	.098	.217	.437	.079	.251	.749	6
	20	.153	.563	.099	.941	.338	.122	.414	.586	15
	22	.073	.076	.134	.863	.664	.396	.486	.514	17
	24	.058	.673	.219	.614	.346	.372	.398	.602	13
	26	.033	.091	.114	.806	.124	.241	.186	.814	5
	27	.140	.003	0	.532	0	.023	.103	.897	1
	30	.004	.479	.176	.698	.063	.017	.357	.643	10
	31	.024	.141	.665	0	.066	.224	.378	.622	11
	37	.301	.460	.108	.494	.038	.425	.345	.655	9
p = 3	4	.678	.384	.993		.132	.241	.412	.588	14
p = 4	5	.598		.249		.434	.745	.525	.475	20
	12	.569		.682		.895	.766	.696	.304	30
	13	.388		.811		.620	.430	.528	.472	22
p = 5	6	.599						.599	.401	27
	36	.506						.506	.494	19
p = 6	7	.527		.554		.515		.525	.475	21
	8	.204		.994		.978		.770	.230	35
	9	.375		.841		.917		.726	.274	31
	10	.316		.985		.881		.745	.255	33
	14	.767		0		0		.176	.824	4
	15	.103		.741		.689		.582	.418	24
	17	.276		.911		.926		.741	.259	32
	21	.285		.518		.778		.549	.451	23
	28	.432		.999		.832		.771	.229	36
	32	.272		.996		.985		.787	.213	37
	33	.132		.999		.994		.755	.245	34
	34	.311		.736		.773		.599	.401	28
p = 7	23	.109		.231	.918	.053	.134	.306	.694	7
p = 8	35	.068	.074	.311	.190	.119		.158	.842	3

Table 6.

Efficiency scores of subunits and DMUs under the VRS technology

Subgroup	DMU	Subunit1	Subunit2	Subunit3	Subunit4	Subunit5	Subunit6	βVRS	1-βVRS	Rank
p = 1	1	.325				.796		.499	.501	25
	25	.429				.944		.640	.360	31
	29	.331				.844		.536	.464	28
p = 2	2	0	.100	0	0	.255	.212	.057	.943	10
	3	0	0	.191	.387	.363	.146	.259	.741	16
	11	.118	.261	0	.338	.166	.373	.125	.875	14
	16	0	0	0	.563	0	.129	.069	.931	11
	18	0	0	.024	0	0	0	.002	.998	3
	19	.066	0	.008	0	.012	0	.012	.988	4
	20	.137	.424	.049	.324	.338	.120	.285	.715	18
	22	.064	.016	.081	.430	.382	.395	.290	.710	19
	24	.057	.371	.154	.411	.287	.257	.254	.746	15
	26	.024	.036	.112	.145	.122	.219	.091	.909	12
	27	.134	0	0	0	0	.010	.014	.986	5
	30	.004	.443	.172	.374	.059	.010	.277	.723	17
	31	.023	.100	.038	0	.033	.158	.051	.949	9
	37	0	0	.005	.382	0	0	.039	.961	6
p = 3	4	0	0	0		0	0	0	1	1
p = 4	5	.587		.219		.398	.518	.462	.538	22
	12	.562		.638		.872	.697	.666	.334	33
	13	.372		.672		.589	.312	.464	.536	23
p = 5	6	.573						.573	.427	30
	36	.457						.457	.543	21
p = 6	7	.363		.364		.504		.448	.552	20
	8	.153		.872		.742		.645	.355	32
	9	.361		.812		.862		.696	.304	34
	10	.164		.570		.709		.527	.473	27
	14	.196		0		0		.045	.955	8
	15	0		.731		.558		.481	.519	24
	17	.272		.832		.880		.699	.301	35
	21	.207		.490		.773		.516	.484	26
	28	0		0		0		0	1	1
	32	.237		.915		.923		.729	.271	37
	33	.066		.964		.923		.704	.296	36
	34	.220		.733		.766		.565	.435	29
p = 7	23	.013		.161	0	.027	.043	.041	.959	7
p = 8	35	.058	0	.275	.132	.102		.108	.892	13

In Table 5 for the CRS technology, the first ranked is DMU₂₇ of subgroup 2, the second ranked is DMU₁₈ of subgroup 2, and the last ranked is DMU₃₂ of subgroup 6. Although no DMU is completely efficient under the CRS technology, the more subunits are closer to efficient, the more efficient the overall DMU is, e.g., DMU₁₈ and DMU₂₇. DMU₁₈ is efficient in subunits 1, 2, 5 and 6, and almost efficient in subunit 3, but inefficient in subunit 4 (β = 0.969), so DMU₁₈ is inefficient overall. Thus, a DMU is efficient overall if and only if all of its subunits are efficient. This suggests that hospital management may need to consider actions to improve the performance of its inefficient subunits. In Table 6 for the VRS technology, two DMUs are efficient overall, namely DMU₄ and DMU₂₈. All subunits of these two DMUs are efficient. For example, DMU₄ is efficient overall, so it performs efficiently in all of its subunits (k = 1, 2, 3, 5 and 6).

Subunits and subgroups contain different DMU sets respectively, and it is worthwhile investigating any correlations between them. As DEA is a non-parametric technique, the Spearman rank correlation coefficient may be more appropriate than others such as the Pearson correlation coefficient to measure the correlation among efficiencies. Since three subgroups (p = 3, 7 and 8) contain only one DMU, their Spearman correlation coefficients are not applicable. Table 7 presents the Spearman rank correlation coefficients of the other five subgroups. The upper and lower parts of Table 7 show the Spearman rank correlations between the subunit efficiencies and the overall efficiencies of DMUs for the CRS and VRS technologies, respectively.

Table 7.

Correlation coefficients of inefficiencies between subunits and subgroups under the CRS and VRS technologies

		Subunit k = 1	k = 2	k = 3	k = 4	k = 5	k = 6
DDF-CRS	Subgroup p = 1	.500				-.500
	p = 2	.139	.559*	.574*	.068	.572*	.640*
	p = 4	-.500		.500		1.000**	.500
	p = 5	1.000**
	p = 6	-.399		.937**		.825**
DDF-VRS	p = 1	1.000**				1.000**
	p = 2	.232	.611*	.573*	.645*	.818**	.576*
	p = 4	-.500		.500		1.000**	.500
	p = 5	1.000**
	p = 6	.291		.949**		.942**

^*The correlation is significant at .05 level (two-tailed); **the correlation is significant at .01 level (two-tailed)

The Spearman correlation coefficients between subunit 5 and subgroup 1 are -0.500 and 1.000 in CRS and VRS, respectively; the correlation coefficients between subunit 1 and subgroup 6 are -0.399 and 0.291 in CRS and VRS, respectively, indicating that the correlation of efficiencies between the subgroups and subunits are different under the CRS and VRS technologies. The descriptive statistics in Table 4 show that the DMU scales are largely different, so the VRS technology may be more reasonable in this context. The following analysis is primarily based on VRS technology.

The correlation coefficients can provide guidance for hospital management decision makers to improve efficiency. For example, both in the CRS and VRS technologies, the correlations between subunit 1 and subgroup 4 are -0.500. The output of subunit 1 is the total inpatient days. The increased efficiency of subunit 1 in group 4 means that more resources are allocated to subunit 1, such as increased beds or total hospital stay. This is inconsistent with the service objectives and functional orientation of subgroup 4 hospitals, which are community-based health care facilities providing routine health care. Therefore, the community-based characteristics of subgroup 4 may explain why the efficiency of subunit 1 is negatively correlated with its overall efficiency.

When only one subunit is inefficient, an effective strategy is to improve the performance of the inefficient subunit. For example, DMU₃₇ is efficient in its subunits 1, 2, 5 and 6, and nearly efficient in subunit 3 (β_VRS = 0.005), but it is more inefficient in subunit 4 (β_VRS = 0.382) under the VRS technology. In this case, DMU₃₇ can try to improve its performance in subunit 4 to become more efficient. Alternatively, if DMU₃₇ as an acute general hospital has shortcomings in providing family medicine specialist clinic services, it could trade these off with other services.

Where two or more subunits have similar inefficiencies, an effective strategy might be to first improve the performance of subunits that are highly correlated with the overall efficiency of the DMU. For example, DMU₂ is efficient in subunits 1, 3 and 4 and is inefficient in subunits 2, 5 and 6 under the VRS technology. Moreover, subunits 5 and 6 of DMU₂ have similar inefficiency (0.255 vs. 0.212), but the correlation coefficient of inefficiency between the overall system and subunit 5 is higher than that of subunit 6 (0.829 vs. 0.563). Therefore, DMU₂ can prioritise the performance of subunit 5 to improve its performance of the overall system by, say, allocating more resources to supply allied health outpatient services.

If the correlation coefficients of efficiencies between the subunits and the overall system are similar, the subunit with high inefficiency should be prioritised to improve. For example, DMU₁₆ is efficient in subunits1, 2, 3 and 5, and inefficient in subunit 4 and 6 with similar correlation coefficients (0.573 vs. 0.576) under the VRS technology, DMU₁₆ can firstly improve the performance of subunit 4, which is more inefficient than subunit 6 (0.563 vs. 0.129), by allocating more resources to the provision of family medicine specialist clinic services.

Comparisons of the four models

The overall efficiency of each DMU for the four models DDF-CRS, CHIRZ, DDF-VRS and LLM are reported in columns 2–5 of Table 8, respectively. The DMU efficiencies of DDF-CRS, DDF-VRS, CHIRZ and LLM models are represented by 1-β_CRS, 1-β_VRS, eCHIRZ and eLLM respectively. It can be seen that the efficiencies of DDF-CRS are very close to CHIRZ, and those of DDF-VRS are very close to LLM. Under the CRS technology, there is no DMU completely efficient for CHIRZ or DDF-CRS; under the VRS technology, two DMUs (namely DMUs 4 and 28) are efficient in both DDF-VRS and LLM and the rest are inefficient.

Table 8.

Efficiency scores of DMUs in the four models

Subgroup	DMU	1-βCRS	eCHIRZ	1-βVRS	eLLM
p = 1	1	.409	.301	.501	.410
	25	.316	.214	.360	.233
	29	.401	.294	.464	.379
p = 2	2	.619	.491	.943	.901
	3	.655	.531	.741	.631
	11	.500	.366	.875	.811
	16	.524	.417	.931	.907
	18	.852	.847	.998	.996
	19	.749	.628	.988	.831
	20	.586	.472	.715	.573
	22	.514	.400	.710	.479
	24	.602	.486	.746	.559
	26	.814	.726	.909	.825
	27	.897	.867	.986	.987
	30	.643	.531	.723	.628
	31	.622	.522	.949	.571
	37	.655	.508	.961	.954
p = 3	4	.588	.456	1	1
p = 4	5	.475	.329	.538	.429
	12	.304	.185	.334	.232
	13	.472	.324	.536	.413
p = 5	6	.401	.251	.427	.295
	36	.494	.328	.543	.333
p = 6	7	.475	.311	.552	.419
	8	.230	.189	.355	.324
	9	.274	.181	.304	.236
	10	.255	.179	.473	.440
	14	.824	.800	.955	.955
	15	.418	.303	.519	.378
	17	.259	.188	.301	.246
	21	.451	.313	.484	.341
	28	.229	.152	1	1
	32	.213	.165	.271	.213
	33	.245	.216	.296	.279
	34	.401	.274	.435	.312
p = 7	23	.694	.609	.959	.920
p = 8	35	.842	.733	.892	.846
Mean		.511	.408	.667	.575
S. D		.198	.204	.257	.279

Figure 2 shows the mean efficiency curves of subgroups for CHIRZ, LLM, DDF-CRS, and DDF-VRS. The efficiency curve of DDF-CRS (1-β_CRS) is close to CHIRZ while that of DDF-VRS (1-β_VRS) is close to LLM, respectively.

Fig. 2.

The average efficiencies of the subgroups

Figure 3 plots the curves of the efficiencies and the number of subunits for the four models. When the number of subunits is one (i.e., producing one output), efficiency is the lowest, with efficiency increasing with increases in the number of subunits; however, DDF-CRS declines when the number of subunits increases from two to three. When the number of subunits is five, the efficiency reaches a maximum and then decreases.

Fig. 3.

Efficiency vs. number of subunits

Figure 4 pairwise compares the efficiency scores between non-DDF and DDF under the same technology, respectively. In Fig. 4(a), the horizontal axis is the efficiency from CHIRZ while the vertical axis is the efficiency from DDF-CRS (1-β_CRS), and the diagonal represents the case of equality between the two; in Fig. 4(b), the horizontal axis is the efficiency from LLM while the vertical axis is DDF-VRS (1-β_VRS). In Fig. 4(a) the majority of points are on or above the diagonal, indicating that the majority of DDF-CRS (1-β_CRS) efficiencies are higher than the CHIRZ efficiencies under the CRS technology. A similar pattern occurs in Fig. 4(b), indicating all efficiencies of DDF-VRS (1-β_VRS) are not lower than LLM.

Fig. 4.

Pairwise comparison of non-DDF and DDF under the same technology. (a) Between CHIRZ and DDF-CRS under CRS (b) Between LLM and DDF-VRS under VRS

CHIRZ and DDF-CRS are CRS technologies while LLM and DDF-VRS are VRS technologies. For completeness, Fig. 5 compares the efficiencies between CRS and VRS for non-DDF and DDF models. In Fig. 5(a), the horizontal axis is the efficiency of CHIRZ (CRS) while the vertical axis is the efficiency of LLM (VRS); in Fig. 5(b), the horizontal axis is the efficiency of DDF-CRS (1-β_CRS) while the vertical axis is the efficiency of DDF-VRS (1-β_VRS). In both graphs, the majority of points lie on or above the diagonal indicating that all four models (CHIRZ, LLM and NH-DDF models) provide results consistent with the assumed technologies.

Fig. 5.

Pairwise comparison between CRS and VRS for non-DDF and DDF models. (a) Between CHIRZ and LLM in non-DDF models (b) Between DDF-CRS and DDF-VRS in DDF models

Distribution analysis

The distributional analysis estimates the densities of the efficiency scores and their components non-parametrically, using kernel-based methods. The results are depicted in Fig. 6. The black, blue, green and red stand for the efficiencies of 1-β_CRS, CHIRZ, 1-β_VRS and LLM, respectively. Here, the gaussian kernel function and bandwidth selected are done via the Silverman rule of thumb [39]. As can be seen from the graph, the distribution of DDF-CRS (1-β_CRS) is close to that of eCHIRZ while DDF-VRS (1-β_VRS) is close to eLLM. Both of DDF-CRS and CHIRZ have one peak, while both DDF-VRS and LLM have two peaks.

Fig. 6.

Estimated densities of efficiency for the four models

We use a more comprehensive comparison for the different models using a test of equality of distributions, from Li [40, 41] and the version adapted to DEA-based efficiency analysis by Simar and Zelenyuk [42]. Li [40] has established that this test statistic is valid for both dependent and independent variables. Fan and Ullah [43] show that the test statistic asymptotically goes to the standard normal, but our study contains only 37 observations. Thus, we do a bootstrap approximation with 1000 replications to find the critical values for the statistic significance.

At the significance level α = 5%, there are statistically significant differences between the four models with respect to the technologies assumed, but not for DDF-CRS (1-β_CRS) and eCHIRZ under the CRS technology and DDF-VRS (1-β_VRS) and eLLM under the VRS technology, as shown in Table 9. In other words, DDF-CRS can be an alternative model for CHIRZ, and DDF-VRS can be an alternative model for LLM, but the DDF model is a general form and the directions align with an input-oriented model .

Table 9.

Li-test results for equality of distributions between models

H0 (f is density)	Test statistics	Bootstrap p-value	Decision
f(1-βCRS) = f(eCHRIZ)	1.190	.080	Do Not Reject H0
f(1-βCRS) = f(1-βVRS)	2.895	.010	Reject H0
f(1-βCRS) = f(LLM)	1.961	.027	Reject H0
f(eCHRIZ) = f(1-βVRS)	4.872	< .001	Reject H0
f(eCHRIZ) = f(LLM)	1.881	.024	Reject H0
f(1-βVRS) = f(LLM)	.700	.257	Do Not Reject H0

Discussion and conclusions

Cook et al. [9] and Li et al. [14] proposed DEA models to evaluate nonhomogeneous DMUs by investigating different output efficiencies with the same kind of inputs. This idea of dividing inputs across multiple subsets of outputs is essentially similar to the approach described by Cook and Zhu [44] for uncovering multiple variable proportionality (MVP). The proposed models in this paper are derived from Cook et al. [9] and Li et al. [14], and provide the CRS and VRS versions of the DDF model for the performance measurement of nonhomogeneous DMUs. The DDF-CRS results are close to that of Cook et al. [9], although the majority of the former’s efficiencies are slightly higher than the latter. The results of the DDF-VRS are close to Li et al. [14], and again the former efficiencies are no less than the latter.

The empirical application provides some insights for management: (i) if the efficiencies of the subunits within the DMU are similar, the subunit that is highly correlated to the DMU should be improved first; (ii) if the correlations are similar, the subunits with lower efficiency should be improved first. The proposed models can help managers choose the direction of performance improvement, and also determine the priority of performance improvement.

Non-homogeneity is a problem especially in the current COVID-19 situation, where hospitals have different strategies to reduce infection. For example, to manage COVID-19 in China, most hospitals have special fever clinics, some hospitals take care of COVID-19 patients, some hospitals chiefly take care of maternity, and some hospitals take care of elderly patients. These hospitals have different outputs, and the level of non-homogeneity suggests that conventional DEA models may not be appropriate to assess healthcare service efficiency. Our models have several advantages. First, they provide the more general DDF models which can easily accommodate the conventional DEA input / output orientation models. Second, they allow managers to readily choose the direction and priority for improvement in the light of managers’ preferences.

There are a number of possible avenues for future research. Although this paper analyses the correlation between subunits’ efficiencies and hospitals’ efficiencies to determine the direction and the priority of efficiency improvement for performance advances, how much improvement in the directions to reach the frontier can be further explored. The outputs in the study were assumed to be desirable but they could be extended to include undesirable outputs [45, 46] in the health care sector, such as mortality and infection rates.

Supplementary Information

Below is the link to the electronic supplementary material.

Funding

This research was funded by the National Natural Science Foundation of China (Grant No. 72171052), the National Social Science Fund of China (Grant No. 20CFX038) and the Natural Science Foundation of Fujian, China (Grant No. 2020J01936).

Data Availability

The data are available in the electronic supplementary material.

Declarations

Ethics statement

This research does not require ethical approval as it relies on the secondary use of anonymous information and does not use any human/animal subjects.

Conflict of interest

The authors declare that they have no conflict of interest.