There are two approaches to measure the efficiency according to the relevant literature, parametric and non-parametric. Non-parametric approaches are most commonly used in healthcare studies, since they have the advantage that the functional form need not be known [6]. The most commonly used non-parametric approach is Data Envelopment Analysis (DEA) [7]. Moreover, DEA has the advantage of handling multiple inputs and outputs, as well as used with any input-output measurement. In studies measuring Healthcare efficiency, DEA is the most used method [8].
Farrell (1957), based on Debreu (1951) and Koopmans (1951), first introduced modern efficiency, and attempted to measure the efficiency of a firm by considering multiple inputs [9,10]. Farrell analyzed and decomposed the efficiency measured, into two components, technical and allocative efficiency, both of which together appraise economic efficiency [11].
Charnes, Cooper and Rhodes (CCR, 1978), based on Farrell, proposed the model of DEA. DEA is a linear programming method that constructs a non-parametric frontier containing all the firms submitted for analysis in order to measure their efficiencies. The firms are called Decision Making Units (DMU’s) and their data, translated as inputs and outputs, are used to measure their efficiencies. The constructed frontier includes all efficient DMU’s while below the frontier all inefficient ones are placed. Technical efficiency depends on the “input-output ratio of productivity” [12] and can be decomposed into pure technical efficiency and scale efficiency. Essentially, technical efficiency refers to the conversion of inputs into outputs according to best practice so that the DMU is as efficient as possible. “Pure technical efficiency and scale efficiency comprise technical efficiency” [2].
Technical efficiency measured by DEA has two orientations according to the relevant literature. The output orientation refers “to the maximum amount of outputs that can be produced by the DMU’s for a given amount of inputs used” [6,13], while the input orientation refers “to the minimum amounts of inputs used by DMU’s in order to obtain a certain level of outputs” [6,13]. The input orientation of DEA binds the outputs produced in order to solve a linear programming equation that minimizes the inputs used, while the output orientation of DEA binds the inputs used to solve a linear programming equation that maximizes the outputs produced.
In addition, two methods of DEA have been proposed, the first one is based on the assumption of constant return to scale (CRS) as was introduced by Charnes, Cooper and Rhodes (CCR, 1978), while the other one is based on the assumption of variable return to scale (VRS) as later introduced by Banker, Charnes and Cooper at 1984. “The CRS method is applied when all DMU’s are operating at an optimal level while, under imperfect competition, the VRS method is applied considering that not all DMU’s are operating at an optimal level, assuming that there are scale efficiencies” [14, 42].
Either CRS or VRS DEA is able to measure the technical efficiencies of the DMU’s included in the analysis but without the availability to include panel data, taking into consideration the impact of time upon the efficiencies. To overcome this limitation, the Malmquist Productivity Index (MPI) can be used along with the DEA. The MPI measures productivity change over time and decomposes it into change due to technology and technical efficiency [15].
Productivity change was initially explained as technology change but later it was accepted that in productivity change are included both, technology and technical efficiency change. MPI was first introduced by Caves et al. (1982), who relied on Shephard’s (1970) distance function, in order to measure the productivity change [15].
The MPI DEA is essentially a non-parametric mathematical programming approach, which according to the literature is the most widely used method to include panel data in the analysis and calculate the indices of total factor productivity, technical efficiency and its components (pure technical efficiency and scale efficiency) and technological change, over time [2]
Productivity is defined as the ratio of an index of outputs over an index of inputs used to produce them [16-18]. Increasing productivity means that more outputs are obtained from the same amount of inputs or less inputs are required to produce the same amount of outputs. The change of productivity over time is called productivity change and shifts over time due to technical efficiency and technical/technological change [2,11,19-21].
In this paper the input-oriented MPI DEA is used. First input-oriented because in the health sector it is impossible to predefine the outputs, but instead the inputs can be predefined and controlled. Secondly, MPI DEA, because panel data are included in the paper [22-24].
Model Specification - Data Envelopment Analysis and Malmquist Productivity Index
The mathematical concepts of DEA and MPI are briefly analyzed below, since the aim of this paper is to evaluate the productivity and efficiency of the 155 Health Centers in Greece during 2016-2018. Extended mathematical analysis of the methods of DEA and MPI and how they are used is presented in the relevant literature.
Before presenting the mathematical background of MPI, DEA will be analyzed. The mathematical analysis refers to an input-oriented DEA under CRS and VRS assumption, since both are needed to estimate MPI.
In the input-oriented mathematical model of CRS, it is assumed that there are “N DMU’s that use K inputs to produce M outputs. Under this assumption there are two matrices, K*N input matrix, referred as X, and M*N output matrix, referred as Y, which both represent the data of all N DMU’s” [25]. In order to measure the efficiency of the DMU’s the literature considers the calculation of the ratio of all outputs over all inputs. T.J. Coelli presented the following mathematical linear programming problem in 1996:
“maxu,v (u’yi/v’xi),
s.t.
u’yj/v’xj≤1, j=1,2,…..,N,
u,v≥0,
Where u is an M*1 vector of output weights and v is a K*1 vector of input weights” [25]. The problem aims to compute the values of u and v, maximizing the efficiency of the DMU’s. As it can be observed there is a constraint indicating that all efficiency measures must lie within the closed interval of (0,1).
To avoid the infinite solutions of the above mathematical formula, “a new constraint, v’xi=1” [25], has been introduced:
“maxμ,ν (μ’yi),
s.t.
v’xi=1,
μ’yj-v’xj≤0, j=1,2,….,N,
μ, ν≥0.
A notation change from u,v to μ,ν converts the first mathematical linear programming problem into a multiplier form” [25].
By applying duality in linear programming, an equivalent form is developed:
“minθ,λ θ,
s.t.
-yi+Yλ≥0,
θxi-Xλ≥0,
λ≥0.
The symbol θ is a scalar and λ is an N*1 vector of constants” [25]. The above model has fewer constraints and is easier to apply. The symbol θ represents the efficiency of the DMU’s and their values are within closed interval of (0,1). Values of 1 mean that the DMU’s operate at an optimal level of efficiency, while values less than 1 mean inefficiencies. The mathematical function has to be solved N times for each DMU [25].
The CRS model is based on the assumption that all DMU’s are operating at an optimal scale. In contrast, VRS model overpasses this assumption considering that there might be scale efficiencies. By adding one more constraint to the CRS model, scale efficiency effects are calculated and technical efficiency is decomposed into pure technical efficiency and scale efficiency for each DMU [42]. The mathematical function transforms as follows:
“minθ,λ θ,
s.t.
-yi+Yλ≥0,
θxi-Xλ≥0,
Ν1’λ=1
λ≥0.
N1 represents an N*1 vector of ones” [25].
The MPI is an extended application of DEA, to measure productivity change over time for each DMU, and analyze it into change owing to technical efficiency and change owing to technology [25].
MPI DEA is used for panel data and there is no need to choose between CRS or VRS approach, since they give the same results. This is because in estimating the MPI DEA, both the CRS and the VRS approaches are used to calculate the various distances that construct the Malmquist Indices [26]. The distances mentioned are essentially the technical efficiencies of each DMU for each period included in the analysis (from t to t+1). Assuming that the MPI is measured at a given period (t), the distances calculated are:
- “previous period (t-1) CRS DEA frontier
- currents period (t) CRS DEA frontier
- next periods (t+1) CRS DEA frontier
- currents period (t) VRS frontier”
[25,27]
There are two orientations for the MPI DEA method, input and output orientations. In input orientation the production is described by calculating the minimal proportional decrease of the input vector, given the output vector, while in the output orientation the production is described by calculating the maximal proportion increase of the output vector, given the input vector [27].
In this paper, panel data for three years (2016-2017-2018) are considered, forming two periods (2016-2017 and 2017-2018). The calculation of MPI involves the estimation of four distances for each of the two periods. DEA analysis by CRS and VRS assumption is performed for each of the three years (2016, 2017 and 2018) measuring the efficiencies of the DMU’s.
By applying the MPI DEA method (DEAP program) “five indices are calculated for each DMU for the two periods:
- Technical efficiency change (relative to CRS technology) - effch
- Technological change - techch
- Pure technical efficiency change (relative to VRS technology) - pech
- Scale efficiency change - sech
- Total factor productivity change – tfpch”
[25]
All the indices are relative to the previous year, which explains the fact that although data are available for the years 2016, Malmquist Indices cannot be estimated as no data are available for the year 2015.
The Malmquist Index was first introduced in 1970 with Shephard’s distance function and has since been widely used in many areas were efficiency needed to be measured with panel data. Färe specified an output-oriented MPI in 1994 [28]. In this paper, the input-oriented MPI is used. Essentially the MPI index of one period, is the geometric mean of two Malmquist Indices calculated for year t and year t+1.
“Mt (yt+1, xt+1, yt, xt )= dt (xt+1, yt+1)/ dt (xt, yt), Mt+1= dt+1 (xt+1, yt+1)/ dt+1 (xt, yt)”
[19,25,28,29].
By computing the geometric mean of the above individual Malmquist Indices, the MPI for one period (from t to t+1) takes the following form:
“M (yt+1, xt+1, yt, xt) = { [ dt (xt+1, yt+1)/ dt (xt, yt)] * [dt+1 (xt+1, yt+1)/ dt+1 (xt, yt)] }1/2”
[19,25,28,29].
The MPI for a period (t to t+1), represents the productivity at the production point (xt+1, yt+1) relative to the production point (xt, yt) for each DMU.
In addition, reshaping the index can the changes due to technical efficiency and technology be explained:
“M (yt+1, xt+1, yt, xt) = {[dt+1 (xt+1, yt+1)/dt (xt, yt)]* [(dt (xt, yt)/dt+1 (xt, yt))*(dt (xt+1, yt+1)/dt+1 (xt+1, yt+1))]1/2}”
[19,25,28,29].
The first fraction of the equation represents technical efficiency change, while the second one technology change, for the period (t to t+1). Moreover, technical efficiency change can be further analyzed into change due to pure technical efficiency and scale inefficiency. Positive total productivity growth from time t to time t+1 means a value greater than one for the index [19,25,28,29].
In order to calculate the MPI equation, the four distances must be calculated by linear programming methods as presented below:
- [dt (xt, yt)]-1 = minθ,λ θ,
s.t. -yit+Ytλ≥0, θxit–Xtλ≥0, λ≥0
- [dt+1 (xt+1, yt+1)]-1 = minθ,λ θ
s.t. -yi,t+1+Yt+1λ≥0, θxi,t+1-Xt+1λ≥0, λ≥0
- [dt (xt+1, yt+1)]-1 = minθ,λ θ,
s.t. -yi,t+1+Ytλ≥0, θxi,t+1-Xtλ≥0, λ≥0
- [dt+1 (xt, yt)]-1 = minθ,λ θ
s.t. -yi,t+Yt+1λ≥0, θxi,t-Xt+1λ≥0, λ≥0
Note that in this paper there must be calculated N(3T-2) linear programming equations [25,29]. Taking into consideration the 155 Health Centers and the 2 time periods that were included into the analysis, 620 Linear Programming equations need to be calculated.
Data
Efforts were made to include all Health Centers of Greece in the analysis of this paper, but due to lack of data, only 155 Health Centers were finally included. Therefore, 52 Health Centers were excluded to avoid random estimation and possibility of bias.
The sample used for the analysis of this paper is homogenous, as it includes the majority of Health Centers of Greece (74,87% of the total), distributed across the seven Health Regions of Greece. The 155 Health Centers use the same categories of inputs generating the same categories of outputs, differing only in the quantities been used. This ensures comparability and validates this paper to measure their productivities and efficiencies for the years 2016, 2017, 2018 with DEA. Furthermore, according to the literature, the requirements for conducting MPI DEA are satisfied, ensuring meaningful results. These requirements include that at least one DMU in the sample consumes and produces each input and output and that each DMU in the sample consumes at least one input and produces at least one output [30,31]. By including the majority of Health Centers in Greece discriminatory power between the efficient and inefficient units is also achieved [32,33].
In the analysis of this paper, 12 outputs were included to measure the productivity change and the change in technical efficiency and technology of each Health Center. The outputs represent the total Health Care services provided by each Health Center:
- Total number of “Nursing Operations” applied – Output1
- Total Number of “Microsurgeries” applied – Output2
- Total Number of “Dental Procedures” applied – Output3
- Total number of “Chronic disease cases” faced – Output4
- Total number of “Emergencies” faced – Output5
- Total Number of “Regular Incidents” faced – Output6
- Total Number of “Urgent Incidents” faced – Output7
- Total Number of “Transcriptions” given – Output8
- Total Number of “Bio pathological and Laboratory exams” applied – Output9
- Total Number of “Test Mantoux” applied – Output10
- Total Number of “Vaccinations for adults” applied – Output11
- Total Number of “Vaccinations for kids and teenagers” applied – Output12
In contrast, 4 inputs were used, representing the total staff employed and occupied in the Health Centers:
- Total “Number of Managers” employed – Input1
- Total “Number of Doctors” employed – Input2
- Total “Number of Nursing Staff” employed – Input3
- Total “Number of non-medical staff” employed – Input4
All inputs and outputs used for this paper are for the years 2016, 2017 and 2018. In Additional File 1 a table is presented, which shows the descriptive statistics of all inputs and outputs used to evaluate the total productivity and efficiency of each of the 155 Health Centers included in the analysis of this paper. The descriptive statistics show the minimum, maximum and mean values, as well as the standard deviation of each input and output included in the analysis.