Stability analysis for DEA models: an empirical example.

Alshare, Khaled ; Whiteside, Mary M.

ABSTRACT

This paper provides a framework for finding stable Data Envelopment Analysis (DEA) efficiency classifications. The approach is innovative in (1) assigning DEA efficiency classifications of efficient (E), inefficient (IE), or unstable (US) based upon cluster analysis of sensitivity scores and (2) comparing different analytical approaches, such as the Cobb-Douglas Production Function and ratio analysis, to DEA. A case study with 36 Jordanian hospitals illustrates the methodology. The DEA hospital classifications are robust with respect to input-output variable selections and surprisingly stable across reference sets for different years. However, differences between constantreturn-to-scale (CCR) and variable-return-to-scale (BCC) DEA models suggest the importance of initial model formulation. Cobb-Douglas efficiency shows considerable agreement with DEA in this case. Ratio analysis has a bias for finding units efficient relative to the other methods.

INTRODUCTION

The purposes of this study are to layout and illustrate a methodology to find stable efficiency classifications. Efficiency literature shows that classification of decision-making units (DMU's) strongly depends on the method used. In particular, the attempt to determine the relative efficiency of various types of hospital is confusing, if not contradictory. For example, traditional ratio analysis has shown that private hospitals are more efficient than their public counterparts. On the other hand, using DEA Valdmanis (1990) reported that public hospitals appeared to be more efficient than private non-profit hospitals. White and Ozcan (1996) again using DEA found that religious hospitals were more efficient than secular (public) nonprofit hospitals. These contradictory outcomes may result simply from the use of analytic procedures that lead to unstable efficiency classifications. Thus, assurance of stable classifications is crucial for drawing credible policy implications from the application of DEA or other efficiency classification analyses.

This paper recommends a novel way to avoid such vulnerable conclusions by aggregating results from several efficiency analyses to achieve a robust efficiency classification. The methodology includes fitting multiple DEA models and comparing to outcomes from other efficiency classification procedures. Central to finding stable DEA classifications are sensitivity scores, which quantify the sensitivity of DEA classifications to changes in relative input-output variable values for particular reference sets and DEA models. Sensitivity scores can be used to cluster DMU's into groups that are efficient (E), inefficient (IE), or unstable (US). Thus, by implication, groups E and IE are stable with respect to DEA efficiency. Similarly, the methodology employs quantitative outcome measures from other efficiency analyses to cluster DMU's into stable or unstable groups. Finally, the disagreements in classification among different analytic procedures are tested for significance using Cochran's test for categorical outcomes and a rank based nonparametric analysis of variance for dependent sample design.

This paper unfolds as follows. The next section provides a background in related research. The subsequent sections layout the methodology and apply the procedure to Jordanian hospitals. Conclusions are discussed at the end of the paper.

METHODS OF EFFICIENCY CLASSIFICATION

Data Envelopment Analysis

DEA is a nonparametric linear programming procedure for determining relative efficiency of DMU's with multiple inputs and multiple outputs. Charnes, Cooper and Rhodes (1978) introduced data envelopment analysis with the CCR model, appropriate for constant-return-to-scale. Banker, Charnes and Cooper (1984) modified the original formulation to create the BCC model, appropriate for variable-return-to-scale. A DEA input-oriented formulation assigns an efficiency score less than one for inefficient units, meaning that a linear combination of other organizations in the sample could produce the same vector of outputs, using a vector of smaller inputs, or a vector of greater outputs, using the same vector of inputs.

The outcome of a DEA analysis provides useful information on how the inputs and the outputs can be adjusted in order to transform inefficient DMU's into efficient ones (Shafere and Bradford, 1995), but no measure of the relative vulnerability of efficient units. However, sensitivity analysis (Charnes et al., 1992, Charnes, Rousseau, and Semple, 1996, Seiford and Zhu, 1998) provides information about the relative stability of the classification, even for efficient units. Thus, meaningful rankings can result among all DMUs. For further results on stability of DEA efficiency, see (Anderson and Petersen, 1993, Zhu, 1996, Seiford and Zhu, 1998). Relevant early applications of DEA and recent applications of sensitivity analysis include (Aida et al., 1998, Boljuncic, 1998, Ozcan, 1992, Read and Thanassoulis, 2000, Rousseau and Semple, 1995, Valdmanis, 1992, Zhu, 2001). An extensive review on sensitivity analysis can be found in Cooper et al., (2001).

Cobb-Douglas Production Function

One of the best-known relations in economics is the relation between a single output and multiple inputs known as the production function. Many industries where output normally increases with input use the Cobb-Douglas model to estimate production and cost functions. The function determines the maximum producible output from the amount of input used in the production process (Allison and English, 1993). Cobb and Douglas (1928) formulated the original model:

Pi = b[L.sup.k.sub.i] [C.sup.1-k.sub.i] [[epsilon].sub.i] (1)

where, [P.sub.i] = Quantity of output,

[L.sub.i] = Quantity of labor input,

[C.sub.i] = Quantity of capital input,

[[epsilon].sub.i] = Error term

b, k = Parameters to be estimated, and

i = [1, ... n], where n is the number of units.

For a given reference set, the values of b and k can be estimated using the least squares method or the maximum likelihood procedure, among others. In the illustration that follows, we use linear programming to:

Minimize [summation] (ln[P.sub.i] - ln[Q.sub.i]) (2)

such that

(ln[P.sub.i] - ln[Q.sub.i]) [greater than or equal to] 0 (3)

Where [Q.sub.i] is observed output to estimate the parameters of the production function (Giokas, 1991).

Efficiency scores are the ratio of the actual production to optimal production predicted by the model. Units with efficiency scores of one are technically efficient.

Efficiency Score [E.sub.i] = [Q.sub.i]/[P.sub.i] (4)

Ratio Analysis

Ratio analysis represents relationships between different specific input and output variables. Each ratio relates one output to one input. There are two main uses of ratio analysis. First is the traditional normative use whereby ratios are compared with a pre-set standard. The second is the use of ratio analysis for forecasting and predicting future variables with the aid of statistical models (Athanassopoulos and Ballantine, 1995). The vast majority of studies have used financial ratios (i.e., return on assets, return on equity, profit, and sales) in conjunction with statistical models (e.g., factor analysis, principal component factor analysis, and discriminant analysis) to evaluate the financial performance of hospitals. Managers better know ratio analyses, characterized by computational simplicity, than other techniques (Athanassopoulos and Ballantine, 1995).

Thanassoulis, Boussofiane and Dyson, (1996) found that ratio analysis and DEA closely agree on the performance of hospitals as a whole. The two methods, however, can disagree on the relative performance of individual hospitals. The authors also found in their study that ratio analysis, unlike DEA, is not suitable for setting targets for hospitals to become more efficient. This is because DEA takes simultaneous account of all inputs and outputs in assessing efficiency, while ratio analysis relates only one input to one output at a time. However, the two methods can complement one another in evaluating efficiency.

DEA Sensitivity Analysis

The sensitivity analysis in this study employs a technique developed by Charnes, Rousseau, and Semple (1996), which provides a single measure of the minimum magnitude of change required simultaneously in input-output variable values to reclassify a DMU. For each DMU this sensitivity measure ([alpha]) can be thought of as the radius of a "circle" of stability. Change of relative total magnitude less than [alpha] will leave the DEA efficiency classification unchanged. The linear programming formulation for sensitivity analysis follows.

CCR Model

Minimize a such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [[lambda].sub.j] > 0 and a is unrestricted.

[??] is the output matrix with the tested unit's output vector removed.

[??] is the input matrix with the tested unit's input vector removed.

[??] is the output vector of the tested unit.

[I.sub.j0] is the input vector of the tested unit.

[alpha] is the radius (percentage) within which the DMU's classification remains unchanged.

[[lambda].sub.j] are the dual variables associated with the constraints representing DMUj, j = 1, ..., n, in the primary equation.

BCC Model

The essential difference between the BCC model and CCR Model is the addition of a new constraint to the linear program formulated in the CCR model. That constraint takes on the following expression:

[n.summation over (j=1)] [[lambda].sub.j] = 1;

METHODOLOGY

Figure 1 displays the proposed research framework for finding stable DEA classifications.

[FIGURE 1 OMITTED]

The methodology consists of two major phases. In the first phase the intent is to find a stable classification (or declare unstable) each DMU within each DEA model under consideration. In a particular application, it may be that the nature of the return to scale is sufficiently known that only one model need be solved. Consider the following design factors:

DEA models (m levels, m [greater than or equal to] 1.)

Input/output variable sets (v levels, v [greater than or equal to] 1),

Repeat multi-variate observations; i.e. reference sets, per DMU (r levels, r [greater than or equal to] 1),

From Figure 1, the iterations on Select DEA model, Select I/O variables, Select DMU set result in the analysis of each DMU [m.sup.*][v.sup.*]r times via DEA. Each analysis yields a sensitivity score, [alpha] ([alpha]>0 is efficient, [alpha]<0 is inefficient). For each iteration, group efficient and inefficient DMU's separately. Then, for the two separate groups, perform univariate cluster analysis on the absolute value of the sensitivity scores a to create stable and unstable categories. It should be noted that that main idea of using the cluster analysis is to objectively obtain two categories (stable and unstable) for each efficient and inefficient group. Thus, the number of cluster was determined to be two in each case. The univariate cluster analysis, using NCSS Software Program, provides two clusters of the units. The first cluster is the stable group, which has high absolute value of sensitivity scores. The second cluster is the unstable group, which has low absolute value of sensitivity scores. These ranked efficiency classifications are efficient and stable (E-S), efficient but not stable (E-US), inefficient but not stable (IE-US) or inefficient and stable (IE-S). For each DMU, summarize the [v.sup.*]r categorizations within each DEA model by a single classification as efficient (E), inefficient (IE), or unstable (US) depending upon whether or not a clear majority of iterations yield either E-S or IES classifications.

The purpose of the second phase of the methodology is to compare efficiency classifications for all DEA models to Cobb-Douglas, ratio analysis or other efficiency classification results. Herein we consider CCR-DEA, BCC-DEA, Cobb-Douglas, and Ratio Analysis; but efficiency techniques to be compared will differ in particular applications.

Although extensions of the original Cobb-Douglas Model, such as a translog model (Christensen, Jorjenson, and Lau, 1971, Christensen, Jorjenson, and Lau, 1973) or seemingly unrelated regression (SUR) (Zellner, 1962), can be used to estimate the parameters when there is more than one dependent variable; Cobb-Douglas models, in this study, can only mimic DEA variable sets with a single output. Thus, repetitions of Cobb-Douglas efficiency analysis will not be entirely analogous to those for DEA. From each Cobb- Douglas analysis, again use the efficiency scores to group DMU's as efficient (scores of 1) or inefficient (scores < 1). Cluster the inefficient DMU's into stable and unstable groups. Since the Cobb-Douglas scores do not allow for the concept of unstable efficiency, the results do not exactly parallel those for DEA. Moreover, even the concept of efficiency is treated somewhat differently by the two procedures. It is this multiplicity of heretofore-incomparable methods that has led to such confusion in applying academic categorizations of efficiency to measuring productivity. If consensus regarding a DMU's efficiency can be derived despite these differences, then the categorization merits the designation of stable.

To obtain a single measure of efficiency from multiple ratios, normalize all ratio scores. Then, conduct pair-wise comparisons using Analytical Hierarchy Procedure (AHP) to assign a weight for each ratio (Anderson, Sweeney, Williams, 1994). Calculate the total score for each DMU by multiplying each ratio by its weight and summing. Based on the total scores, cluster DMU's into three groups; efficient, inefficient, and unstable.

Thus, each DMU has been categorized as efficient, inefficient, or unstable under each competing analysis. Several nonparametric procedures are available to test the significance of any observed disagreements in efficiency classifications. One such is the Cochran test for related observations with a categorical response. In addition, with the classes ordered (Inefficient < Unstable < Efficient), the Friedman (or Quade) test for dependent samples is appropriate. Conover (1999) describes the Cochran, Friedman, and Quade procedures.

APPLICATION: JORDANIAN HOSPITALS

An analysis of public and private Jordanian hospitals provides an illustration of the methodology proposed. The data, as shown in Figure 2, are the most recently available from the Ministry of Health in Jordan. For DEA analysis two reference sets (r=2 observations per DMU) are considered; Year = 1994 and Year = 1995. Figure 2 displays other possible subsets that could be used for relative efficiency classifications.

[FIGURE 2 OMITTED]

Hospital administrators in Jordan identified the most important input and output variables available for consideration as measures of hospital efficiency. Table 1 summarizes the v = 5 input and output variable sets selected. For example, the first model consists of one output (patient days) and two inputs (number of beds and total number of doctors and nurses). Thus, for each of the CCR and BCC models 10 (5x2) efficiency classifications and 10 stability scores are determined.

The resulting cluster analyses for DEA efficient and inefficient hospitals yield the initial categorizations displayed in Table 2. For instance, Hospital 1 is classified as inefficient using the CCR model with the 1994 reference set for the five input-output variable sets considered but stable for only one of these variable sets. Hospital 1 is classified as IE-US for all five variable sets when 1995 data provide the reference set. Using the BCC model and the 1994 reference set, Hospital 14 is efficient for all variable sets and stable for three. With the BCC model and the 1995 reference set, Hospital 14 is again efficient for the five variable sets but stable for only two. After summarizing across the 10 DEA results for each of the CCR and BCC models, place hospitals into final categories per model according to the following criterion:

 if the hospital receives 6 (> 50%) E-S or IE-S (i.e. stable)
 categorizations, classify accordingly as E or IE; otherwise,
 classify as unstable, US.

Hospital 1 under CCR and Hospital 14 under BCC receive fewer than six stable ratings and are classified US for the respective DEA models.

The first four models of Table 1 (single output functions) are used to determine Cobb-Douglas efficiency for each hospital in each year (1994 and 1995). Parameters are estimated using constrained linear programming as previously described in the linear programming sets in (2) and (3).

In equation (4), if the efficiency score ([Q.sub.i]/[P.sub.i]) is equal to one, then the hospital is technically efficient. If the efficiency score is less than one, then the hospital is technically inefficient. Since efficient hospitals could not be classified into stable and unstable (efficiency score =1), only inefficient hospitals were classified into two groups; inefficient stable and inefficient unstable using cluster analysis in the same manner it was employed in the DEA sensitivity analysis. Thus, each hospital is classified eight times (4 different sets of input variables x 2 different years) in one of three categories E, IE-US, IE-S. Table 3 displays these results. In a manner similar to the final DEA classification, Cobb-Douglas classification is stable only if 5 (>50%) of initial classifications are E or IE-S.

Ratio analysis was used in this study as another measure of hospitals' performance to be compared to DEA and Cobb-Douglas results. The following ratios were used in evaluating hospitals' efficiency: Average length of stay; patient days/bed; patient days/ number of doctors; patient days/number of nurses; occupancy rate; and bed turn over. Ratio analysis was performed for the pooled data as before and then repeated for public and private hospital reference sets separately for each year (resulting in two efficiency classifications, pooled and separate, per hospital per year). On average, private hospitals score better than public hospitals in patient days per doctor and patient days per nurse. Public hospitals score better in the other ratios.

Ratio analysis was performed according to the following steps:

1. Normalize all ratio scores.

2. Make pair-wise comparisons of ratios using Analytical Hierarchy Procedure (AHP) to assign weight for each ratio. Hospital administrators made these comparisons.

3. Calculate the total score for each hospital by multiplying each ratio by its weight and summing.

4. Rank hospitals according to their total scores to determine relative efficiency.

5. Classify hospitals based on their total scores, using cluster analysis, into three groups; efficient, inefficient, and unstable.

For the four cases (pooled 1994 and 1995, and separate 1994 and 1995), most hospitals have the same relative efficiency rank in each case. In the pooled case, more than 50% of the hospitals have the same classification in both years. Using separate sample cases, 94.4% of hospitals have the same classification in both years.

Finally, classify hospitals Efficient (E), Inefficient (IE), or unstable (US) as before using E or IE only for a clear majority. For example, H1 was classified as efficient four times (i.e., 100% efficient), therefore, classify as efficient.

Table 4 summarizes classifications across efficiency approaches. Cochran's statistic is used to test the hypothesis of no difference in efficiency classifications among the four approaches. (If a DMU is classified as E, then Cochran's score is 1; else Cochran's score is 0.)

The probability of the observed value of Cochran's statistic is less than .001; thus, the differences are significant among the four methods when classifying as either efficient or not. Similarly, the Friedman test (for ordered classes) yields a probability less than .001; indicating that the differences when classifying three ways: efficient, unstable, or inefficient also are significant. As anticipated, ratio analysis results in more DMU's being labeled efficient.

When the Cochran test is repeated for the two DEA models and the Cobb-Douglas method, removing the ratio analysis classifications, the observed (efficient or not) classes are not significantly different among these three methods. However, when the Friedman test is repeated without ratio analysis, the differences in the three-way classifications (E, US, and IE) remain significant (p-value = .006). CCR-DEA and Cobb-Douglas agree in efficiency classifications, but BCC-DEA differs from each of the others, with an observed bias toward classifying a hospital as unstable rather than inefficient. All told, the degree of agreement is striking among the three methods. For hospitals 7, 12, 16, 18, 19, 26, 27, 33, and 34, evidence of inefficiency is substantial. Hospitals 14 and 21 are arguably efficient. The remaining hospitals seem unstable with regard to an efficiency classification. The only clear contradictions are for Hospital 36.

CONCLUSIONS

This paper proposes a way to find stable DEA efficiency classifications. The technique provides researchers with a toolbox to use in specific applications that includes DEA sensitivity analysis, cluster analysis, and nonparametric tests of differences among several related samples with either categorical or ordinal response variables. Such a methodology is a powerful, practical antidote to the lack of well-developed stochastic theory for DEA.

The methodology is applied to data from public and private hospitals in Jordan, provided by the Minister of Health for that country. Despite significant differences in efficiency classifications among methods, considerable consensus emerges with DEA and Cobb-Douglas efficiency in classifying hospitals as efficient, unstable, or inefficient. It is interesting to note that efficiency classifications vary as much with differing DEA models as with different input-output variable sets. For example, the CCR and BCC models agreed on a final classification of efficient, unstable, or inefficient for 75% of the total number of hospitals, while the five input-output combinations showed initial efficient, not-efficient (whether or not sable) agreement of 94%, 86%, 78%, and 78% for each of the DEA models in each of the years considered. It can be argued that, in DEA applications, efficiency scores are more sensitive to the DEA model type (CCR versus BCC) than to input-output combinations.

In comparing the results of the DEA models to those for Cobb-Douglas and ratio analysis, observe that the concept of efficiency differs under each method. However, CCR and Cobb-Douglas models agreed on classification of 94% of the total number of hospitals. Interestingly, CCR, Cobb-Douglas, and ratio analysis converge for the stable hospitals. Their only clear area of difference in classification is the bias toward efficiency exhibited by ratio analysis. Sensitivity analysis with DEA provides the means for categorizing the stability of the evaluated units' efficiency.

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Khaled Alshare, Emporia State University

Mary M. Whiteside, The University of Texas at Arlington

Table 1. Input-Output Combinations

 Model I Model II Model III

Outputs

Patient days X X X
Occupancy rate

Inputs

Number of beds X X
No. doctors and nurses X X
Number of doctors X
Number of nurses X
Admissions X

 Model IV Model V

Outputs
Patient days X X
Occupancy rate X

Inputs
Number of beds X
No. doctors and nurses X
Number of doctors X
Number of nurses X
Admissions X

Table 2. Efficiency-Stability Classification of Hospitals Using
DEA Models Across Input-Output Models

 CCR 94 BCC 94

 Efficient Not- Efficient Not-
 efficient efficient

Hospital S US S US S US S US

H1 1 4 1 4
H2 1 4 1 1 3
H3 1 4 5
H4 1 4 5
H5 1 4 5
H6 1 4 5
H7 5 5
H8 1 4 5
H9 1 4 4 1
H10 2 3 3 2
H11 2 3 2 3
H12 5 2 3
H13 1 4 5
H14 3 2 3 2
H15 3 2 2 3
H16 3 2 2 3
H17 2 1 2 3 2
H18 5 0 3 2
H19 5 1 4
H20 2 3 2 3
H21 1 4 5
H22 2 3 5
H23 2 3 5
H24 2 3 5
H25 2 3 5
H26 4 1 3 2
H27 4 1 5
H28 2 3 1 4
H29 2 3 5
H30 2 3 5
H31 2 3 1 4
H32 2 3 2 3
H33 5 0 5 0
H34 4 1 2 3
H35 2 3 2 3
H36 5 3 2

 CCR 95 BCC 95

 Efficient Not- Efficient Not-
 efficient efficient

Hospital S US S US S US S US

H1 5 5
H2 1 4 2 3
H3 5 1 4
H4 5 2 3
H5 3 2 3 2
H6 5 2 3
H7 5 5
H8 2 3 2 3
H9 5 5
H10 1 2 2 1 3 1
H11 1 2 2 3 2
H12 5 4 1
H13 5 5
H14 4 1 2 3
H15 5 2 1 2
H16 2 3 2 3
H17 2 3 1 3 1
H18 5 0 3 2
H19 3 2 5
H20 4 1 2 3
H21 1 4 5
H22 5 5
H23 5 5
H24 2 3 1 4
H25 5 5
H26 4 1 4 1
H27 2 3 4 1
H28 5 0 4 1
H29 5 5
H30 1 4 5
H31 2 3 2 3
H32 2 3 2 3
H33 5 0 4 1
H34 5 0 4 1
H35 5 5
H36 5 3 2

Table 3. Efficiency-Stability Classification of Hospitals
Using Cobb-Douglas Models

Hospital Efficient Not-efficient Not-stable

H1 1 7
H2 8
H3 8
H4 8
H5 2 2 4
H6 1 7
H7 7 1
H8 1 7
H9 3 5
H10 2 6
H11 4 4
H12 8
H13 8
H14 5 3
H15 8
H16 2 6
H17 2 4 2
H18 8
H19 8
H20 8
H21 7 1
H22 1 7
H23 8
H24 3 5
H25 8
H26 7 1
H27 6 2
H28 5 3
H29 3 5
H30 3 5
H31 3 5
H32 4 4
H33 8
H34 8
H35 3 5
H36 8

Table 4. Classification of Hospitals across Approaches

Hospital CCR BCC Cobb-Douglas Ratio Analysis

H1 US US US E
H2 US US US E
H3 US US US E
H4 US US US US
H5 US US US E
H6 US US US US
H7 IE IE IE IE
H8 US US US E
H9 US US US US
H10 US US US E
H11 US US US US
H12 IE IE IE US
H13 US US US US
H14 E US E E
H15 IE US IE US
H16 IE US IE IE
H17 US US IE US
H18 IE US IE IE
H19 IE US IE IE
H20 IE US IE US
H21 US E E E
H22 US US US US
H23 US US US US
H24 US US US E
H25 US US US US
H26 IE IE IE US
H27 IE IE IE US
H28 IE US IE US
H29 US US US E
H30 US US US E
H31 US US US E
H32 US US US US
H33 IE IE IE US
H34 IE IE IE US
H35 US US US US
H36 IE E IE IE