Measuring school efficiency in Taiwan’s remote islands: A Comparison of
DEA and SFA
Li-Ju Chen1 and He-Kai Chen 2
1
Professor, Department of Education, National Kaohsiung Normal University,
Taiwan
t1466@nknucc.nknu.edu.tw
2
Doctoral candidate, Department of Education, National Kaohsiung Normal
University, Taiwan
nio203@gmail.com
Introduction
In the field of efficiency measurement, frontiers have been estimated using many
different methods over the past 40 years. The two principal methods are the Data
Envelopment Analysis (DEA), and the stochastic frontier analysis (SFA) which involve
mathematical programming and econometric methods, respectively (Coelli, 1996b). DEA
and SFA both operate as a production function converting inputs into outputs, and shed the
light on education expenditure allocation policy-making.
DEA proposed by Chames, Cooper, and Rhodes (1978) is the non-parametric
mathematical programming approach to frontier estimation, and has been adopted
extensively for measuring school efficiency recently in Taiwan. However, the results from
DEA analysis can only portray the relative efficiency among schools, not absolute efficiency
of each school. Moreover, DEA ignores the effect of random shock, which could be
attributed to statistical errors instead of real inefficiency of decision making unit. In contrast,
the SFA proposed by Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck
(1977), is categorized as the parametric approach and is more appealing to researchers
because it allows to assume that deviations from the frontier may reflect not only
inefficiencies but also noise in the data, and can separate statistic error components from the
inefficiency term on the process (Bogetoft & Otto, 2011).
The focus of this study is to benchmark school efficiency of different sizes with Data
Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) models. The SFA is
newly applied in efficiency measuring of primary schools in Taiwan, where DEA is the
prevailing methodology. The authors investigate the efficiency scores generated by the two
models, and cluster schools into subgroups with different features. By comparing the
characteristics of subgroups with one another, the feasibility of DEA and SFA application
under different scales of schools is provided.
In order to ensure data integrity and homogeneity of input, the measurement of school
efficiency should base on the same set of school data. Under the premise, Penghu, the only
island county in Taiwan area, which includes 40 elementary schools spreading upon 7 of 100
small islands, is chosen as the sample. The purpose of this study is as follows:
1. Measuring the school efficiency with SFA and DEA.
2. Clustering the schools based on the efficiency scores and discriminating the characteristics among
subgroups.
3. Exploring the diversity between the efficiency scores and ranks generated by DEA and SFA.
4. Investigating the characteristics of parameters estimated by SFA and DEA in different subgroups.
Research Design
1.Input and output variables:
The data consists of school input and output during 2011-2012 school year. The input
variables include school expenditures and 4th, 5th, and 6th graders’ family household
expenses on education. The school expenditures conclude the educator wages(x1 ), the
instructional materials(x2 ), and the award & subsidy(x3 ), while the family expenses on
education contain the extra-curriculum of cram school( x4 ),and the educational
expenditure(x5 ). The output variable is represented by the scores from standardized student
assessment conducted annually by Penghu Education Department.
2. SFA model specification
The stochastic frontier models combine the inefficiency term u and the error term v. The
former can be characterized as inefficiency.
The base model after a log transformation is as follow:
y k = f(x k ; β) + v k − uk ,
v k ~N(0, σ2v ), uk ~ N+ (0, σ2u ), k =1, ..., K.
The SFA program Frontier was proposed by Coelli(1996a), and can be executed under
DOS environment.
3. DEA model specification
The DEA model in this study belongs to input orientation and assumed constant returns to
scale (CRS). The multiplier form of the linear programming problem is as follow:
s
hk    rYrk
Max
r 1
m
s.t.
v X
i 1
i
ik
1
s
m
r 1
i 1
 urYrj   vi X ij  0 , j  1,, n
u r ,vi    0, r  1,, s; i  1,, m
Duality in linear programming
Min
s
 m

Z k       si   sr 
r 1
 i 1

n
s.t.
 X
j 1
j
ij
n
y 
j 1
rj
j
 xik  si  0, i  1,, m
 sr  Yrk , r  1,, s
 j , si , sr  0,
j  1,, n i  1,, m r  1,, s
The DEA program DEAP was proposed by Coelli(1996b), and can be executed under
DOS environment.
4. Procedure of this research
Educational
expenditure
provided by
school
Educational
expenditure
provided by
family
Input
variable
DEA efficiency
measure
DEA
efficiency
scores
Output
variable
SFA efficiency
measure
Correlation between
efficiency scores
Correlation between ranks
Summary of Input Slacks
Peer Count Summary
group1
Scores derived
from Annually
achievement test
SFA
efficiency
scores
Cluster analysis
Parameters estimation
group2
group3
group4
Characteristics analysis of each group
Figure 1 Flowchart of the study
Results
1.The measurement of school efficiency by DEA and SFA
Table 1 school efficiency scores, ranks and clusters under DEA and SFA estimation
school
Number of
class
Number of
student
DFA efficiency
scores
DEA
rank
SFA efficiency
scores
SFA
rank
cluster
S11
6
86
0.727
30
0.905
26
1
S12
6
55
0.605
36
0.929
22
1
S20
6
60
0.558
37
0.816
37
1
S21
6
37
0.502
38
0.931
20
1
S23
6
39
0.728
28
0.855
33
1
S24
6
14
0.182
40
0.765
40
1
S30
6
39
0.608
35
0.894
29
1
S32
6
51
0.728
28
0.923
24
1
S33
6
29
0.609
34
0.925
23
1
S36
6
42
0.858
22
0.896
28
1
S37
6
38
0.759
26
0.82
36
1
S39
6
78
0.843
24
0.813
38
1
S09
6
74
0.623
32
0.95
10
2
S16
6
29
0.469
39
0.943
14
2
S17
6
58
0.885
19
0.964
7
2
S19
6
108
0.852
23
0.938
17
2
S22
6
30
0.61
33
0.966
5
2
S25
6
121
0.886
18
0.932
19
2
S26
6
51
0.641
31
0.944
13
2
S29
6
45
0.86
21
0.982
1
2
S31
6
33
0.757
27
0.973
2
2
S34
6
54
0.776
25
0.946
11
2
S38
5
13
0.875
20
0.935
18
2
S03
26
680
1
1
0.857
32
3
S04
11
223
1
1
0.923
25
3
S07
12
234
0.927
13
0.929
21
3
S08
6
126
1
1
0.902
27
3
S10
6
127
0.91
15
0.853
34
3
S13
6
94
1
1
0.892
30
3
S18
6
119
1
1
0.868
31
3
S27
6
42
0.899
17
0.844
35
3
Group
school
Number of
class
Number of
student
DFA efficiency
scores
DEA
rank
SFA efficiency
scores
SFA
rank
cluster
S28
6
55
1
1
0.785
39
3
S01
33
491
0.924
14
0.941
16
4
S02
23
591
0.91
15
0.955
8
4
S05
26
681
1
1
0.97
3
4
S06
6
143
0.94
12
0.944
12
4
S14
5
12
1
1
0.968
4
4
S15
9
176
1
1
0.942
15
4
S35
6
116
1
1
0.953
9
4
S40
6
50
1
1
0.965
6
4
125
0.811
Mean
Group
0.911
※Results：
1. According to DEA and SFA efficiency scores, there are 4 groups clustered by k-means analysis and each
one shows different feature.
2. School lists are sorted by the clusters, each of which contains different number of schools.
3. Both the rank and efficiency scores of DEA and SFA in schools are quite different, group1 features
lower both DEA and SFA scores than average, group2 features lower DEA scores and higher SFA scores,
group3 features higher DEA scores and lower SFA scores, group4 features both higher DEA and SFA
scores .
Figure 1 The distribution of DEA and SFA efficiency scores
※Results：
1. The shape of SFA and DEA distributions are skewed to the left.
2. Most schools are measured high performance.
Table 2 The Pearson coefficient of correlation among number of students, classes, output, DEA , and SFA
efficiency scores
Variables
DEA efficiency scores
SFA efficiency scores
Variables
DEA efficiency scores
SFA efficiency scores
**
Number of students
.406
.129
Number of classes
.309
.092
Output(Y)
.209
.844**
Input(X1)
-.618**
-.191
Input(X2)
-.465**
-.111
Input(X3)
-.781**
-.173
Input(X4)
.194
.168
Input(X5)
-.145
.138
DEA efficiency scores
1
.213
SFA efficiency scores
.213
1
**. The correlation is significant under level 0.01 (2-tailed).
※Results：
1. DEA efficiency scores show positive correlation with number of students, while SFA does no.
2. The efficiency scores of DEA and SFA show no significant correlation with the number of classes in
schools .
3. The correlation between DEA and SFA efficiency scores isn’t significantly related.
4. DEA efficiency scores show high correlation with inputs, while SFA efficiency scores show high
correlation with the output.
1
0.95
S22
S09
S26
S12
S33
S16
S21
S31
S34
S32
S11
0.9
S29
S05
S14
S40
S17
S02 S35
S06
S19S38 S01 S15
S25 S07
S04
S23
0.85
S10
S27
S37
S20
S08
S13
S36
S30
S18
S03
S39
0.8
S28
S24
0.75
0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 2 the scatter plot of school efficiency
※Results：
1. The X-axis represents the DEA efficiency scores, the Y-axis represents SFA efficiency scores.
1
2. The lines of average values of DEA and SFA cross, and divide 40 schools into 4 groups, similar to
clustering analysis.
Table 3 Characteristics of input and output in different clusters
Cluster
Average DEA
efficiency
scores
Average SFA
efficiency
scores
𝑌̅
̅̅̅
X1
̅̅̅2
X
Group1
Group2
Group3
Group4
Average
0.642
0.749
0.971
0.972
0.811
0.873
0.952
0.873
0.955
0.911
66
75
67
75
70
401,164
358,592
174,320
234,433
305,071
3,386
3,073
1,193
1,877
2,505
̅̅̅3
X
̅̅̅
X4
778
729
212
357
553
9,655
9,681
20,334
26,682
15,470
̅̅̅
X5
10,491
11,020
11,494
14,280
11,620
※Results：
1.
2.
3.
The Comparison of the average scores of 4 subgroups with total schools displays the characteristics of
DEA and SFA estimation. Schools in group1 and group3 have similar SFA efficiency scores and outputs,
in spite of divergence of its inputs. On the contrary, the DEA efficiency scores in group1 and goup3
show that schools in group3 is more efficient than schools in group1. The same with group2 and
group4.
The DEA efficiency scores in schools of group3 are similar to group4, but the SFA efficiency scores of
group4 are higher than group3. On the average, inputs and the output of group4 are higher than those of
group3.
As schools in goup1 and group2 locate at rural region, the scales of school is small. The costs per
student from public sector (government) decline as school size enlarged, and student costs from private
sector (family) increase, as the large-scaled schools tend to be located at county center.
Table 4 Attribute of different clusters
̅
Cluster School scale
𝒀
̅𝐗̅̅𝟏̅
̅𝐗̅̅̅𝟐
̅𝐗̅̅𝟑̅
̅̅
𝐗̅𝟒̅
̅̅̅̅
𝐗𝟓
Group1
Small
Low
High
High
High
Low
Low
Group2
Small
High
High
High
High
Low
Low
Group3
Large
Low
Low
Low
Low
High
High
Group4
Large
High
Low
Low
Low
High
High
※Results：
1. The efficiency scores measured by DEA and SFA vary with the scale of schools, the higher the DEA
scores, the larger the school scale .
2. DEA is more sensitive to scales. When school size is larger, the higher DEA efficiency scores are
measured. On the other hand, SFA keeps stable when measuring different sizes of schools.
3. Inputs in both models are the same, but the SFA is more sensitive to output value than DEA.
Table 5
One-way ANOVA on School efficiency scores in different clusters
ANOVA
Efficiency scores
Sum of
Squares
Df
Mean Square
F
Sig.
DFA
Between Groups
.821
3
.274
Within Groups
.594
36
.016
1.415
39
Between Groups
.065
3
.022
Within Groups
.055
36
.002
Total
.120
39
Total
SFA
16.589
.000
14.097
.000
※Results：
1. Both the DEA and SFA efficiency scores of 4 subgroups exhibit significant differences between one
another.
Table6 The comparison of the extreme efficiency scores and parameters from each groups
DEA
score
S18(3)
S40(4)
S16(2)
S24(1)
SFA score
Number of
students
Y
X1
X2
X3
X4
X5
1(1)
1(1)
0.496(39)
0.868(31)
0.965(6)
0.943(14)
119
50
29
65
73
75
115,720
903
247,797 1,894
530,153 3,257
202 12,672
120
0
1,034 16,029
7,803
5,655
15,551
0.182(40)
0.765(40)
14
57
1,108,315 8,667
3,000 17,333
22,900
※Results：
1. 2 most efficient schools S18 and S40 which DEA efficiency scores equal to 1 are the main peer targets,
account for 17 and 20, respectively. However, S18 performs lower SFA scores than average(see Table 3)
owing to its feature of lower output.
2. DEA measures S16 and S24 the inefficient schools. However, S16 performs much better than S24 under
SFA measurement.
Table 7
Parameters estimation by SFA
Per student
β0
β1
β2
β3
β4
coefficient
3.884
0.132
-0.088
-0.008
0.011
standard-error
0.648
0.053
0.035
0.014
0.007
t-ratio
5.993
2.476
-2.513
-0.559
1.567
-0.062
0.017
0.866
0.047
0.006
0.140
-1.322
2.714
6.172
β5
2
σ
𝛄
※Results：
1. γ=0.866 means there is 86.6% of residuals due to production inefficiency.
2. Parameters listed in Table 7 show that input(x1 ), and input(x4 ) are contributive to school efficiency(βi >
0).
3. According to negative values of parameters β2 , β3 , and β5 , Input(x2 ), input(x3 ), and input(x5 ) show
negative effects on school efficiency.
Conclusion
Based on the results of empirical data, the measurement of school efficiency with SFA is
inconsistent with DEA. These two kinds of efficiency scores provide information for
clustering schools with different features on input and output. Besides, the more rural region
the school locates the larger relevance appears with the public resource in student
achievement while the more urbanized region the school sites, the lesser relevance. By
comparing the DEA technical inefficiency patterns with SFA in each subgroup, some school
improvement strategies are possible to be proposed. Some of the schools which locate at the
rural area displays higher efficiency therefore we do not recommend that the authority close
down those schools for their high educational cost.
According to the parameters of SFA, it's shown that the input and output which this study
adopts is appropriate as most of the residuals come from production inefficiency. The
parameters of DEA provide information for optimum input and output, while the SFA
parameters provide the information for factors that truly affect the efficiency scores. SFA is
appropriate for measuring schools under a variety of school sizes, which are likely
under-estimated by DEA as the per-student expenditure is high. Some schools with little
input and output are also labeled efficient by DEA, though. But, it is difficult to convince the
public how well these schools perform, and disobeying to the spirit of pursuing excellence.
On the other hand, SFA shows high correlation with the output and is easy to explain the
reasons of being the example of other schools.
References
Aigner, D. J., Lovell, C. A. K. & Schmidt, P. (1977). Formulation and estimation of stochastic frontier
production function models. Journal of Econometrics, 6, 21-37.
Bogetoft, P. & Otto, L. (2011). Benchmarking with DEA, SFA, and R. New York, NY: Springer.
Chames, A., Cooper, W. W. and Rhodes, E. (1978), Measuring the Efficiency of Decision Making Units,
European Journal of Operational Research, 2, 429-444.
Coelli, T.J. (1996a), A Guide to FRONTIER Version 4.1: A Computer Program for Frontier Production
Function Estimation, CEPA Working Paper 96/07, Department of Econometrics, University of New
England, Armidale.
Coelli, T.J. (1996b), A Guide to DEAP Version 2.1: A Data Envelopment Analysis (Computer) Program,
CEPA Working Paper 96/08, Department of Econometrics, University of New England, Armidale.
Messusen, W., & Van den Broeck, J. (1977). Efficiency estimation from Cobb-Douglas production function
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