Amparo Seijas Díaz Universidad de A Coruña
Reference: Received 29th April 2003; Published 04th September 2003. ISSN 15791475
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Abstract
 The objective of this study is to evaluate the performance of the secondary education centres of A Coruña in Spain using a production function, which allows us to measure their efficiency. To this end, data envelopment analysis is used, since it is inherently flexible and therefore most suited to the peculiarities of the education process. The results clearly demonstrate that the technical efficiency of the centres analysed is extremely high and that the factors, which do not come under the auspices of the manager i.e. the noncontrollable factors, are fundamental to the efficiency of the centre.
Resumen
 El objetivo de este estudio consiste en evaluar el rendimiento de los centros de educación secundaria en A Coruña (España) utilizando una función de producción que nos permite medir su eficiencia. Para este fin, manejamos un análisis envolvente de datos, ya que resulta inherentemente flexible, y por lo tanto es el que mejor se ajusta a las peculiaridades del proceso educativo. Los resultados muestran claramente que la eficiencia técnica de los centros analizados es extremadamente alta y que los factores que no vienen auspiciados por el gestor, por ejemplo los factores no controlables, son fundamentales para la eficiencia del mismo.

1. INTRODUCTION
Interest in evaluating the efficiency of educational processes is mainly due to the importance that education has on the economy as a whole, and its impact on the welfare of the population. Investment in human capital, increased productivity, and higher salaries earned by a more highly educated population, all decisively contribute to increasing a country´s wealth. The positive externalities generated by education also constitute an important asset for society as a whole, since raised educational levels help to combat poverty and favour social cohesion.
Within the field of education economics, the evaluation process itself has tended to focus on the internal efficiency of the schools. Within this context, from the beginning of the mid 1970´s, there have been certain key studies that focus on the efficiency of nonuniversity education centres via the estimation of a production function that utilises nonparametric techniques.1 In Spain, this analytical tendency is relatively recent and it has been the work of Pedraja and Salinas (1996), and Mancebón (1996,1998) and Muñiz (2000) who have been at the forefront of research in this field.
The idea that underlies these approaches is that the production process of any educational centre may be specified by using a production function capable of reflecting a technical relationship between the group of productive factors, which when correctly combined provide certain outputs. The production function determines the maximum level of output that may be obtained given a specific quantity of inputs and a specific level of technology, that is, the minimum quantity of inputs needed to obtain a given level of outputs.
The objective of this study therefore, is to evaluate the performance of the secondary schools in the province of A Coruña during the academic years 95/96, 96/97, 97/98 and 98/99, using a production function that allows us to measure the level of technical efficiency. The results unequivocally reveal that technical efficiency was high and that the noncontrollable or "context" variables played a crucial role in achieving this efficiency.2
This research has been organised in the following way. The first stage involves establishing the production function, which will be the means by which efficiency is measured. The second stage involves outlining the technique used for carrying out the estimation, describing the sample parameters, and defining the variables. The following stage of the research involves an evaluation of the performance of a sample of secondary education centres, and the subsequent testing of the robustness of the results. The final section of the paper offers the study´s main conclusions.
2. CONCEPTUAL FRAMEWORK
The study and measurement of the efficiency of educational institutions should be based on an analysis of the appropriate production function, which reflects the technical relationship between the groups of productive factors that, in combination, dictate specific levels of output.
Often, educational production functions are based on dubious or unrealistic criteria, which attempt to represent, in a simplified way, certain productive organisations. In these kinds of models, the decision makers are generally aware of the in which a specific number of inputs are used in order to obtain different outputs. All of these variables are easily quantifiable and the relationship between them is strictly deterministic, which means that given a particular quantity and group of inputs the result is always the same quantity of output. Further, all of the inputs may be substituted without any type of restriction.
The idiosyncrasies inherent in the educational sector mean that our starting point must differ significantly from that described above. It would seem logical to digress since, in the case of education, the decisionmakers are unaware of the production function, and this must therefore be estimated using imperfect data. Further, many of the important inputs cannot be substituted, and in many cases the production process is subject to a certain degree of uncertainty.
The identification and measurement of the outputs and inputs of the education sector is a somewhat difficult task given the peculiarities inherent in many of the variables involved in the production process. Because of the very essence of education, many facets of the production process are, by their very nature, both intangible and multiple, and include cognitive abilities, attitudes, and social behaviour and other qualitative attributes associated with scholastic output.
The human learning process is generally cumulative in nature, in that educational results are not simply the product of a given moment in time but are acquired through the accumulation of learning over previous periods. It must also be remembered that the public education system differs from the private system in that the profit motive does not dictate the supply of services in the public sector. This is because objectives are of a more social bent, and prices, if they exist, are not linked to the characteristics that define competitive markets.
Educational output does not depend solely on those factors that are linked to the educational centres themselves. Other factors such as the characteristics of the individuals and their environments play key roles in the educational process. It should also be mentioned that the pupils´ nonscholastic environment is not influenced by those responsible for educational material, which adds a further difficulty to the task of designing a suitable production function.
It is within this context that Hanushek (1972, 1979) states that scholastic performance depends on a combination of four groups of factors, which are; family background, the innate abilities and other peculiarities specific to the individual student him or herself, the student´s peer group and school input. Most pertinent literature expresses the education production function as a process that is synthesised by using the following relationship:
Where is a vector of the educational achievement of the ith student at a given moment in time t; is a vector that takes in family characteristics for the ith student at a given moment in time t; is a vector that reflects the student´s peer group at a given point in time t; is a vector that reflects the student´s "initial endowments" along with other factors inherent in the student, and finally, which is a vector of school inputs relevant to the ith student at a given moment in time t.
In short, any attempt to estimate the productive performance of the educational UDM should involve a rigorous analysis of each of the elements mentioned in the previous paragraph.3 We have tried to bear this precept in mind when choosing the variables used to define the production process fundamental to the educational system being analysed.
3. METHODOLOGY AND MODEL
Given the special nature of the education sector, a technique is needed which is flexible enough to define the area enclosed by the production frontier in order to calculate the technical efficiency indices for the schools in this analysis. Data envelopment analysis (DEA) fulfils this role, since all that is needed is a set of observations that produce similar outputs from a set of common inputs, without the necessity of knowing the type or quantity of technology that underlies the production process.4
The DEA technique was initially developed by Charnes, Cooper and Rodhes (1978), and may be considered as a multiple outputs application of traditional ratios analysis as proposed by Farrell in (1957). The technique is used in order to facilitate the construction of the efficiency frontier and assumes constant returns to scale, convexity, and the free disposability of inputs y outputs.
Banker, Charnes y Cooper (1984) subsequently developed a model which was very similar to the model described above, but which differed in that they relaxed the supposition of constant returns to scale, and designed a frontier that was more flexible and thus better adapted to the different scales of production that the UDM might take.5 Some years later Banker y Morey (1986), besides incorporating the supposition of variable returns to scale, also dealt with the problem of those variables which could not be controlled by the manager.
The DEA models in question fall into two basic categories, those that deal with inputs and those that deal with outputs. These different formulations do not always offer the same results and this is indeed the case with all of those specifications that are based on BCC models. The underlying supposition here is that the choice between the two structures will depend on the particular circumstances of each piece of research.
The heterogeneity of the performance of the education centres and those factors, which lie outside the control of the managers, are elements of undoubted interest and potential obstacles which must be taken into consideration when we come to defining the specifications of our DEA model. In this sense, the variable returnstoscale assumption is fundamental when it comes to resolving the problem of heterogeneity. This assumption allows us to obtain a flexible frontier, one, which is capable of adapting to the specific behaviour of each individual school, while at the same time being able to relay information about their optimum size. The factors that lie beyond the control of the managers will be addressed by recourse to the development of mathematical programs orientated in terms of outputs or more elaborate models in which some of the variables, either outputs or inputs are determined exogenously.
Most of the research in the education sector that adopts DEA analysis uses an outputorientated version of the model. This is because it is impossible for those responsible for education management to control some of the resources that are involved in the production process or alternatively because the values of these resources are determined by decision makers operating at higher level than that of the school itself. This analysis uses both types of variables. On the one hand there is the socioeconomic index and the average school mark which takes in the previous two years of academic study, and these together represent the nondiscretional factors, while on the other, there are the centre´s operating expenses and the numbers of teaching staff. Although these variables are by nature discretional, it is, in the end, the educational authorities that determine the actual numbers and quantities of the said variables.
Thus the outputorientated version of the BCC was the obvious choice, and further, this version of the programme provides information about the level of pure technical efficiency. It is therefore capable of measuring the optimum relationship between inputs and outputs, and isolates the effects that the scale of production might provoke with respect to the results of the efficiency indicators.
4. THE SAMPLE AND THE SELECTION OF VARIABLES
The definition and selection of the UDM that form part of the sample is of paramount importance since they serve as a means of reflecting the productive behaviour of the secondary education centres in the province of A Coruña, Spain. Therefore, certain factors should be taken into consideration, three of the most important of which are: the level of homogeneity of the observations, the number of units to be included, and the geographical and temporal framework of the analysis (see Golany and Roll).
DEA methodology requires that the distinct observable UDM´s be comparable. The various centres being analysed and compared therefore must be carrying out similar tasks, attempting to fulfil the same objectives, and be subject to the same market conditions. These requisites lead us to exclude all private education centres from the sample along with all of those public centres that do not impart the final year of the secondary education cycle. Finally, the variables that determine the behaviour of the centres being analysed must be homogeneous, although there will, of course, be differences in terms of intensity and magnitude. To this end those centres that are organised under the auspices of the most recent education legislation are not included, given that they constitute a clear potential source of statistical interference.6
The DEA efficiency analysis is also conditioned by the necessity of defining the geographical boundaries within which the centres that go to make up the sample are located together with the temporal framework of the analysis. In our particular case the physical geographical area that forms the boundaries of the study is the province of A Coruña. A Coruña is an area which is especially dynamic and which satisfactorily integrates the range of characteristics that define the present socioeconomic status of the autonomous region of Galicia. The temporal framework of the analysis is made up of the academic years 95/96, 96/97, 97/98 and 98/99. Table 1 therefore, provides the definitive number of centres that make up the sample.
TABLE 1 SECONDARY EDUCATION CENTRES 

Academic Year 9596 
Academic Year 9697 
Academic Year 9798 
Academic Year 9899 
TOTAL NUMBER OF CENTRES 
TOTAL NUMBER OF CENTRES IN THE SAMPLE 
TOTAL NUMBER OF CENTRES 
TOTAL NUMBER OF CENTRES IN THE SAMPLE 
TOTAL NUMBER OF CENTRES 
TOTAL NUMBER OF CENTRES IN THE SAMPLE 
TOTAL NUMBER OF CENTRES 
TOTAL NUMBER OF CENTRES IN THE SAMPLE 
CENTRES 
54 
53 
52 
51 
47 
47 
47 
47 
Source: Own elaboration based on information provided by education authority.
When estimating the productive behaviour of any UDM using DEA models one of the most important problems is the selection of the variables. This is because the results obtained using this technique can be very sensitive to the specifications inherent in empirical models. In order to overcome this difficulty it becomes necessary to manage all the statistical information at our disposal and to choose those variables which best reflect the situation we are attempting to analyse by means of specific statistical techniques and including basic DEA models.
This analysis utilizes a wide range of indicators in order to determine the variables that define the productive process taking place in the secondary schools that go to make up our sample. The variables that were chosen were those, which best measure the various facets of educational output and which take in; family background, peer group, initial endowments and other strictly scholastic factors. In each and every case all the pertinent operations are carried out in order to obtain the optimum final indicators.
Below, we list each of the elements that determine the production function for the education centres that are the subject of this study.
a) Education centre Output
Education is a process that affects "individuals". It is imperative therefore, that the results of the measurements carried out with respect to the centres are both qualitative as well as quantitative in nature. The majority of the studies on the education production function use the marks obtained by students in certain standardised tests. In Great Britain for example these tests would correspond to the general certificate of secondary education exams normally taken at the age of 16. These results are used as form of reflecting the quality of scholastic performance. However, to a lesser extent, there are studies that attempt to measure educational results using indicators such as the number of students that manage to pass a particular schoolyear course, students that "graduate" from the centre in which they were studying, attendance rates, and the levels of students who continue in, or give up formal education (see Burkhead, Fox and Holland, 1967), all of which supposedly quantitatively measure the results of individual education centres.
This work utilizes statistical information made available by the education council in order to measure both the quantitative and qualitative aspects of the output of the centres. Specifically, we make use of the average marks of the students in each centre in each of the subjects evaluated in the university entrance examinations, and the number of students who sit the exams that correspond to each of these subjects. Data is also used that corresponds to the total number of students registered for the final year of the secondary education. From this data we make an approximation of the output of the public secondary schools in the province of A Coruña for the schoolyear courses in 95/96, 96/97, 97/98 and 98/99, using the following variables:
· The average mark obtained in the University Entrance Exam (known in Spain as PAAU tests) in the subjects of "technical science" and "health sciences".
· The average mark obtained in the University Entrance Exam in the subjects of "humanities" and "social sciences".
· The number of pupils that passed the University Entrance Exam as a proportion of the total number of students enrolled in those courses.
The first 2 of these variables measure certain qualitative aspects of scholastic performance whilst the third measures those, which are quantitative. The third variable is based on the number of students enrolled in the final year of compulsory secondary education, rather than the number of students who take the University Entrance Exam. This we do in order to eliminate the potentially biased effects of the strategic results of certain centres.
b) The variables that determine scholastic performance
i) Family background and peer group
The socioeconomic level of the students and the student´s peer group are the most important factors that go to explaining a student´s scholastic performance. These factors therefore, must be taken into consideration and reflected most carefully in any kind of empirical study, in spite of the difficulties inherent in the demarcation and weighting given to these kinds of concepts.
To this end we have opted to analyse a set of variables of an economic, social, and demographic nature at a municipal level in order to construct a series of indicators that mirror the socioeconomic characteristics of the environment in which the secondary schools are located and which are the object of this study.
Given the relatively high number of variables that we must deal with and the need to demarcate the context within which these variables operate, it would seem logical to look for a way of simplifying the information available without prejudicing the logical structures that underlie the data sets. This we attempt to do by using a factorial analysis, specifically a principal components analysis.
From the information obtained at a municipal level using the factor analysis, we proceed to create a synthetic indicator that measures the socioeconomic aspects of the municipal area. To this end we use the weighted sum of the values of the factors as an index, and for the weighting itself we use the square root of the percentage of the variance relevant to each of the factors (see Aznar Grasa, 1976). We then add the factors that are relative to the educational districts. The weighting in this case is the percentage of the total population eligible for primary and secondary education within the given municipal area and allows us to obtain a synthetic index for each of the districts .
ii) initial endowments
The production process that takes place within the auspices of the education system is characterised by the peculiarity that the process develops within the actual student (See Becker, 1975) and thus the qualities inherent in the student affect the results obtained by the school. This analysis utilizes the average marks obtained in the school evaluation reports,7 as it constitutes the only reliable statistical information available which allows us to evaluate the "quality" of the students contained in the sample.
iii) School factors
The "School factors" variable is quantified by measuring operating expenses (excluding those connected with personnel), and the number of teachers in each centre. The educational inputs analysed in our empirical model for the school course years 95/96, 96/97, 97/98 and 98/99 are the following:
· The operating expenses (excluding personnel cost) divided by the total number of students enrolled in these centres.
· The total number of teachers divided by the total number of students enrolled.
5. ANALYSIS OF RESULTS
The efficiency indices calculated for the centres that make up our sample in the academic year 95/96, reveal that of the 53 centres examined, 31 were in fact inefficient, a proportion that represents 58% of the total (table 2). For the academic years 96/97 and 97/98, the percentage of inefficient centres was 55% and in 98/99 45%. Further, on average, the centres, which were classified as "inefficient", were close to being "efficient" since they obtained a value of 90.58%. (See table 2)
TABLE 2 MAIN RESULTS 
Academic Year 9596 

EFFICIENT 
INNEFICIENT 
TOTAL 
AVERAGE EFFICIENCY* 
URBAN (U) 
8(38%) 
13(62%) 
21 
93,73% 
SEMIURBAN (SU) 
8(35%) 
15(65%) 
23 
92,02% 
SEMIRURAL (SR) 
6(67%) 
3(33%) 
9 
93,18% 
TOTAL CENTRES 
22 
31 
53 
 
PERCENTAGE 
42% 
58% 
100% 
92,84% 
Academic Year 9697 

EFFICIENT 
INNEFICIENT 
TOTAL 
AVERAGE EFFICIENCY* 
URBAN (U) 
10(50%) 
10(50%) 
20 
92,82% 
SEMIURBAN (SU) 
10(45%) 
12(55%) 
22 
89,64% 
SEMIRURAL (SR) 
3(33%) 
6(67%) 
9 
92,40% 
TOTAL CENTRES 
23 
28 
51 
 
PERCENTAGE 
45% 
55% 
100% 
91,37% 
Academic Year 9798 

EFFICIENT 
INNEFICIENT 
TOTAL 
AVERAGE EFFICIENCY* 
URBAN (U) 
6(35%) 
11(65%) 
17 
92,93% 
SEMIURBAN (SU) 
10(45%) 
12(55%) 
22 
88,60% 
SEMIRURAL (SR) 
5(62%) 
3(38%) 
8 
89,88% 
TOTAL CENTRES 
21 
26 
47 
 
PERCENTAGE 
45% 
55% 
100% 
90,58% 
Academic Year 9899 

EFFICIENT 
INNEFICIENT 
TOTAL 
AVERAGE EFFICIENCY* 
URBAN (U) 
9(53%) 
8(47%) 
17 
96,28% 
SEMIURBAN (SU) 
10(45%) 
12(55%) 
22 
91,58% 
SEMIRURAL (SR) 
7(88%) 
1(12%) 
8 
89,88%** 
TOTAL CENTRES 
26 
21 
47 
 
PERCENTAGE 
55% 
45% 
100% 
93,35% 
The factors that seem to significantly contribute to explaining these high levels of efficiency are the context variables i.e. the synthetic index for each of the districts together with the attributes that define the characteristics of the students, that is the average marks obtained in their school reports. We test this reasoning by specifying 4 new BCC models. In each of these new models one of the original inputs is eliminated thus providing us with 4 variants (table 3).8 The overall, average efficiency, which includes all of the education centres in the sample, constitutes the specification which is most notably affected by the elimination of the average marks obtained by the students and the synthetic index.
TABLA 3 VARIOUS SPECIFICATIONS FOR THE DEA MODEL 
Academic Year 9596 

DEA 
DEA (ELIMINATION TEACHERS STUDENTS 
DEA (ELIMINATION OPERATING EXPENSES 
DEA (ELIMINATION AVERAGE MARK) 
DEA (ELIMINATION SYNTHETIC INDEX) 
AVERAGE 
95,8 
95,7 
95,7 
93,4 
94,1 
Academic Year 9697 

DEA 
DEA (ELIMINATION TEACHERS STUDENTS 
DEA (ELIMINATION OPERATING EXPENSES 
DEA (ELIMINATION AVERAGE MARK) 
DEA (ELIMINATION SYNTHETIC INDEX) 
AVERAGE 
95,2 
95,1 
95,1 
91,8 
93,7 
Academic Year 9798 

DEA 
DEA (ELIMINATION TEACHERS STUDENTS 
DEA (ELIMINATION OPERATING EXPENSES 
DEA (ELIMINATION AVERAGE MARK) 
DEA (ELIMINATION SYNTHETIC INDEX) 
AVERAGE 
94,8 
94,5 
94,6 
92,9 
93,0 
Academic Year 9899 

DEA 
DEA (ELIMINATION TEACHERS STUDENTS 
DEA (ELIMINATION OPERATING EXPENSES 
DEA (ELIMINATION AVERAGE MARK) 
DEA (ELIMINATION SYNTHETIC INDEX) 
AVERAGE 
97,0 
96,5 
96,4 
94,2 
94,2 
Source: Own elaboration
The educational centres that fall into each one of the subsamples are classified as, urban, semiurban and semirural. Using this classification as a basis we proceeded to analyse the distribution of efficient and inefficient centres accordingly. Table 2 shows that there is no important difference in terms of the distribution of efficient and inefficient schools in the urban and semiurban areas. This homogeneity is particularly pronounced for the academic years 95/96 and 96/97. Quite the reverse is the case of the centres that are located in semirural areas, in which efficient centres predominate although the academic year 96/97 provides a notable exception.
The reflections offered above should make it fairly obvious that the "context", in spite of being a determining factor when it comes to evaluating the efficiency indices, is not central to explaining the distribution of the centres and the various levels of efficiency that different types geographical location obtain.9 The results would seem to suggest that the secondary schools situated in semirural areas are capable of overcoming the disadvantages inherent in their geographical locations.
The nature or characteristics of the students themselves do not, a priori, appear to explain these differences in distribution either. This is because the average marks of the student reports do not appear to be substantially different from one centre to another. Further, the operating expenses or overheads do not allow us to isolate a particular pattern of behaviour that might allow us to better define the distribution of the efficient and inefficient centres. The ratio of students per teacher on the other hand, is clearly reflective of increased levels of efficiency in those secondary schools located in semirural areas. Thus, it would seem logical to posit that the teacherperstudent ratio (together with other factors not included in this model) go a long way to explaining the above tendency.
TABLA 4
AVERAGE VALUES FOR THE DISTINCT VARIABLES USED FOR THE EDUCATION CENTRES 
Academic Year 9596 

TEACHERS/ STUDENTS ENROLLED 
AVERAGE MARKS OBTAINED IN THE SCHOOL 
OPERATING EXPENSES/ STUDENTS ENROLLED 
SYNTHETIC INDEX 
AVERAGE MARK (TECHNICAL SCIENCE; HEALTH SCIENCE) 
AVERAGE MARK (HUMANITIES) 
PUPILS THAT PASSED THE UNIVERSITY ENTRANCE EXAM/ STUDENTS ENROLLED IN THE COURSE 
URBAN (U) 
0,15 
6,75 
19.518,99 
45,08 
5,11 
5,12 
0,42 
SEMIURBAN (SU) 
0,12 
6,86 
23.970,98 
33,95 
5,12 
5,16 
0,41 
SEMIRURAL (SR) 
0,13 
7,00 
20.544,13 
26,29 
5,25 
5,10 
0,40 
Academic Year 9697 

TEACHERS/ STUDENTS ENROLLED 
AVERAGE MARKS OBTAINED IN THE SCHOOL 
OPERATING EXPENSES/ STUDENTS ENROLLED 
SYNTHETIC INDEX 
AVERAGE MARK (TECHNICAL SCIENCE; HEALTH SCIENCE) 
AVERAGE MARK (HUMANITIES) 
PUPILS THAT PASSED THE UNIVERSITY ENTRANCE EXAM/ STUDENTS ENROLLED IN THE COURSE 
URBAN (U) 
0,30 
6,73 
48,956,34 
44,94 
4,74 
4,86 
0,45 
SEMIURBAN (SU) 
0,08 
6,89 
13.794,69 
34,11 
4,86 
4,99 
0,41 
SEMIRURAL (SR) 
0,16 
6,93 
26.900,97 
26,29 
4,76 
4,91 
0,44 
Academic Year 9798 

TEACHERS/ STUDENTS ENROLLED 
AVERAGE MARKS OBTAINED IN THE SCHOOL 
OPERATING EXPENSES/ STUDENTS ENROLLED 
SYNTHETIC INDEX 
AVERAGE MARK (TECHNICAL SCIENCE; HEALTH SCIENCE) 
AVERAGE MARK (HUMANITIES) 
PUPILS THAT PASSED THE UNIVERSITY ENTRANCE EXAM/ STUDENTS ENROLLED IN THE COURSE 
URBAN (U) 
0,07 
6,83 
10.481,25 
45,14 
5,05 
5.02 
0,43 
SEMIURBAN (SU) 
0,08 
6,90 
14.849,51 
34,11 
4,93 
4,88 
0,44 
SEMIRURAL (SR) 
0,14 
7,09 
22.036,98 
26,80 
5,09 
4,96 
0,44 
Academic Year 9899 

TEACHERS/ STUDENTS ENROLLED 
AVERAGE MARKS OBTAINED IN THE SCHOOL 
OPERATING EXPENSES/ STUDENTS ENROLLED 
SYNTHETIC INDEX 
AVERAGE MARK (TECHNICAL SCIENCE; HEALTH SCIENCE) 
AVERAGE MARK (HUMANITIES) 
PUPILS THAT PASSED THE UNIVERSITY ENTRANCE EXAM/ STUDENTS ENROLLED IN THE COURSE 
URBAN (U) 
0,09 
6,76 
11.225,16 
45,14 
5,16 
5,31 
0,42 
SEMIURBAN (SU) 
0,10 
6,89 
14.017,74 
34,11 
5,17 
5,31 
0,43 
SEMIRURAL (SR) 
0,15 
6,99 
23.704,80 
26,80 
5,38 
5,22 
0,39 
Source: Own elaboration
6. ANALYSIS OF THE SENSITIVITY OF THE RESULT
Data envelopment analysis constitutes a nonparametric, deterministic technique and is the technique we have chosen, but there is no statistical tool available that allows us to evaluate the reliability of the results obtained. Thus the values obtained may be particularly sensitive to measurement errors and random interference. The robustness of the results may be verified however by studying the sensitivity of the results to certain alternative specifications for the production function.
In order to contrast the validity of the results, we set up 8 different BCC models for the subsamples corresponding to the academic years 95/96, 96/97, 97/98 y 98/99. Each one of these models utilizes diverse variables, and these are given in Table 5. These variations allow us to observe whether the "ranking" of the centres obtained using the BCC model remains the same when there are changes in the inputs and outputs. To this end, the Spearman correlation coefficient (see McCarthy and Yaisawarng, 1993) is used for the various academic years that make up the study.
TABLE 5 SENSITIVITY ANALYSIS (Various specifications for the BBC Model) 

DEA1 
DEA2 
DEA3 
DEA4 
DEA5 
DEA6 
DEA7 
DEA8 
DEA9 
Teachers/students enrolled 
* 
* 
* 
* 
* 
* 
* 
* 
* 
Average marks obtained in the school 
* 

* 
* 
* 
* 

* 
* 
Average marks obtained in the school (only June) 

* 






* 
Average marks obtained in the school (only September) 









Operating spenses/students enrolled 
* 
* 
* 
* 
* 
* 
* 
* 
* 
Synthetic index 
* 
* 


* 
* 
* 
* 
* 
F1 (Level of district urbanization) 



* 





F2 (Labor and demographic dynamism) 



* 





Average mark (Technical science; health science) 
* 
* 
* 
* 

* 
* 


Average mark (Technical science; health science) (only June) 




* 



* 
Average mark (Technical science; health science) (only September) 









Average mark (Humanities) 
* 
* 
* 
* 

* 
* 


Average mark (only June) 




* 



* 
Average mark (only September) 









Pupils that passed the University Entrance Exam over students enrolled in that course 
* 
* 
* 
* 
* 

* 
* 

Pupils that passed the University Entrance Exam over students enrolled in that course (only June) 









Pupils that passed the University Entrance Exam over students enrolled in that course (only September) 









Pupils that passed the University Entrance Exam over studentswho take exam 





* 


* 
Source: Own elaboration
Table 6 contains the results obtained for the Spearman correlation coefficient for the schools for the academic years 95/96, 96/97, 97/98 and 98/99. This data allows us to check whether the changes that occurred in the variables significantly alter the results obtained using the original model (DEA1). For the academic years 95/96, 96/97 and 98/99 the correlation coefficients provide high values in all cases except where a particular relevant variable was omitted. Within this group of models the DEA3, DEA7, and DEA8 stand out as being the most significant if we look at table 6. In the academic year 97/98, the results of the Spearman correlation coefficients produce high values for all of the specifications.
TABLE 6 SENSITIVITY ANALYSIS (Coef. Correlation Spearman) 
CURSO 9596 
DEA1 
DEA2 
DEA3 
DEA4 
DEA5 
DEA6 
DEA7 
DEA8 
DEA9 
1 
0,884 
0,751 
0,848 
0,932 
0,927 
0,638 
0,705 
0,814 
CURSO 9697 
DEA1 
DEA2 
DEA3 
DEA4 
DEA5 
DEA6 
DEA7 
DEA8 
DEA9 
1 
0,963 
0,860 
0,804 
0,959 
0,911 
0,648 
0,705 
0,912 
CURSO 9798 
DEA1 
DEA2 
DEA3 
DEA4 
DEA5 
DEA6 
DEA7 
DEA8 
DEA9 
1 
0,889 
0,820 
0,787 
0,967 
0,910 
0,836 
0,840 
0,839 
CURSO 9899 
DEA1 
DEA2 
DEA3 
DEA4 
DEA5 
DEA6 
DEA7 
DEA8 
DEA9 
1 
0,861 
0,492 
0,784 
0,936 
0,821 
0,700 
0,753 
0,733 
Source: Own elaboration
Finally, the sensitivity analysis allows us to conclude that the changes in the variables that define the production function for each of the centres does not significantly alter the results. This implies that the efficiency indices obtained for the original model are not a consequence of the models´ specifications.
7. CONCLUSIONS
The main conclusions that may be obtained from this research are as follows:
1.  The DEA technique, when used as a means of measuring technical efficiency, is highly suitable for the evaluation of educational performance.
2.  BCC models are especially useful for estimating efficiency because of their flexibility when it comes to adapting the individual behaviour of each centre and informing on the optimum dimensions of these centres.
3.  The levels of efficiency of the centres that make up the sample are remarkably high.
4.  Those factors that are beyond the control of the manager play a very important role in the determination of educational results.
5.  In the case of the sample analysed, the distribution of efficient and inefficient centres is in no way related to the type of geographical environment i.e. urban, semiurban, and semirural.
6.  The sensitivity analysis carried out allows us to check that the indices obtained in using the BCC model are robust and are not simply a consequence of the specification adopted by the model.
Footnotes
1 See Bessent y Bessent (1980); Charnes, Cooper and Rhodes (1981); Bessent, Bessent, Elam and Long (1984); Jesson, Mayston and Smith (1987); Smith and Mayston (1987); Mayston Jesson (1988); Färe,Grosskoproduction function and Weber (1989); Norman and Stoker (1991); Ray (1991), Ganley and Cubbim (1992); McCarthy and Yaisawarng (1993); Thanassoulis and Dunstan (1994); Lovell, Walters and Wood (1994); Chalos and Cherian (1995), Ruggiero, Duncombe and Miner (1995), Engert (1996), Ruggiero (1996a, 1996b), Noulas and Ketkar (1998), Thanassoulis (1999), Ruggiero and Vitaliano (1999) and Mancebón and Mar Molinero (2000)
2 In our particular case, the education centres do not constitute units of decision in terms of inputs, given that this control is exercised by agents that are external to the centres themselves, although of course these inputs are managed by the education centres. The case with respect to the outputs however is rather different given that those responsible for the education centres can make and take decisions with respect to the quantity and quality of their products. This leads us to adopt the terminology, "units of decision and management" (UDM) on referring to the work of Erias, Fernández, Prado y Dopico (1998) in which the same terminology is used in order to refer to hospital centres.
3 See the work of Cohn, E. and Milamans, S. D. (1975), in which each of the elements that go to make up the education production function are described.
4 The frontier defined using these characteristics is defined by Farrel (1957), Better practise
5 The lineal programme developed by Banker, Charnes and Cooper (1984) is known as a BCC model.
6 The Primary and Secondary school systems in Spain underwent a gradual change after the passing of the 1990 education act. The most important points of this legislation include; an increase in the minimum school.leaving age from 14 to 16, changes in the actual structure of the school courses themselves, and finally, certain important modifications in the university entrance procedure. All three of these factors exacerbate the statistical problems inherent in comparing centres governed by two different strands of legislation.
7 The marks that Spanish students obtain in order to gain access to university are made up of the overall average mark obtained on completing secondary education (50%) together with a further exam aimed at measuring the student´s aptitude for tertiary education (50%).
8 It should be underlined that the elimination of some of the variables when it comes to calculating the efficiency indices will provoke a reduction in the number of results for the new specifications. Thus we only emphasise the relative alterations, which in each model provokes the elimination of one of its inputs.
9 The synthetic index was elaborated according to the social, demographic, and economic characteristics of the municipal area allowing us to determine the localization of the centres.
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About the AuthorAutor: Amparo Seijas Díaz Dirección: Universidad de A Coruña Correo electrónico: asdeai@udc.es

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