Abstract
Measuring the relative efficiency of a finite fixed set of service-producing units (hospitals, state services, libraries, banks,...) is an important purpose of Data Envelopment Analysis (DEA). We illustrate an innovative way to measure this efficiency using stochastic indexes of the quality from these services. The indexes obtained from the opinion-satisfaction of the customers are estimators, from the statistical view point, of the quality of the service received (outputs); while, the quality of the offered service is estimated with opinion-satisfaction indexes of service providers (inputs). The estimation of these indicators is only possible by asking a customer and provider sample, in each service, through surveys. The technical efficiency score, obtained using the classic DEA models and estimated quality indicators, is an estimator of the unknown population efficiency that would be obtained if in each one of the services, interviews from all their customers and all their providers were available. With the object of achieving the best precision in the estimate, we propose results to determine the sample size of customers and providers needed so that with their answers can achieve a fixed accuracy in the estimation of the population efficiency of these service-producing units through the use of a novel one bootstrap confidence interval. Using this bootstrap methodology and quality opinion indexes obtained from two surveys, one of doctors and another of patients, we analyze the efficiency in the health care system of Spain.
Highlights
Data Envelopment Analysis (DEA) has become a widely used technique to compare the efficiency of serviceproducing units because it handles the multiple outputs characteristic of public sector production, is nonparametric and does not require input price data [34]
We propose to determine the provider sample size in the service-producing units (SPUs) as i.e., if we consider items to estimate the customer opinion indexes the provider sample size in the SPU is determined as where is the sample size necessary to achieve
Output-oriented CCR and BCC model we look at the SPUs in which the expected efficiency confidence interval contains the one, 1, 2, 4, 7, 10, 11, 12, these SPUs coincide with the efficient population units
Summary
Data Envelopment Analysis (DEA) has become a widely used technique to compare the efficiency of serviceproducing units because it handles the multiple outputs characteristic of public sector production, is nonparametric and does not require input price data [34]. This population efficiency is an unknown non-evaluable parameter, since it would be necessary to use the indexes obtained with the opinion of all the customers and all the providers of all the services as data and this is a census of the entire population [40] This statistical analysis of the DEA efficiency gives rise to the problem of determining the sample size of customers and providers needed to guarantee an a priori fixed accuracy in the estimation, with confidence interval, of the DEA efficiency of each public service, which is the object of our investigation. The population efficiency, or census efficiency, is unknown because, in order to know it, it would be necessary to interview all the providers and all the customers (census) of each one of the public services and, with this population data, to obtain the opinion indexes and use them as input/output data in the classic LP model CCR or BCC of Table 1; in real applications, these censuses are completely non-viable. A rigorous proof of all the results are provided in the Appendix I
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