Data envelopment analysis efficiency in the public sector using provider and customer opinion: An application to the Spanish health system.

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.

Classification of Samples with Order-Restricted Discriminant Rules.

In recent years, mass spectrometry techniques have helped proteomics to become a powerful tool for the early diagnosis of cancer, as they help to discover protein profiles specific to each pathological state. One of the questions where proteomics is giving useful practical results is that of classifying patients into one of the possible severity levels of an illness, based on some features measured on the patient. This classification is usually made using one of the many discrimination procedures available in statistical literature. We present in this chapter recently developed restricted discriminant rules that use additional information in terms of orderings on the means, and we illustrate how to apply them to mass spectrometry data using R package dawai. Specifically, we use proteomic prostate cancer data, and we describe all steps needed, including data preprocessing and feature extraction, to build a discriminant rule that classifies samples in one of several disease stages, thus helping diagnosis. The restricted discriminant rules are compared with some standard classifiers that do not take into account the additional information, showing better performance in terms of error rates.

Performance and estimation of the true error rate of classification rules built with additional information. An application to a cancer trial.

Classification rules that incorporate additional information usually present in discrimination problems are receiving certain attention during the last years as they perform better than the usual rules. Fernández, M. A., C. Rueda and B. Salvador (2006): "Incorporating additional information to normal linear discriminant rules," J. Am. Stat. Assoc., 101, 569-577, proved that these rules have lower total misclassification probability than the usual Fisher's rule. In this paper we consider two issues; on the one hand, we compare these rules with those based on shrinkage estimators of the mean proposed by Tong, T., L. Chen and H. Zhao (2012): "Improved mean estimation and its application to diagonal discriminant analysis," Bioinformatics, 28(4): 531-537. with regard to four criteria: total misclassification probability, area under ROC curve, well-calibratedness and refinement; on the other hand, we consider the estimation of the true error rate, which is a very interesting parameter in applications. We prove results on the apparent error rate of the rules that expose the need of new estimators of their true error rate. We propose four such new estimators. Two of them are defined incorporating the additional information into the leave-one-out-bootstrap. The other two are the corresponding cross-validation after bootstrap versions. We compare these estimators with the usual ones in a simulation study and in a cancer trial application, showing the good behavior of the rules that incorporate additional information and of the new leave-one-out bootstrap estimators of their true error rate.