Data envelopment analysis with common-denominator ratio variables: An application to education with methodological extensions
The use of ratio variables in convex Data Envelopment Analysis (DEA) models has long been recognized as a methodological issue, as ratios may be incompatible with the convexity assumptions underlying standard DEA. However, in this paper, we prove that if all variables are ratio variables sharing a common denominator, then convex combinations of feasible activities are also feasible. Moreover, we establish a result demonstrating the equivalence between DEA models with common-denominator ratio variables under variable returns to scale and DEA models with volume (non-ratio) variables under constant returns to scale. These significant results enable the development of a framework for evaluating the efficiency of the Organisation for Economic Co-operation and Development (OECD) countries based on the results of the Programme for International Student Assessment (PISA) report, using mean performance scores as outputs. In this framework, we give some methodological extensions, such as the incorporation of the index of economic social and cultural status (ESCS) as an input, thereby enabling fairer comparisons with countries with a lower socio-economic level. Furthermore, we introduce different methods for estimating directions of improvement and calculating targets appropriate to the difficulty of improving each performance score. Finally, we review and introduce several novel contributions to emerging methodologies that can complement classical radial and directional models, such as efficient frontier estimation with adaptive constrained enveloping splines (ACES), stochastic chance-constrained models, and fuzzy models. All these methodologies can be used to analyse data from other PISA or similar reports, allowing non-specialists to implement DEA appropriately.