Are private schools better than public schools? Appraisal for Ireland by   methods for observational studies
  In observational studies the assignment of units to treatments is not under control. Consequently, the estimation and comparison of treatment effects based on the empirical distribution of the responses can be biased since the units exposed to the various treatments could differ in important unknown pretreatment characteristics, which are related to the response. An important example studied in this article is the question of whether private schools offer better quality of education than public schools. In order to address this question, we use data collected in the year 2000 by OECD for the Programme for International Student Assessment (PISA). Focusing for illustration on scores in mathematics of 15-year-old pupils in Ireland, we find that the raw average score of pupils in private schools is higher than of pupils in public schools. However, application of a newly proposed method for observational studies suggests that the less able pupils tend to enroll in public schools, such that their lower scores are not necessarily an indication of bad quality of the public schools. Indeed, when comparing the average score in the two types of schools after adjusting for the enrollment effects, we find quite surprisingly that public schools perform better on average. This outcome is supported by the methods of instrumental variables and latent variables, commonly used by econometricians for analyzing and evaluating social programs. 
Danny Pfeffermann, Victoria Landsman
stat.AP