Estimation of treatment effects in observational studies by recovering the assignment probabilities and the population model

In observational studies the assignment of units to treatments is with unknown probabilities. Consequently, estimation and comparison of treatment effects based on the empirical distributions of the response under the various treatments can be biased since units exposed to one treatment could differ in important but unknown characteristics from units exposed to other treatments. In this article we study the plausibility of analyzing observational data by deriving the parametric distribution of the observed response under a given treatment as a function of the distribution that would be obtained under a strongly ignorable assignment, and the assignment process, which is modeled as a function of the observed data (the response and covariate values). The use of this approach is founded by showing that the sample distribution of the observed responses is identifiable under some general conditions. The goodness of fit of this distribution can be tested by using standard test statistics since it refers to the observed data, but we also develop a new test. The proposed approach allows also testing the assumptions underlying the use of methods that employ instrumental variables, or methods that use propensity scores with a given set of covariates.We assess the performance of the proposed approach and compare it to existing approaches using data collected in the year 2000 by OECD for the Programme for International Student Assessment (PISA). In the present application we compare students’ scores in mathematics between public and private schools in Ireland and conclude, somewhat surprisingly, that the public schools perform better than the private schools. This finding is supported by one of the existing methods as well.