An analysis of test scores adjusted for the regression-to-the-mean effect based on incomplete data

Abstract

In this study, we consider a situation where students who obtained low scores in their test results are given a remedial class and administered an after-remedial test to check the learning effects of the class. A beta-binomial distribution is assumed for the model of the test scores, namely, the number of correct answers in a test consisting of n questions. In addition, we consider the more common situation where the after-remedial test consists of m questions. We present the expectation value and variance of the score of the after-remedial test and the difference from the before-remedial test. Moreover, a degree of incompleteness in the before-remedial data is classified into three situations: selection, censoring, and truncation. For each situation, practical estimation procedures of the beta binomial distribution parameters are provided to fit the model by using the moment estimators. In every situation, we find that the statistical test adjusted for the regression-to-the-mean effect is appropriate. Thus, the result suggests that the application of proper tests is important for assessing the learning effect.