Rasch Model

The Rasch model is a psychometric model used in the social sciences to analyze categorical response data, usually collected using a content knowledge test or attitudinal questionnaire, in order to assess the extent to which a set of persons has a certain level of an attribute of interest (e.g., mathematical proficiency or level of anxiety) and the extent to which a positive answer to the questions or statements demands a certain level of that attribute (e.g., difficulty of a mathematical question or level of anxiety required in order to agree or endorse a statement). Originally developed in the 1950s by Danish mathematician Georg Rasch for the analysis of dichotomous responses to intelligence tests, the Rasch model in its most basic form states that the probability that a person can correctly answer a test question—or that he or she endorses a given statement—can be modeled as a function of the difference between an effect associated with the person (e.g., a student’s mathematical proficiency or potentially any person property) and a question effect (e.g., the item difficulty or level of agreement demanded by a statement), such that both effects are in the same scale and

$$\begin{array}{l}\text{Probability of person}p\text{correctly answering item}\\ i=f\left(\text{Person}p\text{proficiency }-\text{ item}i\text{difficulty}\right).\end{array}$$

In other words, the Rasch model considers the performance of a person on a question to be a product of the trade-off between a person effect (oftentimes interpreted as the level of person proficiency) and an item effect (commonly interpreted as the difficulty of the item). The Rasch model is [Page 1371]widely applied in the social sciences, particularly in the context of psychometric analysis of educational testing, where it is often considered to be a special case within item response theory (IRT), and, more generally, as a special case of a generalized linear model (GLM) in statistics. Although it can formally be understood as a special case within item response theory or GLMs, the Rasch model was developed by Rasch under a specific set of theoretical commitments regarding the nature of measurement in the social sciences and the requirements that a model must fulfill in order to be used for measurement, which have historically set apart the use and development of the Rasch model and its extensions from the wider psychometric and statistical literature.