This paper is concerned with a theoretical discussion of two simple, discrete probability distributions for the analyses of two-way categorical data and an application for testing independence hypothesis. The two probability models are both members of the exponential family. These are the hypergeometric distribution and the dichotomous Rasch Model (Rasch, 1960). The hypergeometric distribution determines the probability distribution of the simplest case of a two-way data structure, a 2 x 2 contingency table with mutually exclusive categories (or cells) given the marginal sums of the table. From the dichotomous Rasch Model (Rasch, 1960) the probability distribution of (0,1) matrices with given marginals can be derived. The relations between the two probability models will be discussed and it will be shown that under some restrictions the former results as a special case of the latter. The paper particularly points at the (mathematical) elegance of the two models under consideration. In both cases the discrete probability distribution of the contingency table and the (0,1) matrix respectively with given marginal sums can be determined by combinatorial considerations only. For both probability models the marginal sums of the table and the (0,1) matrix respectively are sufficient statistics. Both probability models are central distributions depending on the given values of the marginal sums (of the table and the matrix respectively) only. It will also be shown that the central distribution determined by the Rasch Model can be extended to a distribution that is not centralized. With regard to the hypergeometric distribution the extension to a non-central distribution was already proposed by Ronald A. Fisher (1954). The resulting probability model was later called extended hypergeometric distribution (Harkness, 1965). As far as applications are concerned the goal of this paper is to give an idea of how to extend the possible fields of applications of the Rasch Model. It will be shown that the Rasch Model can be applied in general for the analyses of two- way categorical observations for the sake of testing simple independence hypothesis of two categorical variables, in analogy to Fisher’s well known exact test based on the hypergeometric distribution. ... continued in Methodology

... This makes the model applicable in many fields of experimental educational, psychological, medical, and behavioral research, as well as even more general in the natural sciences, so that it may not only be applied as a measurement model for calibrating and constructing psychological or educational tests. An example of testing such independence hypothesis with regard to educational research will be given. The paper also aims at encouraging lecturers to introduce the Rasch Model already in basic statistics courses together with other simple, discrete probability models such as, for example, the Bernoulli-, binomial-, and hypergeometric distribution.