Robust Nonparametric Group Comparisons With Ordinally Scaled Variables In Psychology 
                                        by Andras Vargha
Institute of Psychology, E”tv”s Lor nd University, Budapest, Hungary

Empirical studies carried out in the social and behavioral domain provide enough evidence that several types of non-normally distributed variables occur widely in psychology and related areas.  Often these variables are not only non-normal, but of qualitative nature where only ordinality holds. Still we may want to compare several treatments or groups to decide if data obtained from a specific source (treatment, situation, population, etc.) are generally greater or smaller than those coming from other sources.

The aim of the present paper is to give an overview of nonparametric statistical procedures and the underlying mathematical models that can be used for a stochastic comparison of several groups with respect to ordinally scaled variables.  In the two-group case stochastic equality (STE) is defined by 
P(X > Y) = P(X < Y).  In the multigroup case each group is assigned a stochastic treatment effect, which indicates the extent of stochastic dominance of the population represented by this group over the
union of all populations.  Then, stochastic homogeneity (STH) of the populations to be compared with respect to the given dependent variable is defined as the equality of these stochastic treatment effects.

The paper overviews a number of statistical procedures for testing the null hypotheses of STE and STH, and presents the results of a recent Monte Carlo study.  Finally, the paper demonstrates the difficulties that arise in generalizing the concept of STH to multiway layouts.  These problems are mostly due to the intransitivity of STE, which may occasionally lead to curious circumstances, a couple of which are listed as follows.

- The set of populations A, B, and C is stochastically homogeneous, while A < B, B < C, and C < A hold in terms of pairwise stochastic comparisons.

- If the model of two-way stochastic comparisons is defined by means of stochastic treatment effects assigned each combination of the levels of the two grouping factors (that is each cell), then a two-way stochastic interaction may be present when there is a complete lack of two-way interaction in the corresponding parametric ANOVA model even for normally distributed dependent variables, and vice versa.

- In the above model it may occur that all cells have identical treatment effects, implying a complete lack of stochastic main effects and stochastic interaction, and at the same time the levels of factor B are stochastically heterogeneous at each level of factor A.