  The analysis of contingency tables is a powerful statistical tool used in
experiments with categorical variables. This study improves parts of the theory
underlying the use of contingency tables. Specifically, the linkage
disequilibrium parameter as a measure of two-way interactions applied to
three-way tables makes it possible to quantify Simpsons paradox by a simple
formula. With tests on three-way interactions, there is only one that
determines whether the partial interactions of all variables agree or whether
there is at least one variable whose partial interactions disagree. To date,
there has been no test available that determines whether the partial
interactions of a certain variable agree or disagree, and the presented work
closes this gap. This work reveals the relation of the multiplicative and the
additive measure of a three-way interaction. Another contribution addresses the
question of which cells in a contingency table are fixed when the first- and
second-order marginal totals are given. The proposed procedure not only detects
fixed zero counts but also fixed positive counts. This impacts the
determination of the degrees of freedom. Furthermore, limitations of methods
that simulate contingency tables with given pairwise associations are
addressed.
