Three-way contingency tables

Abstract

A three-way contingency table is an array of observed values x_ijk, i=1,…, I, j=1,…,J, k=1,…,K of I×J×K random variables, arranged in I rows, J columns and K layers. As model for the corresponding random variables, we choose

$$ {X_{111, \cdots ,}}{X_{IJK}} \sim M\left( {n;{\pi _{111,L,}}{\pi _{IJK}}} \right), $$

(3.1)

that is a multinomial distribution with number parameter n and probability parameters π_ijk, where

$$ n = x \ldots = \sum\limits_{i} {\sum\limits_{j} {\sum\limits_{k} {{{x}_{{ijk}}}} } } $$