Cross-Classified Data

Abstract

This chapter describes the use of classical likelihood methods and loglinear models to analyze cross-classified data. Cross-classified data arise when a random sample W ¹, W ²,…, W ^m, say, is drawn from a discrete d-variate distribution where each trial W ^k=(W ^k₁ ,...,W ^k _d )’ has common joint probability mass function:

$${p_i}: = P[W_1^k = {i_1}, \ldots ,W_d^k = {i_d}],$$

(4.1.1)

. Here the support of W ^k _j is taken to be $\{ 1, \ldots ,{L_j}\}$ without loss of generality. The symbol W, without a superscript, will be used to denote a generic classification variable with probability mass function (4.1.1). By sufficiency, the data can be summarized as the counts $\{ {Y_i}:i \in x\}$ in a d-dimensional contingency table where Y _i is the number of vectors W which equal i. Thus the counts $\{ {Y_i}:i \in x\}$ have the M _t ( m, p ) multinomial distribution where $p = \{ {p_i}:i \in X\} ,\sum\nolimits_{i \in x} {{p_i}} = 1,t = \Pi _{j = 1}^d{L_j},$, and $m = \sum\nolimits_{i \in x} {{Y_{i\cdot }}}$.