Cross-Classified Data

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 W1, W2,…, Wm, say, is drawn from a discrete d-variate distribution where each trial Wk=(W₁k,...,W _d k)’ has common joint probability mass function: 4.1.1 $${p_i}: = P[W_1^k = {i_1}, \ldots ,W_d^k = {i_d}],$$ . Here the support of W _j k 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 $$m = \sum\nolimits_{i \in x} {{Y_{i\cdot }}} $$ .