Association Model

GENERAL SETUP FOR A TWO-WAY CROSS-CLASSIFIED TABLE

For the cell of the ith row and the jth column (i = 1,…, I, and j = 1,…, J) in a two-way table of R and C, let fij denote the observed frequency and Fij the expected frequency under some model. Without loss of generality, a log-linear model for the table can be written as follows:

where μ is the main effect, μR is the row effect, μC is the column effect, and μRC is the interaction effect on the logarithm of the expected frequency. All the parameters in equation (1) are subject to ANOVA-type normalization constraints (see Powers & Xie, 2000, pp. 108–110). It is common to leave μR and μC unconstrained and estimated nonparametrically. This practice is also called the “saturation” of the marginal distributions of the row and column variables. What is of special interest is μRC: At one extreme, μRC may all be zero, resulting in an independence model. At another extreme, μRC may be “saturated,” taking (I − 1)(J − 1) degrees of freedom, yielding exact predictions (Fij = fij for all i and j).

[Page 30]

Typically, the researcher is interested in fitting models between the two extreme cases by altering specifications for μRC. It is easy to show that all odds ratios in a two-way table are functions of the interaction parameters (μRC). Let θij denote a local log-odds ratio for a 2 × 2 subtable formed from four adjacent cells obtained from two adjacent row categories and two adjacent column categories:

Let us assume that the row and column variables are ordinal on some scales x and y. The scales may be observed or latent. A linear-by-linear association model is as follows: