Log-Linear Models for Contingency Tables

A simple chi-squared test on a contingency table can tell you two categorical variables aren't independent, but it can't easily extend to three or more variables, or tell you which combination of variables is driving the pattern. Log-linear models solve this by treating the expected count in each cell, $\mu_{ijk}$, the way ordinary regression treats a continuous outcome — except the "outcome" being modeled is $\log \mu_{ijk}$, and the "predictors" are indicator terms for each category and each combination of categories, exactly mirroring the main-effect-plus-interaction structure of an ANOVA table.

For a two-way table, the saturated model is $\log \mu_{ij} = \lambda + \lambda^A_i + \lambda^B_j + \lambda^{AB}_{ij}$: an overall level $\lambda$, a main effect for row category $i$, a main effect for column category $j$, and an interaction term $\lambda^{AB}_{ij}$ capturing anything beyond independence. Testing whether that interaction term can be dropped (set to zero) without hurting fit is mathematically the same test as an independence chi-squared test on the two-way table — log-linear models are a strict generalization, not a different technique. Once a third variable $C$ is added, the model can include a three-way interaction $\lambda^{ABC}_{ijk}$, and dropping just that term while keeping all two-way interactions tests specifically for whether the association between $A$ and $B$ is the same across every level of $C$ — a more precise question than any single chi-squared test can answer.

Model fit is compared using deviance (a likelihood-ratio statistic), and nested models are compared by testing whether the drop in deviance from adding a term is large relative to its degrees of freedom, exactly as with nested regression models.