Log-linear model

Additive main effects encode mutual independence in contingency-table Poisson models. An interaction permits association of its factors.

For cell counts

Y_{ij}

in a two-way contingency table, the independence model is

Y_{ij} \sim Pois (μ_{ij}), lo g μ_{ij} = λ + λ_{i}^{(1)} + λ_{j}^{(2)} .

The absence of an interaction makes

μ_{ij}

factor as a row effect times a column effect. Its maximum-likelihood fitted values are

μ_{ij} = \frac{y _{i +} y _{+ j}}{y _{++}} .

Saturated log-linear model

A saturated log-linear model has enough parameters to reproduce every observed contingency-table cell count and therefore has zero residual deviance.

For treatment groups Control, LD, and SD and binary outcomes, equal LD and SD efficacy can be imposed while retaining separate group totals by using separate treatment main effects but one shared treated-versus-control interaction with outcome. Comparing this five-parameter model with the saturated six-parameter model gives a one-degree-of-freedom analysis of deviance for nested generalized linear models.