Log-linear model

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

Independent Poisson distributions with a separate nuisance intercept in every stratum become multinomial distributions after conditioning on their totals. Profiling the intercept gives $e^{{\hat{\gamma }}_s}=N_s/{\sum }_l{e}^{x_s^T{\beta }_l}$. Substitution produces the same parameter-dependent log-likelihood as the conditional multinomial model. Both approaches fit ${\hat{\mu }}_{sl}=N_s{\hat{p}}_{sl}$. This exact equivalence differs from approximating a single binomial event count by a rare-event Poisson distribution.

For independent Poisson cell counts in stratified two-by-two contingency tables, the model $\log {\mu }_{abj}={\gamma }_j+{\alpha }_{aj}+{\beta }_{bj}$ allows both margins to vary by stratum but excludes within-stratum association. Its fitted means are the displayed products of margins, because the Poisson regression score equations match those margins. Every stratum's odds ratio is one. With positive nondegenerate margins, $J$ strata give $3J$ parameters and $J$ residual degrees of freedom. A common association adds one parameter; stratum-specific associations saturate the tables.

For $c>0$ and a centered normal distribution prior, this conditional log-posterior has second derivative $-c{e}^{\alpha }-s^{-2}<0$. It is strictly concave but does not belong to a normal distribution or gamma distribution family. A random-walk Metropolis algorithm, or a locally scaled proposal around its unique mode with the full proposal correction, can update it. Gaussian conjugacy is unavailable because the Poisson distribution likelihood contains an exponential rate.

For cell counts $Y_{ij}$ in a two-way contingency table, the independence model isThe absence of an interaction makes ${\mu }_{ij}$ factor as a row effect times a column effect. Its maximum-likelihood fitted values are

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.