Stratified two-by-two conditional independence model (${\hat{\mu }}_{abj}=n_{a+j}{n}_{+bj}/n_{++j}$)

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.