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The printed formula expands to the main effects of school type, acceptance and college, together with school-type-by-college and acceptance-by-college interactions. It contains neither a school-type-by-acceptance interaction nor the corresponding three-way interaction. Its Poisson regression model is thereforeFor every college the mean table has odds ratioThus it tests conditional independence of school type and acceptance given college, while permitting both margins to vary freely across colleges. It does not merely assert a common, possibly nonzero association.

The maximum likelihood estimation score equations match every school-type-by-college and acceptance-by-college margin. Solving them gives the stratified two-by-two conditional independence model's fitted countsIndeed these counts factor by row and column and have exactly the observed margins. For nondegenerate tables there are three free mean parameters per college: one total and two independent margin proportions. There are $100$ cells and $75$ parameters, hence 25 residual degrees of freedom. The residual deviance iswith zero-count contributions interpreted as zero. The Poisson linear terms cancel because the fitted totals equal the observed totals. Under the null and suitable large-count regularity, compare $G^2$ with ${\chi }_{25}^2$. Sparse cells, structural zeros or degenerate margins require a different calibration, such as an appropriate conditional test or simulated reference distribution.

A large residual deviance is evidence of school-type/acceptance association in at least some colleges. A small one indicates compatibility with conditional independence, not proof of identical admission processes or absence of selection on qualifications. Residuals and individual college contrasts locate departures. Adding the school-type-by-acceptance term gives a common odds ratio, one extra parameter and 24 residual degrees of freedom. Adding the full three-way term permits each college's association and saturates the tables. The first comparison tests a common association; the second checks whether a common association is adequate. No frequencies are supplied here to evaluate these tests numerically.