Connectivity for 3 x 3 x K contingency tables
We consider an exact sequential conditional test for three-way conditional test of no interaction.
At each time ( \tau ), the test uses as the conditional inference frame the set
(\mathcal{F}(H_\tau)) of all tables with the same three two-way marginal tables as the obtained table (H_\tau).
For (3\times 3\times K) tables, we propose a method to construct (\mathcal{F}) ((H_\tau)) from (\mathcal{F}(H_{\tau-1})). This enables us to perform efficiently the sequential exact conditional test. The subset (\mathcal{S}_\tau) of (\mathcal{F}(H_{\tau})) consisting of (s+H_\tau-H_{\tau-1}) for (s \in \mathcal{F}(H_{\tau-1})) contains (H_{\tau}), where the operations (+) and (-) are defined elementwise. Our argument is based on the minimal Markov basis for (3 \times 3 \times K) contingency tables and we give a minimal subset (\mathcal{M}) of some Markov basis
which has the property that (\mathcal{F}(H_{\tau}=\{s-m \mid s \in \mathcal{S}_\tau, m\in \mathcal{M}\}).
