Connectivity for 3 x 3 x K contingency tables

We consider an exact sequential conditional test for three-way conditional test of nointeraction. At each time , the test uses as the conditional inference frame the set F(H ) of alltables with the same three two-way marginal tables as the obtained table H . For 33K tables,we propose a method to construct F(H ) from F(H􀀀1). This enables us to perform ecientlythe sequential exact conditional test. The subset S of F(H ) consisting of s + H 􀀀 H􀀀1 fors 2 F(H􀀀1) contains H , where the operations + and 􀀀 are dened elementwise. Our argumentis based on the minimal Markov basis for 3 3 K contingency tables and we give a minimalsubset M of some Markov basis which has the property that F(H ) = fs 􀀀 m j s 2 S ;m 2 Mg.