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.

A Multistep Iterative Proportional Fitting Procedure to Estimate Cohortwise Interregional Migration Tables Where Only Inconsistent Marginals are Known

The interregional cohort survival model developed by Rogers is an excellent one that includes all of the three population processes: birth-death, aging, and interregional migration. Rogers's model, however, has been rarely implemented because it requires detailed data concerning cohortwise (that is by age and sex) interregional migration tables which are not usually available as published data. The most usual case is that only marginal tables that are from different sources can be obtained. However, in this case, those marginal tables necessarily show inconsistency in the sense that they do not have identical common submarginals. This inconsistency prevents the standard iterative proportional fitting (IPF) procedure from converging to the estimate of the complete migration table which conforms to the given observed marginals. Thus to implement Rogers's model some method is needed to estimate the complete migration table where only inconsistent marginals are available. In this paper a multistep IPF procedure is proposed for that purpose and an actual application of the proposed method is shown. The multistep IPF procedure has universal applicability to a wide class of general problems concerned with the estimation of a joint table under inconsistent marginals.