Tests of marginal homogeneity in a two-way contingency table given by [1], [3], and [13] do not seem to lend themselves easily to extension to the problem of m-way marginal homogeneity in an N-way r \times r \times \cdots \times r contingency table, m < N. The principle of minimum discrimination information estimation and the associated minimum discrimination information statistic applied in [5] to the problem of marginal homogeneity in an r \times r contingency table can be easily extended to the case of a multidimensional contingency table. Estimates of the cell entries under the hypotheses of m-way marginal homogeneity are given. Relationships among the tests of homogeneity for m-way, m = 1, 2, \cdots, N - 1, marginals are given by an analysis of information. Numerical results are given for two sample 3 \times 3 \times 3 tables, and two 5 \times 5 tables.