Two-Dimensional Contingency Tables with Both Completely and Partially Cross-Classified Data

Models are developed for the analysis of contingency table data with sup~lemental marginal totals.The method of maximum likelihood is used to estimate the parameters in the models, and the expected cell values for goodness-of-fit statistics.The value of utilizing the suppiemental margins is discussed in terms of asymptotic variances and the consistency of estimates.The approach developed is illustrated in a 2 x 2 table example.-~ of the one we adopt in this paper.Reinfurt [1970], Koch and Reinfurt [1970],and Koch, Imrey and Reinfurt [1972] use a modified minimum chi-squared approach to the contingency table version of the Blumenthal-Hocking-0xspring problem.In this paper we describe methods for obtaining maximum likelihood estimates for expected cell values in contingency tables with partially cross-classified data.We consider two models for the basic cross-classification (unrestricted and independeme) .and a special class of "random" mechanisms which produce the partially cross-classified data, explaining how they relate to the completely cross-classified data.First we obtain maximum likelihood estimates for the parameters of the different models, then we use these to get estimated expected cell values and associated goodness-of-fit statistics.