The Analysis of Partially Categorized Contingency Data

The literature on categorized data has been primarily concerned with the analysis of contingency tables. In the usual situation, it is assumed that we have sampled a multinomial population in which the classes are described according to two or more categories and the data are summarized in a contingency table with corresponding dimension. Specifically, in a two-way table with r rows and c columns, it is assumed that the data in the rc cells follow a multinomial distribution with cell probabilities 7iij . The usual analysis consists of estimating the rii and testing certain hypotheses on these parameters. In this paper, we consider situations in which the data have been taken from such a multinomial population, but, because of partial categorization of some of the observations, the summary is in the form of two or more related tables. For example, we may have data ni;, i = 1, 2; j = 1, 2, 3, in a 2 X 3 table from a population with six classes described by two categories with two and three classes, respectively. Let us assume that in addition, we have data mij , i = 1, 2; j = 1, 2, in a 2 X 2 table whose description is derived from the original table in the sense that the first two column classifications have been combined. For the purpose of gaining additional precision in the analysis, it is desired to combine the information in the two sets of data. If the data nii are assumed to follow a multinomial distribution with parameters irij, = 1, 2; j = 1, 2, 3, then the data mij are also multinomial with parameters (7ril + 7r12), 713 , (X21 + r22), r23 . Assuming that the two sets of data are independent, we may obtain maximum likelihood estimates of the parameters from the combined data using the procedure described by Hocking and Oxspring 1971 (hereafter abbreviated (HO)). The purpose of this paper is to illustrate the application of (HO) to the estimation of parameters from contingency data when, in addition to the basic table, we have tables which arise because row and/or column classifications have been combined. In some cases, the partially categorized data arises because of the nature of the data and the manner in which it was collected. Alternatively, the experiment may have been intentionally designed