Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical Responses

Advancements in Marginal Modeling for Categorical Data

“…The procedures and insights presented here owe much to the work of Lang and Agresti (1994), and of Grizzle Starmer and Koch (1969). Bishop et al (1975) and especially Haber (1985) developed the first more general, but still rather restricted maximum likelihood procedures for marginal models.…”

“…The restrictions may pertain to the marginal tables, but also to the dependencies in the joint table (Lang and Agresti 1994;Croon et al 2000;Vermunt et al 2001). In this way, one can estimate and test, for example, a model for Table 1 in which simultaneously marginal homogeneity is assumed for the marginals and a linear by linear (uniform) association for the turnover table itself.…”

Generalized Exponential Symmetry Model and Orthogonal Decomposition of Symmetry for Square Tables

“…For the analysis of contingency tables, Lang and Agresti [10] and Lang [11] considered the simultaneous modeling of the joint distribution and of the marginal distribution. Aitchison [12] discussed the asymptotic separability, which is equivalent to the orthogonality in Read [6] and the independence in Darroch and Silvey [13], of test statistic for the goodness-of-fit of two models (also see Lang and Agresti [10]; Lang [11]; Tomizawa and Tahata [7]; Tahata and Tomizawa [14]).…”

“…Aitchison [12] discussed the asymptotic separability, which is equivalent to the orthogonality in Read [6] and the independence in Darroch and Silvey [13], of test statistic for the goodness-of-fit of two models (also see Lang and Agresti [10]; Lang [11]; Tomizawa and Tahata [7]; Tahata and Tomizawa [14]). …”

“…Note that the matrix X is full column rank which is K . In a similar manner to Haber [15], Lang and Agresti [10], we denote the linear space spanned by the columns of the matrix X by S(X) with the dimension K . Note that X 1 1 r (r+1)/2 = 1 r 2 where 1 t is the t 1 vector of 1 elements, thus 1 r 2 …”