Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical Responses

Abstract

We discuss model-fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses. The models are members of a broad class of generalized logit and loglinear models. We fit them by improving a maximum likelihood algorithm that uses Lagrange's method of undetermined multipliers and a Newton-Raphson iterative scheme. We also discuss goodness-of-fit tests and adjusted residuals, and give asymptotic distributions of model parameter estimators. For this class of models, inferences are equivalent for Poisson and multinomial sampling assumptions. Simultaneous models for joint and marginal distributions may be useful in a variety of applications, including studies dealing with longitudinal data, multiple indicators in opinion research, cross-over designs, social mobility, and inter-rater agreement. The models are illustrated for one such application, using data from a recent General Social Survey regarding opinions about various types of government spending.

Double linear diagonals-parameter symmetry and decomposition of double symmetry for square tables

“…Also note that A 3 1 R(R+2)/4 = 1 R 2 . We denote the linear space spanned by the columns of the matrix A by S(A) with the dimension K , in a similar manner to Haber (1985); Lang and Agresti (1994), and Tomizawa and Tahata (2007). Let U be an R 2 × d 1 , where d 1 = R 2 − K = (3R 2 − 2R − 8)/4, full column rank matrix such that the linear space spanned by the column of U , i.e., S(U ), is the orthogonal complement of the space S(A).…”

Orthogonal decomposition of point-symmetry for multiway tables

“…Note that the matrix X has full column rank which is K. In a similar manner to Haber (1985), and Lang and Agresti (1994), we denote the linear space spanned by the columns of the matrix X by S(X) with the dimension K. Let U be an r 2 × d 1 full column rank matrix, where d 1 = r 2 − K = (r − 1) 2 /2, such that the linear space spanned by the columns of U , i.e., S(U ), is the orthogonal complement of the space S(X).…”

“…Aitchison (1962) discussed the asymptotic separability, which is equivalent to the orthogonality in Read (1977) and the independence in Darroch and Silvey (1963) of the test statistic for goodness of fit of two models (also see Lang and Agresti 1994;Lang 1996;Tomizawa, 1992Tomizawa, , 1993Tomizawa and Tahata 2007). For the r T table, we shall consider the orthogonality (i.e., separability or independence) of test statistics for decomposition of the P T model into the QP T h and MP T h models.…”

“…For the analysis of contingency tables, Lang and Agresti (1994) and Lang (1996) considered the simultaneous modeling of the joint distribution and of the marginal Table 2 Numbers of degrees of freedom for various point-symmetry models applied to the r T table…”

Analyzing Change Among Developmental Stages with Categorical Models

“…A practical restriction is to equating parameters for effects XA and XB. The resulting model is called quasi-symmetric latent-class (QLC) model (Lang and Agresti, 1994),…”