Joseph B. Lang

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 Lagranges 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 s...

A Bigger Piece of the Pie? State Corrections Spending and the Politics of Social Order

The dramatic increase in American state prison populations during the past three decades has sparked considerable research interest. Empirical research has most often examined changes in prison admissions or populations, but few studies have considered shifts in state corrections budgets. This study examines variation in annual, state-level corrections expenditures as a proportion of state expenditures from 1980 to 1998, drawing together existing theoretical arguments about criminal punishment under a common rubric that focuses on state responsibility for the maintenance of social order and the need for state officials to maintain office through popular election. From this view, partisan politics, economic and racial threats, citizen preferences, fiscal considerations, policy priorities, and crime are important explanations of corrections spending because they affect strategies for maintaining social order, garnering votes, and maintaining political office. Findings generally support this perspective. Partisan politics, racial threats, state economic prosperity, and budgetary priorities all play a role in determining state corrections expenditures.

Homogeneous Linear Predictor Models for Contingency Tables

Maximum likelihood (ML) inference for the class of homogeneous linear predictor (HLP) models for contingency tables is described. HLP models constrain the expected table counts m through L(m) = Xβ, where the link L is allowed to be a many-to-one, nonlinear function. Generalized log-linear, association trend, marginal cumulative probit, and conditional marginal homogeneity models are given as specific examples. ML fit results, which include point estimates, goodness-of-fit statistics, and asymptotic-based approximate distributions, are described and compared for equivalent HLP models. The results are valid for a wide variety of sampling plans including combinations of product multinomial and Poisson sampling. An important practical implication of this article is that the implementation of ML fitting and theory is straightforward, and an attractive alternative to weighted least squares estimation, for HLP models.

Association-Marginal Modeling of Multivariate Categorical Responses: A Maximum Likelihood Approach

Abstract Generalized log-linear models can be used to describe the association structure and/or the marginal distributions of multivariate categorical responses. We simultaneously model the association structure and marginal distributions using association-marginal (AM) models, which are specially formulated generalized log-linear models that combine two models: an association (A) model, which describes the association among all the responses; and a marginal (M) model, which describes the marginal distributions of the responses. Because the models composite link function is not required to be invertible, a large class of models can be entertained and model specification is typically straightforward. We propose a “mixed freedom/constraint” parameterization that exploits the special structure of an AM model. Using this parameterization, maximum likelihood fitting is straightforward and typically feasible for large, sparse tables. When a parsimonious association model is used, the size of the fitting proble...

On the Partitioning of Goodness-of-Fit Statistics for Multivariate Categorical Response Models

Abstract Numerical and asymptotic stochastic partitioning of goodness-of-fit statistics are considered for a broad class of simultaneous multivariate categorical response models. These simultaneous models impose constraints on the joint and marginal distributions of categorical response variables. Under certain conditions, the tenability of the corresponding simultaneous hypothesis can be assessed by separately testing the two subhypotheses: one regarding the joint distributions and the other regarding the marginal distributions. Specifically, easily verifiable sufficient conditions are introduced that allow us to partition the overall goodness-of-fit statistic into two interesting goodness-of-fit statistics: one for testing whether the joint distribution model holds and the other for testing whether the marginal distribution model holds. Moreover, it is proven that when the sufficient conditions hold and the simultaneous hypothesis is true, the two component goodness-of-fit statistics are asymptotically ...

Bayesian ordinal and binary regression models with a parametric family of mixture links

An ordinal and binary regression model with parametric link is introduced. The link is a member of a one-parameter family of “mixture links”, a family that comprises smooth mixtures of the extreme minimum-value, extreme maximum-value, and logistic distributions. A Bayesian version of this flexible model serves as a vehicle for introducing a priori information regarding the choice of link. Owing to non-conjugacy, posterior and predictive distributions are approximated using Markov chain Monte Carlo simulation methods. Link-independent, Bayesian interpretations of covariate effects are described. The method is illustrated through the analyses of several data sets.

Application of Association-Marginal Models to the Study of Social Mobility

Researchers studying social mobility are often interested in examining both the association between origins and destinations and the relationship between the marginal distributions of origins and destinations. Often, this has resulted in an attempt to partition various models into components of, or derive indexes for, exchange/circulation mobility and structural mobility. As an alternative, or perhaps supplement, to such concerns, here the authors present a relatively simple but useful way to directly and simultaneously model the association between origins and destinations on one hand and, on the other hand, the relationship between origin and destination marginal distributions.

Comparing marginal distributions of large, sparse contingency tables

Abstract The feasibility of maximum likelihood (ML) analyses of models for marginal distributions of contingency tables diminishes as the numbers of margins and response categories increases. This article describes alternative approaches that are much more feasible. We recommend a “pseudo ML” approach that obtains model parameter estimates by treating repeated responses as independent and uses a jackknife to estimate the covariance matrix of those estimates. We test marginal homogeneity using a Wald statistic, or by adapting the efficient score statistic from the independent-samples case. We illustrate these approaches with a seven-dimensional table having 78 125 cells, and we give simulation results that show no substantive loss of efficiency from using pseudo ML estimates.

Improved tests of independence in singly-ordered two-way contingency tables

A new approach is described for improving statistical tests of independence between two categorical variables R and C , where C is ordinal and R may or may not be ordinal. Common tests of independence that exploit the ordinality of C use a restricted-alternative approach. A different, relaxed-null approach to improving tests of independence is considered. Specifically, the M -moment score test is introduced and shown to be an attractive alternative to well known restricted-alternative tests, such as the row-means Cochran-Mantel-Haenszel test, the Kruskal-Wallis test, and the likelihood-ratio test based on the cumulative-logit row-effects model or the log-linear row-effects model. Unlike these restricted-alternative tests, which are designed to detect location shifts, the M -moment score test is designed to be powerful for detecting shifts in any of the first M conditional moments of C across the values of R . Using multinomial-Poisson homogeneous modeling theory, the M -moment score tests are shown to be computationally and conceptually simple, with an attractive complement consistency property. Results of a simulation study compare the M -moment score test to several other commonly-used tests on the basis of their operating characteristics.

Order restricted inference for hypotheses concerning qualitative dispersion

We discuss likelihood-based order restricted inference for hypotheses concerning the qualitative dispersion associated with a probability vector; levels of dispersion are compared using the concept of Schur majorization. In particular, we show how to compute the maximum likelihood estimates of a probability vector under certain equality and inequality constraints on the dispersions, and show that the corresponding estimators are consistent. Asymptotic null distributions of the likelihood ratio statistics are also derived. Two data sets are analyzed using the methods developed in this paper.