On the closure of relational models

Relational models for contingency tables are generalizations of log-linear models, allowing effects associated with arbitrary subsets of cells in a possibly incomplete table, and not necessarily containing the overall effect. In this generality, the MLEs under Poisson and multinomial sampling are not always identical. This paper deals with the theory of maximum likelihood estimation in the case when there are observed zeros in the data. A unique MLE to such data is shown to always exist in the set of pointwise limits of sequences of distributions in the original model. This set is equal to the closure of the original model with respect to the Bregman information divergence. The same variant of iterative scaling may be used to compute the MLE in the original model and in its closure

Relational models for contingency tables

The paper considers general multiplicative models for complete and incomplete contingency tables that generalize log-linear and several other models and are entirely coordinate free. Sufficient conditions of the existence of maximum likelihood estimates under these models are given, and it is shown that the usual equivalence between multinomial and Poisson likelihoods holds if and only if an overall effect is present in the model. If such an effect is not assumed, the model becomes a curved exponential family and a related mixed parameterization is given that relies on non-homogeneous odds ratios. Several examples are presented to illustrate the properties and use of such models

On the application of discrete marginal graphical models

Graphical models are defined by general and possibly complex conditional independence assumptions and are well suited to model direct and indirect associations and effects that are of central importance in many problems of sociology. Such relevance is apparent in research on social mobility. This article provides a unified view of many of the graphical models discussed in a largely scattered literature. The marginal modeling framework proposed here relies on parameters that capture aspects of associations among the variables that are relevant for the graph and, depending on the substantive problem at hand, may lead to deeper insight than other approaches. In this context, model search, which uses a sequence of nested models, means the restriction of increasing subsets of parameters. As a special case, general path models for categorical data are introduced. These models are applied to the social status attainment process, generating substantive results and gaining new insights into the difference between liberal and conservative welfare systems. To help others use these models, all details of the analyses are posted on the Web site for this article at http://nemethr.web.elte.hu/discrete-graphical-models/. Researchers can thus easily modify the analyses to their own data and models

Rejoinder : On the Application of Discrete Marginal Graphical Models

Tematic issue on marginal modeling. Rejoinder to dicussants

Discrete Graphical Models in Social Mobility Research - A Comparative Analysis of American, Czechoslovakian and Hungarian Mobility before the Collapse of State Socialism

Variants of path models have been widely used for the analysis of the social status attainment process. The methods presented here differ from earlier approaches in several ways. Social status is considered a categorical variable and path models are developed starting from graphical models, using the marginal log-linear approach. Overall model fit may be tested by standard techniques. Under these models, the status attainment process is completely characterized by a set of parameters that measure the strengths of the relevant effects. This is in sharp contrast with estimating and interpreting ad hoc parameters, without paying attention to overall model fit and to other effects influencing the process. The method is applied to the social status attainment process in the USA, Hungary and Czechoslovakia at the end of the last century, and shows that policies in the latter socialist countries to prevent status inheritance had little success

On the use of historical estimates

The use of historical, i.e., already existing, estimates in current studies is common in a wide variety of application areas. Nevertheless, despite their routine use, the uncertainty associated with historical estimates is rarely properly accounted for in the analysis. In this communication, we review common practices and then provide a mathematical formulation and a principled frequentist methodology for addressing the problem of drawing inferences in the presence of historical estimates. Three distinct variants are investigated in detail; the corresponding limiting distributions are found and compared. The design of future studies, given historical data, is also explored and relations with a variety of other well-studied statistical problems discussed