Tamás Rudas

A Monte Carlo comparison of the small sample behaviour of the Pearson, the likelihood ratio and the Cressie-Read statistics

There has been a long debate on the applicability of the chi-square approximation to statistics based on small sample sizes. Most of the work done in this area is concerned with the Pearson chi-square and the likelihood ratio stastics. This paper reports results of a simulation study on the small sample behaviour of these statistics adding a statistic suggested by Cressie and Read (1984). Simulated percentage points and approximate 95% confidence intervals for the true percentage points are given for 13 contingency tables fitting log-linear models. It is found that the behaviour of the Cressie-Read statistic is very similar to that of the Pearson chi-square statistic. Both seem to be applicable for smaller sample sizes than suggested by other authors.

Parameterization and estimation of path models for categorical data

The paper discusses statistical models for categorical data based on directed acyclic graphs (DAGs) assuming that only effects associated with the arrows of the graph exist. Graphical models based on DAGs are similar, but allow the existence of effects not directly associated with any of the arrows. Graphical models based on DAGs are marginal models and are best parameterized by using hierarchical marginal log-linear parameters. Path models are defined here by assuming that all hierarchical marginal log-linear parameters not associated with an arrow are zero, providing a parameterization with straightforward interpretation. The paper gives a brief review of log-linear, graphical and marginal models, presents a method for the maximum likelihood estimation of path models and illustrates the use of path models, with special emphasis on the interpretation of estimated parameter values, to real data.

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.

Coordinate-free analysis of trends in British social mobility

This paper is intended to make a contribution to the ongoing debate about declining social mobility in Great Britain by analyzing mobility tables based on data from the 1991 British Household Panel Survey and the 2005 General Household Survey. The models proposed here generalize Hausers levels models and allow for a semi-parametric analysis of change in social mobility. The cell frequencies are assumed to be equal to the product of three effects: the effect of the fathers position for the given year, the effect of the sons position for the given year, and the mobility effect related to the difference between the fathers and the sons positions. A generalization of the iterative proportional fitting procedure is proposed and applied to computing the maximum likelihood estimates of the cell frequencies. The standard errors of the estimated parameters are computed under the product-multinomial sampling assumption. The results indicate opposing trends of mobility between the two timepoints. Fewer steps up or down in the society became less likely, while more steps became somewhat more likely.

Prescribed conditional interaction structure models with application to the analysis of mobility tables

The present paper considers some new models for the analysis of multidimensional contigency tables. Although the theoretical background used here appeared already in Haberman (1974), prescribed conditional interaction (PCIN) models were introduced by Rudas (1987) and their mathematical properties were worked out by Leimer and Rudas (1988). These models are defined by prescribing the values of certain conditional interactions in the contingency table. Conditional interaction is defined here as the logarithm of an appropriately defined conditional odds ratio. This conditional odds ratio is a conditional version of a generalization of the well known odds ratio of a 2×2 table and that of the three factor interaction term of a 2×2×2 table and applies to any number of dimensions and any number of categories of the variables. The well known log-linear (LL) models are special PCIN models. Estimated frequencies under PCIN models and tests of fit can be computed using existing statistical software (e.g. BMDP). The paper describes the class of PCIN models and compares it to the class of association models of Goodman (1981). As LL models are widely used in the analysis of social mobility tables, application of more general PCIN models is illustrated.

Invariant Hierarchical Clustering Schemes

Summary. A general parametric scheme of hierarchical clustering procedures with invariance under monotone transformations of similarity values and invariance under numeration of objects is described. This scheme consists of two steps: correction of given similarity values between objects and transitive closure of obtained valued relation. Some theoretical properties of considered scheme are studied. Different parametric classes of clustering procedures from this scheme based on perceptions like “keep similarity classes,” “break bridges between clusters,” etc. are considered. Several examples are used to illustrate the application of proposed clustering procedures to analysis of similarity structures of data.

Canonical representation of log-linear models

ABSTRACT Log-linear models for the distribution on a contingency table are represented as the intersection of only two kinds of log-linear models. One assuming that a certain group of the variables, if conditioned on all other variables, has a jointly independent distribution and another one assuming that a certain group of the variables, if conditioned on all other variables, has no highest order interaction. The subsets entering into these models are uniquely determined by the original log-linear model. This canonical representation suggests considering joint conditional independence and conditional no highest order association as the elementary building blocks of log-linear models.

The π* Index as a New Alternative for Assessing Goodness of Fit of Logistic Regression

In this paper the 7π* index of fit introduced by Rudas et al. [9] is applied to the model of logistic regression. First, the original definition of π* is given with its interpretation, then a review is given on logistic regression focusing on how to assess model fit in traditional ways. Assessing fit often requires grouping of the data and the main part of this paper is concerned with methods for grouping the data and choosing computational technics. These are illustrated using a standard set of data.

Divergence minimization under prior inequality constraints

Motivated by problems in robust statistics we first give a simple proof of the following: Given a probability measure P and positive measures /spl mu/</spl nu/, the /spl gamma/-divergence from P of probability measures Q satisfying /spl mu//spl les/Q or /spl mu//spl les/Q/spl les//spl nu/ is minimized by an explicitly determined Q/sup */ not depending on (the convex function) /spl gamma/. Next we address /spl gamma/-divergence minimization under the above inequality constraint and additional moment constraints.