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Marginal Models: An Overview

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Trends and Challenges in Categorical Data Analysis

Part of the book series: Statistics for Social and Behavioral Sciences ((SSBS))

Abstract

Marginal models involve restrictions on the conditional and marginal association structure of a set of categorical variables. They generalize log-linear models for contingency tables, which are the fundamental tools for modelling the conditional association structure. This chapter gives an overview of the development of marginal models during the past 20 years. After providing some motivating examples, the first few sections focus on the definition and characteristics of marginal models. Specifically, we show how their fundamental properties can be understood from the properties of marginal log-linear parameterizations. Algorithms for estimating marginal models are discussed, focussing on the maximum likelihood and the generalized estimating equations approaches. It is shown how marginal models can help to understand directed graphical and path models, and a description is given of marginal models with latent variables.

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Notes

  1. 1.

    The design also has disadvantages, of course. These include panel attrition and the fact that, even if originally selected appropriately, with passing time the sample will become different in composition from the current population.

  2. 2.

    See Bergsma et al. [12], Chapter 5, for applications in causal analysis and the relationship with structural equation models.

  3. 3.

    The neighbours of a node A are those nodes with which A is connected by an edge.

  4. 4.

    For alternative parameterizations see Sect. 3.5.

  5. 5.

    Note that formula (iii) in Theorem 3.1 in Ghosh and Vellaisamy [38] appears to have a typo.

  6. 6.

    A model is called smooth if it admits a smooth parameterization.

  7. 7.

    The notations for the variables are going to be clarified later.

  8. 8.

    Setting a parameter to zero means setting it zero for all category combinations of the variables in the effect.

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Authors and Affiliations

  1. Department of Statistics, Faculty of Social Sciences, Eötvös Loránd University, Budapest, Hungary

    Tamás Rudas

  2. Department of Statistics, London School of Economics and Political Science, London, UK

    Wicher Bergsma

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  1. Tamás Rudas
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  2. Wicher Bergsma
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Correspondence to Wicher Bergsma .

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Editors and Affiliations

  1. Department of Mathematics, RWTH Aachen University, Aachen, Germany

    Maria Kateri

  2. Department of Statistics, London School of Econ. & Polit. Science, London, UK

    Irini Moustaki

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Rudas, T., Bergsma, W. (2023). Marginal Models: An Overview. In: Kateri, M., Moustaki, I. (eds) Trends and Challenges in Categorical Data Analysis. Statistics for Social and Behavioral Sciences. Springer, Cham. https://doi.org/10.1007/978-3-031-31186-4_3

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