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Local and Overall Deviance R-Squared Measures for Mixtures of Generalized Linear Models

Author

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  • Roberto Mari

    (Università di Catania)

  • Salvatore Ingrassia

    (Università di Catania)

  • Antonio Punzo

    (Università di Catania)

Abstract

In generalized linear models (GLMs), measures of lack of fit are typically defined as the deviance between two nested models, and a deviance-based R2 is commonly used to evaluate the fit. In this paper, we extend deviance measures to mixtures of GLMs, whose parameters are estimated by maximum likelihood (ML) via the EM algorithm. Such measures are defined both locally, i.e., at cluster-level, and globally, i.e., with reference to the whole sample. At the cluster-level, we propose a normalized two-term decomposition of the local deviance into explained, and unexplained local deviances. At the sample-level, we introduce an additive normalized decomposition of the total deviance into three terms, where each evaluates a different aspect of the fitted model: (1) the cluster separation on the dependent variable, (2) the proportion of the total deviance explained by the fitted model, and (3) the proportion of the total deviance which remains unexplained. We use both local and global decompositions to define, respectively, local and overall deviance R2 measures for mixtures of GLMs, which we illustrate—for Gaussian, Poisson and binomial responses—by means of a simulation study. The proposed fit measures are then used to assess, and interpret clusters of COVID-19 spread in Italy in two time points.

Suggested Citation

  • Roberto Mari & Salvatore Ingrassia & Antonio Punzo, 2023. "Local and Overall Deviance R-Squared Measures for Mixtures of Generalized Linear Models," Journal of Classification, Springer;The Classification Society, vol. 40(2), pages 233-266, July.
    • Handle: RePEc:spr:jclass:v:40:y:2023:i:2:d:10.1007_s00357-023-09432-4
    DOI: 10.1007/s00357-023-09432-4
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    References listed on IDEAS

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    1. Salvatore Ingrassia & Antonio Punzo & Giorgio Vittadini & Simona Minotti, 2015. "Erratum to: The Generalized Linear Mixed Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 32(2), pages 327-355, July.
    2. J. Cerdeira & M. Martins & Pedro Silva, 2012. "A Combinatorial Approach to Assess the Separability of Clusters," Journal of Classification, Springer;The Classification Society, vol. 29(1), pages 7-22, April.
    3. Leisch, Friedrich, 2004. "FlexMix: A General Framework for Finite Mixture Models and Latent Class Regression in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 11(i08).
    4. Salvatore Ingrassia & Antonio Punzo, 2020. "Cluster Validation for Mixtures of Regressions via the Total Sum of Squares Decomposition," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 526-547, July.
    5. Grün, Bettina & Leisch, Friedrich, 2008. "FlexMix Version 2: Finite Mixtures with Concomitant Variables and Varying and Constant Parameters," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 28(i04).
    6. Lawrence Hubert & Phipps Arabie, 1985. "Comparing partitions," Journal of Classification, Springer;The Classification Society, vol. 2(1), pages 193-218, December.
    7. Cameron, A Colin & Windmeijer, Frank A G, 1996. "R-Squared Measures for Count Data Regression Models with Applications to Health-Care Utilization," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(2), pages 209-220, April.
    8. Salvatore Ingrassia & Antonio Punzo & Giorgio Vittadini & Simona Minotti, 2015. "The Generalized Linear Mixed Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 32(1), pages 85-113, April.
    9. Michel Wedel & Wayne DeSarbo, 1995. "A mixture likelihood approach for generalized linear models," Journal of Classification, Springer;The Classification Society, vol. 12(1), pages 21-55, March.
    10. Biernacki, Christophe & Celeux, Gilles & Govaert, Gerard, 2003. "Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 561-575, January.
    11. Colin Cameron, A. & Windmeijer, Frank A. G., 1997. "An R-squared measure of goodness of fit for some common nonlinear regression models," Journal of Econometrics, Elsevier, vol. 77(2), pages 329-342, April.
    12. Celeux, Gilles & Govaert, Gerard, 1992. "A classification EM algorithm for clustering and two stochastic versions," Computational Statistics & Data Analysis, Elsevier, vol. 14(3), pages 315-332, October.
    13. Costas Panagiotakis, 2015. "Point Clustering via Voting Maximization," Journal of Classification, Springer;The Classification Society, vol. 32(2), pages 212-240, July.
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