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Statistical inference in constrained latent class models for multinomial data based on $$\phi $$ ϕ -divergence measures

Author

Listed:
  • A. Felipe

    (Complutense University of Madrid)

  • N. Martín

    (Complutense University of Madrid)

  • P. Miranda

    (Complutense University of Madrid)

  • L. Pardo

    (Complutense University of Madrid)

Abstract

In this paper we explore the possibilities of applying $$\phi $$ ϕ -divergence measures in inferential problems in the field of latent class models (LCMs) for multinomial data. We first treat the problem of estimating the model parameters. As explained below, minimum $$\phi $$ ϕ -divergence estimators (M $$\phi $$ ϕ Es) considered in this paper are a natural extension of the maximum likelihood estimator (MLE), the usual estimator for this problem; we study the asymptotic properties of M $$\phi $$ ϕ Es, showing that they share the same asymptotic distribution as the MLE. To compare the efficiency of the M $$\phi $$ ϕ Es when the sample size is not big enough to apply the asymptotic results, we have carried out an extensive simulation study; from this study, we conclude that there are estimators in this family that are competitive with the MLE. Next, we deal with the problem of testing whether a LCM for multinomial data fits a data set; again, $$\phi $$ ϕ -divergence measures can be used to generate a family of test statistics generalizing both the classical likelihood ratio test and the chi-squared test statistics. Finally, we treat the problem of choosing the best model out of a sequence of nested LCMs; as before, $$\phi $$ ϕ -divergence measures can handle the problem and we derive a family of $$\phi $$ ϕ -divergence test statistics based on them; we study the asymptotic behavior of these test statistics, showing that it is the same as the classical test statistics. A simulation study for small and moderate sample sizes shows that there are some test statistics in the family that can compete with the classical likelihood ratio and the chi-squared test statistics.

Suggested Citation

  • A. Felipe & N. Martín & P. Miranda & L. Pardo, 2018. "Statistical inference in constrained latent class models for multinomial data based on $$\phi $$ ϕ -divergence measures," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(3), pages 605-636, September.
    • Handle: RePEc:spr:advdac:v:12:y:2018:i:3:d:10.1007_s11634-017-0289-7
    DOI: 10.1007/s11634-017-0289-7
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    References listed on IDEAS

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    1. Ewa Genge, 2014. "A latent class analysis of the public attitude towards the euro adoption in Poland," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 8(4), pages 427-442, December.
    2. Bartolucci F. & Forcina A., 2002. "Extended RC Association Models Allowing for Order Restrictions and Marginal Modeling," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1192-1199, December.
    3. A. Felipe & P. Miranda & L. Pardo, 2015. "Minimum $$\phi $$ ϕ -Divergence Estimation in Constrained Latent Class Models for Binary Data," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 1020-1042, December.
    4. Albert Satorra & Peter Bentler, 2010. "Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic," Psychometrika, Springer;The Psychometric Society, vol. 75(2), pages 243-248, June.
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