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Estimation of generalized linear latent variable models

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  • Philippe Huber
  • Elvezio Ronchetti
  • Maria‐Pia Victoria‐Feser

Abstract

Summary. Generalized linear latent variable models (GLLVMs), as defined by Bartholomew and Knott, enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log‐likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and can lead to biased estimators. We propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as an M‐estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent finite sample properties, in particular when compared with a well‐established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology.

Suggested Citation

  • Philippe Huber & Elvezio Ronchetti & Maria‐Pia Victoria‐Feser, 2004. "Estimation of generalized linear latent variable models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(4), pages 893-908, November.
    • Handle: RePEc:bla:jorssb:v:66:y:2004:i:4:p:893-908
    DOI: 10.1111/j.1467-9868.2004.05627.x
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    1. Y Chen & X Li, 2022. "Determining the number of factors in high-dimensional generalized latent factor models [Eigenvalue ratio test for the number of factors]," Biometrika, Biometrika Trust, vol. 109(3), pages 769-782.
    2. Björn Andersson & Tao Xin, 2021. "Estimation of Latent Regression Item Response Theory Models Using a Second-Order Laplace Approximation," Journal of Educational and Behavioral Statistics, , vol. 46(2), pages 244-265, April.
    3. Brendan P.M. McCabe & Gael Martin & Keith Freeland, 2010. "A Quasi-locally Most powerful Test for Correlation in the conditional Variance of Positive Data," Monash Econometrics and Business Statistics Working Papers 2/10, Monash University, Department of Econometrics and Business Statistics.
    4. Bianconcini, Silvia & Cagnone, Silvia, 2012. "Estimation of generalized linear latent variable models via fully exponential Laplace approximation," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 183-193.
    5. Jenni Niku & Francis K. C. Hui & Sara Taskinen & David I. Warton, 2021. "Analyzing environmental‐trait interactions in ecological communities with fourth‐corner latent variable models," Environmetrics, John Wiley & Sons, Ltd., vol. 32(6), September.
    6. Yanyuan Ma & Marc G. Genton, 2010. "Explicit estimating equations for semiparametric generalized linear latent variable models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(4), pages 475-495, September.
    7. Chen, Yunxiao & Li, Xiaoou, 2022. "Determining the number of factors in high-dimensional generalized latent factor models," LSE Research Online Documents on Economics 111574, London School of Economics and Political Science, LSE Library.
    8. Hui, Francis K.C., 2017. "Model-based simultaneous clustering and ordination of multivariate abundance data in ecology," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 1-10.
    9. Wu, Jianmin & Bentler, Peter M., 2013. "Limited information estimation in binary factor analysis: A review and extension," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 392-403.
    10. Cai, Jing-Heng & Song, Xin-Yuan & Lam, Kwok-Hap & Ip, Edward Hak-Sing, 2011. "A mixture of generalized latent variable models for mixed mode and heterogeneous data," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 2889-2907, November.
    11. Xu Ning & Francis K. C. Hui & Alan H. Welsh, 2023. "A double fixed rank kriging approach to spatial regression models with covariate measurement error," Environmetrics, John Wiley & Sons, Ltd., vol. 34(1), February.
    12. Ling Zhou & Huazhen Lin & Yi-Chen Lin, 2016. "Education, Intelligence, and Well-Being: Evidence from a Semiparametric Latent Variable Transformation Model for Multiple Outcomes of Mixed Types," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 125(3), pages 1011-1033, February.
    13. Riki Herliansyah & Ruth King & Stuart King, 2022. "Laplace Approximations for Capture–Recapture Models in the Presence of Individual Heterogeneity," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 27(3), pages 401-418, September.
    14. Vassilis Vasdekis & Silvia Cagnone & Irini Moustaki, 2012. "A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses," Psychometrika, Springer;The Psychometric Society, vol. 77(3), pages 425-441, July.
    15. Silvia Cagnone & Paola Monari, 2013. "Latent variable models for ordinal data by using the adaptive quadrature approximation," Computational Statistics, Springer, vol. 28(2), pages 597-619, April.
    16. Christopher J. Urban & Daniel J. Bauer, 2021. "A Deep Learning Algorithm for High-Dimensional Exploratory Item Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 86(1), pages 1-29, March.
    17. Leila Amiri & Mojtaba Khazaei & Mojtaba Ganjali, 2018. "A mixture latent variable model for modeling mixed data in heterogeneous populations and its applications," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 102(1), pages 95-115, January.
    18. Jenni Niku & David I. Warton & Francis K. C. Hui & Sara Taskinen, 2017. "Generalized Linear Latent Variable Models for Multivariate Count and Biomass Data in Ecology," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 22(4), pages 498-522, December.
    19. Lee, Donghwan & Lee, Youngjo & Paik, Myunghee Cho & Kenward, Michael G., 2013. "Robust inference using hierarchical likelihood approach for heavy-tailed longitudinal outcomes with missing data: An alternative to inverse probability weighted generalized estimating equations," Computational Statistics & Data Analysis, Elsevier, vol. 59(C), pages 171-179.
    20. Li Cai, 2010. "High-dimensional Exploratory Item Factor Analysis by A Metropolis–Hastings Robbins–Monro Algorithm," Psychometrika, Springer;The Psychometric Society, vol. 75(1), pages 33-57, March.
    21. Jaeun Choi & Jianwen Cai & Donglin Zeng, 2017. "Penalized Likelihood Approach for Simultaneous Analysis of Survival Time and Binary Longitudinal Outcome," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 190-216, November.
    22. Gaillard, Gabrielle, 2007. "Equity and Health," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 8(4), pages 67-80.
    23. Vitoratou, Silia & Ntzoufras, Ioannis & Moustaki, Irini, 2016. "Explaining the behavior of joint and marginal Monte Carlo estimators in latent variable models with independence assumptions," LSE Research Online Documents on Economics 57685, London School of Economics and Political Science, LSE Library.
    24. Ting Fung Ma & Fangfang Wang & Jun Zhu, 2023. "On generalized latent factor modeling and inference for high‐dimensional binomial data," Biometrics, The International Biometric Society, vol. 79(3), pages 2311-2320, September.
    25. Andersson, Björn & Jin, Shaobo & Zhang, Maoxin, 2023. "Fast estimation of multiple group generalized linear latent variable models for categorical observed variables," Computational Statistics & Data Analysis, Elsevier, vol. 182(C).

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