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Extending Tests of Random Effects to Assess for Measurement Invariance in Factor Models

Author

Listed:
  • Zhenzhen Zhang

    (University of Michigan)

  • Thomas M. Braun

    (University of Michigan)

  • Karen E. Peterson

    (University of Michigan)

  • Howard Hu

    (University of Washington)

  • Martha M. Téllez-Rojo

    (National Institute of Public Health)

  • Brisa N. Sánchez

    (University of Michigan)

Abstract

Factor analysis models are widely used in health research to summarize hard-to-measure predictor or outcome variable constructs. For example, in the ELEMENT study, factor models are used to summarize lead exposure biomarkers which are thought to indirectly measure prenatal exposure to lead. Classic latent factor models are fitted assuming that factor loadings are constant across all covariate levels (e.g., maternal age in ELEMENT); that is, measurement invariance (MI) is assumed. When the MI is not met, measurement bias is introduced. Traditionally, MI is examined by defining subgroups of the data based on covariates, fitting multi-group factor analysis, and testing differences in factor loadings across covariate groups. In this paper, we develop novel tests of measurement invariance by modeling the factor loadings as varying coefficients, i.e., letting the factor loading vary across continuous covariate values instead of groups. These varying coefficients are estimated using penalized splines, where spline coefficients are penalized by treating them as random coefficients. The test of MI is then carried out by conducting a likelihood ratio test for the null hypothesis that the variance of the random spline coefficients equals zero. We use a Monte Carlo EM algorithm for estimation, and obtain the likelihood using Monte Carlo integration. Using simulations, we compare the Type I error and power of our testing approach and the multi-group testing method. We apply the proposed methods to summarize data on prenatal biomarkers of lead exposure from the ELEMENT study and find violations of MI due to maternal age.

Suggested Citation

  • Zhenzhen Zhang & Thomas M. Braun & Karen E. Peterson & Howard Hu & Martha M. Téllez-Rojo & Brisa N. Sánchez, 2018. "Extending Tests of Random Effects to Assess for Measurement Invariance in Factor Models," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 10(3), pages 634-650, December.
    • Handle: RePEc:spr:stabio:v:10:y:2018:i:3:d:10.1007_s12561-018-9222-7
    DOI: 10.1007/s12561-018-9222-7
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    References listed on IDEAS

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