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Multilevel Heterogeneous Factor Analysis and Application to Ecological Momentary Assessment

Published online by Cambridge University Press:  01 January 2025

Junhao Pan ,
Edward Haksing Ip Open the ORCID record for Edward Haksing Ip [Opens in a new window]  and
Laurette Dubé
Junhao Pan
Affiliation:
Sun Yat-sen University
Edward Haksing Ip*
Affiliation:
Wake Forest School of Medicine
Laurette Dubé
Affiliation:
McGill University
*
Correspondence should be made to Edward Haksing Ip, Wake Forest School of Medicine, Winston-Salem, USA. Email: eip@wakehealth.edu URL: https://www.phs.wakehealth.edu/public/profile.cfm?staffid=57C5A0EA-1FF4-425F-BFE1-86EFB3A3CF72
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Abstract

Ansari et al. (Psychometrika 67:49–77, 2002) applied a multilevel heterogeneous model for confirmatory factor analysis to repeated measurements on individuals. While the mean and factor loadings in this model vary across individuals, its factor structure is invariant. Allowing the individual-level residuals to be correlated is an important means to alleviate the restriction imposed by configural invariance. We relax the diagonality assumption of residual covariance matrix and estimate it using a formal Bayesian Lasso method. The approach improves goodness of fit and avoids ad hoc one-at-a-time manipulation of entries in the covariance matrix via modification indexes. We illustrate the approach using simulation studies and real data from an ecological momentary assessment.

Keywords

confirmatory factor analysisecological momentary assessmentresidual covariance matrixrandom effectsMCMC methods
Type
Theory and Methods
Information
Psychometrika , Volume 85 , Issue 1 , March 2020 , pp. 75 - 100
Copyright
Copyright © 2019 The Psychometric Society

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References

Ansari, A., Jedidi, K., &Dubé, L., (2002). Heterogeneous factor analysis models: A Bayesian approach. Psychometrika, 67 49–77CrossRefGoogle Scholar
Arminger, G., & Stein, P., (1997). Finite mixtures of covariance structure models with regressors. Sociological Methods and Research, 26 148–182CrossRefGoogle Scholar
Asparouhov, T., Hamaker, E.L., & Muthén, B., (2017). Dynamic structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 25(3), 359–388CrossRefGoogle Scholar
Averill, J.R., Catlin, G., Chon, K.K.,(1990). Rules of Hope,New York, NY:SpringerCrossRefGoogle Scholar
Benjamini, Y., & Yekutieli, D., (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics, 29 1165–1188CrossRefGoogle Scholar
Bollen, K.A., Structural equation models, 1989 New York:WileyGoogle Scholar
Borkenau, P., & Ostendorf, F., (1998). The Big Five as states: How useful is the five-factor model to describe intra-individual variations over time?. Journal of Personality Research, 32 202–221CrossRefGoogle Scholar
Bringmann, L.F., Vissers, N., Wichers, M., Geschwind, N., Kuppens, P., Peeters, F., etal (2013). A network approach to psychopathology: New insights into clinical longitudinal data. PLoS ONE, 8(4), e60188CrossRefGoogle ScholarPubMed
Browne, M.W.,& Nesselroade, J.R., Maydeu-Olivares, A., McArdle, J.J., (2005). Representing psychological processes with dynamic factor models: Some promising uses and extensions of ARMA time series models.Contemporary psychometrics: A Festschrift for Roderick P. McDonald, Mahwah, NJ:Erlbaum 415–452Google Scholar
Chow, S.M.,Tang, N., Yuan, Y., Song, X., & Zhu, H., (2011). Bayesian estimation of semiparametric dynamic latent variable models using the dirichlet process prior. British Journal of Mathematical and Statistical Psychology, 64(1)69–106CrossRefGoogle Scholar
Csikszentmihalyi, M., & Larson, R., (1987). Validity and reliability of the experience-sampling method. Journal of Nervous and Mental Disease, 175 526–536CrossRefGoogle ScholarPubMed
De Roover, K.D.,Vermunt, J.K.,Timmerman, M.E.,&Ceulemans, E.,(2017). Mixture simultaneous factor analysis for latent variables between higher level units of multilevel data. Structural Equation Modeling: A Multidisciplinary Journal, 24(4), 506–523CrossRefGoogle Scholar
Diener, E., & Emmons, R.A., (1984). The independence of positive and negative affect. Journal of Personality and Social Psychology, 47 1105–11176520704CrossRefGoogle ScholarPubMed
Dunn, E.C.,Masyn, K.E.,Johston, W.R.,&Subramanian, S.V., (2015). Modeling contextual effects using individual-level data and without aggregation: An illustration of multilevel factor analysis (MLFA) with collective efficacy. Population Health Metrics, 13 12–22CrossRefGoogle ScholarPubMed
Gelfand, A.E.,Gilks, W.R.,Richardson, S., Spiegelhalter, D.J., (1996). Model determination using sampling-based methods.Markov Chain Monte Carlo in Practice, London:Chapman & Hall 145–161Google Scholar
Gelman, A., & Gilks, W.R.,Richardson, S., Spiegelharter, D.J., (1996). Inference and monitoring convergence.Markov Chain Monte Carlo in Practice, London:Chapman & Hall 131–144Google Scholar
Gelman, A., Roberts, G.O.,Gilks, W.R., Bernardo, J.M.,Berger, J.O.,Dawid, A.P., etal(1996). Efficient metropolis jumping rules.Bayesian statistics, New York:Oxford University Press 599–607CrossRefGoogle Scholar
Gelman, A., & Meng, X.L., (1998). Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statistical Science, 13 163–185CrossRefGoogle Scholar
Geman, S., & Geman, D., (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741CrossRefGoogle ScholarPubMed
Gilks, W.R.,Richardson, S., & Spiegelhalter, D.J., Markov chain Monte Carlo in practice, 1996 London:Chapman & HallGoogle Scholar
Geisser, S., & Eddy, E., (1979). A predictive approach to model selection. Journal of the American Statistical Association, 74, 153–160CrossRefGoogle Scholar
Goldstein, H., & Browne, W., Maydeu-Olivares, A., McArdle, J.J.,(2005). Multilevel factor analysis models for continuous and discrete data.Contemporary psychometrics: A festschrift for Roderick P McDonald, Mahwah, MJ:Erlbaum 453–475Google Scholar
Goldstein, H., Healy, MJR,& Rasbash, J., (1994). Multilevel time series models with applications to repeated measure data. Statistics in Medicine, 13, 1643–1655CrossRefGoogle Scholar
Hans, C.,(2009). Bayesian lasso regression. Biometrika, 96(4), 835–845CrossRefGoogle Scholar
Hastings, W.K.,(1970). Monte Carlo sampling methods using Markov chains and their application. Biometrika, 57, 97–109CrossRefGoogle Scholar
Heck, R.H.,Thomas, S.L.,Heck, R.H.,(1999). Multilevel modeling with SEM.Introduction to multilevel modeling techniques, Mahwah, NJ:Lawrence Erlbaum Associates Inc 89–127CrossRefGoogle Scholar
Jöreskog, K.G.,&Sörbom, D., LISREL VI user’s guide, 1984 Mooresville, IN:Scientific SoftwareGoogle Scholar
Kaplan, D., (1990). Evaluating and modifying covariance structure models: A review and recommendation. Multivariate Behavioral Research, 25, 137–155CrossRefGoogle ScholarPubMed
Kass, R.E.,&Raftery, A.E.,(1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795CrossRefGoogle Scholar
Kelderman, H.,&Molenaar, P.C.,(2007). The effect of individual differences in factor loadings on the standard factor model. Multivariate Behavioral Research, 42(3), 435–456CrossRefGoogle Scholar
Khondker, Z.S.,Zhu, H.T.,Chu, H.T.,Lin, W.L.,&Ibrahim, J.G.,(2013). The Bayesian covariance lasso. Statistics and Its Interface, 6(2), 243–259Google ScholarPubMed
Krone, T., Albers, C.J.,Kuppens, P., &Timmerman, M.E.,(2018). A multivariate statistical model for emotion dynamics. Emotion, 18(5), 739–754CrossRefGoogle ScholarPubMed
Lee, S.Y., Structural equation modelling: A Bayesian approach, 2007 New York:WileyCrossRefGoogle Scholar
Lee, S.Y.,&Song, X.Y., Basic and advanced Bayesian structural equation modeling: With applications in the medical and behavioral sciences, 2012 New York:WileyGoogle Scholar
Longford, N.T.,&Muthén, B.O.,(1992). Factor analysis for clustered observations. Psychometrika, 57, 581–597CrossRefGoogle Scholar
Lopes, H.F.,&West, M., (2004). Bayesian model assessment in factor analysis. Statistica Sinica, 14, 41–67Google Scholar
Lu, Z.H.,Chow, S.M.,&Loken, E., (2016). Bayesian factor analysis as a variable-selection problem: Alternative priors and consequences. Multivariate Behavioral Research, 51(4), 519–539CrossRefGoogle ScholarPubMed
Lu, J., Huet, C., &Dubé, L., (2011). Emotional reinforcement as a protective factor for healthy eating in home settings. The American Journal of Clinical Nutrition, 94(1), 254–261CrossRefGoogle ScholarPubMed
Maydeu-Olivares, A., &Coffman, D.L.,(2006). Random intercept item factor analysis. Psychological Methods, 11(4), 344–362CrossRefGoogle ScholarPubMed
MacCallum, R.C.,Hoyle, R.H.,(1995). Model specification: Procedures, strategies, and related issues.Structural equation modeling: Concepts, issues, and applications, Thousand Oaks, CA:SAGEGoogle Scholar
MacCallum, R.C.,Roznowski, M., &Necowitz, L.B.,(1992). Model modifications in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111(3), 490–504CrossRefGoogle ScholarPubMed
McArdle, J.J.,(1982). Structural equation modeling of an individual system: Preliminary results from ‘A case study in episodic alcoholism’. Department of Psychology, University of Denver. (Unpublished manuscript).Google Scholar
Metropolis, N., Rosenbluth, A.W.,Rosenbluth, M.N.,&Teller, A.H.,Teller, E., (1953). Equations of state calculations by fast computing machine. Journal of Chemical Physics, 21, 1087–1091CrossRefGoogle Scholar
Merz, E.L.,&Roesch, S.C.,(2011). Modeling trait and state variation using multilevel factor analysis with panas daily diary data. Journal of Research in Personality, 45(1), 2–9CrossRefGoogle ScholarPubMed
Molenaar, PCM (1985). A dynamic factor model for the analysis of multivariate time series. Psychometrika, 50(2), 181–202CrossRefGoogle Scholar
Molenaar, PCM, &Campbell, C.G.,(2009). The new person-specific paradigm in psychology. Current Directions in Psychological Science, 18(2), 112–117CrossRefGoogle Scholar
Muthén, B.O.,(1991). Multilevel factor analysis of class and student achievement components. Journal of Educational Measurement, 28, 338–354CrossRefGoogle Scholar
Muthén, B.O.,(1994). Multilevel covariance structure analysis. Sociological methods and Research, 22, 376–398CrossRefGoogle Scholar
Muthén, B.O.,Hancock, G.R.,Samuelsen, K.M.,(2007). Latent variable hybrids: Overview of old and new models.Advances in latent variable mixture models, Charlotte, NC:Information Age 1–24Google Scholar
Muthén, L.K., & Muthén, B.O. (1998–2013). Mplus User’s Guide., 7th Edition. Los Angeles, CA: Muthén & Muthén.Google Scholar
Nesselroade, J.R.,McArdle, J.J.,Aggen, S.H.,Meyers, J.M.,Moskowitz, D.M.,&Hershberger, S.L.,(2002). Alternative dynamic factor models for multivariate time-series analyses.Modeling intraindividual variability with repeated measures data: Advances and techniques, Mahwah, NJ:Erlbaum 235–265Google Scholar
O’Malley, A.J.,&Zaslavsky, A.M.,(2008). Domain-level covariance analysis for multilevel survey data with structured nonresponse. Journal of the American Statistical Association, 103(484), 1405–1418CrossRefGoogle Scholar
Park, T., &Casella, G., (2008). The Bayesian lasso. Journal of the American Statistical Association, 103(482), 681–686CrossRefGoogle Scholar
Pan, J.H.,Ip, E.H.,&Dubé, L.,(2017). An alternative to post-hoc model modification in confirmatory factor analysis: The Bayesian Lasso. Psychological Methods, 22(4), 687–704292658485745070CrossRefGoogle ScholarPubMed
Reise, S.P.,Kim, D.S.,Mansolf, M., &Widaman, K.F.,(2016). Is the bifactor model a better model or is it just better at modeling implausible responses? Application of iteratively reweighted least squares to the rosenberg self-esteem scale. Multivariate Behavioral Research, 51(6), 818–838278345095312782Google ScholarPubMed
Reise, S.P.,Ventura, J., Nuechterlein, K.H.,&Kim, K.H.,(2005). An illustration of multilevel factor analysis. Journal of Personality Assessment, 84, 126–13615799887CrossRefGoogle ScholarPubMed
Schmukle, S.C.,Egloff, B., &Burns, L.R.,(2002). The relationship between positive and negative affect in the positive and negative affect schedule. Journal of Research in Personality, 36(5), 463–475CrossRefGoogle Scholar
Shiffman, S., Stone, A., Turkkan, J., Jobe, J., etal (2000). Real-time self-report of momentary states in the natural environment: Computerized Ecological Momentary Assessment.The science of self-report: Implications for research and practice, Mahwah, NJ:Lawrence Erlbaum Associates 277–296Google Scholar
Song, H., &Ferrer, E., (2012). Bayesian estimation of random coefficient dynamic factor models. Multivariate Behavioral Research, 47(1), 26–60CrossRefGoogle Scholar
Song, X.Y.,Tang, N.S.,Chow, S.M.,(2012). A bayesian approach for generalized random coefficient structural equation models for longitudinal data with adjacent time effects. Computational Statistics & Data Analysis, 56(12), 4190–4203CrossRefGoogle Scholar
Sörbom, D., (1989). Model modification. Psychometrika, 54(3), 371–384CrossRefGoogle Scholar
Stakhovych, S., Bijmolt, T.H.,&Wedel, M., (2012). Spatial dependence and heterogeneity in Bayesian factor analysis: A cross-national investigation of Schwartz values. Multivariate Behavioral Research, 47(6), 803–839CrossRefGoogle ScholarPubMed
Tanner, M.A.,&Wong, W.H.,(1987). The calculation of posterior distributions by data augmentation (with discussion). Journal of the American Statistical Association, 82, 528–550CrossRefGoogle Scholar
Wang, H., (2012). Bayesian graphical lasso models and efficient posterior computation. Bayesian Analysis, 7(4), 867–886CrossRefGoogle Scholar
Yung, Y.F.,(1997). Finite mixtures in confirmatory factor analysis models. Psychometrika, 62, 297–330CrossRefGoogle Scholar
Zhang, Z., &Nesselroade, J.R.,(2007). Bayesian estimation of categorical dynamic factor models. Multivariate Behavioral Research, 42, 729–756CrossRefGoogle Scholar