Hostname: page-component-599cfd5f84-9drbd Total loading time: 0 Render date: 2025-01-07T05:24:57.346Z Has data issue: false hasContentIssue false
  • English
  • Français

A Finite Mixture of Nonlinear Random Coefficient Models for Continuous Repeated Measures Data

Published online by Cambridge University Press:  01 January 2025

Nidhi Kohli ,
Jeffrey R. Harring  and
Cengiz Zopluoglu
Nidhi Kohli*
Affiliation:
University of Minnesota
Jeffrey R. Harring
Affiliation:
University of Maryland
Cengiz Zopluoglu
Affiliation:
University of Miami
*
Correspondence should be made to Nidhi Kohli, Quantitative Methods in Education Program, Department of Educational Psychology, University of Minnesota, 161 Education Sciences Bldg., 56 East River Road, Minneapolis, MN 55455, USA. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation–maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.

Keywords

estimationfinite mixture modelsnonlinearrandom coefficient models
Type
Original Paper
Information
Psychometrika , Volume 81 , Issue 3 , September 2016 , pp. 851 - 880
Copyright
Copyright © 2015 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aitkin, M., & Rubin, D. B. (1985). Estimation and hypothesis testing in finite mixture models. Journal of the Royal Statistical Society Series D (Methodological), 47, 67–75.CrossRefGoogle Scholar
Arminger, G., & Stein, P., Wittenberg, J. (1999). Mixtures of conditional mean and covariance structure models. Psychometrika, 64, 475–494.CrossRefGoogle Scholar
Blozis, S. A. (2004). Structured latent curve models for the study of change in multivariate repeated measures data. Psychological Methods, 9, 334–353.CrossRefGoogle Scholar
Blozis, S. A., & Cudeck, R. (1999). Linear mixed-effects models with latent variable covariates. Journal of Educational and Behavioral Statistics, 24, 245–270.CrossRefGoogle Scholar
Boos, D., & Brownie, C. (1991). Mixture models for continuous data in dose-response studies when some animals are unaffected by treatment. Biometrics, 47, 1489–1504.CrossRefGoogle ScholarPubMed
Codd, C. L., & Cudeck, R. (2014). Nonlinear random-effects mixture models for repeated measures. Psychometrika, 79, 60–83.CrossRefGoogle ScholarPubMed
Crowder, M., & Hand, D. (1990). Analysis of repeated measures, London: Chapman and HallGoogle Scholar
Cudeck, R. (1996). Mixed-effects models in the study of individual differences with repeated measures data. Multivariate Behavioral Research, 31, 371–403.CrossRefGoogle Scholar
Cudeck, R., & du Toit, S. H. C. (2002). A nonlinear form of quadratic regression with interpretable parameters. Multivariate Behavioral Research, 37, 501–519.CrossRefGoogle ScholarPubMed
Cudeck, R., & Harring, J. R. (2007). The analysis of nonlinear patterns of change with random coefficient models. Annual Review of Psychology, 58, 615–637.CrossRefGoogle ScholarPubMed
Cudeck, R., Harring, J. R., Chow, S-M, Ferrer, E., & Hsieh, F. (2010). Developing a random coefficient model for nonlinear repeated measures data. Statistical methods for modeling human dynamics: An interdisciplinary dialogue, New York, USA: RoutledgeGoogle Scholar
Cudeck, R., & Klebe, K. J. (2002). Multiphase mixed-effects models for repeated measures data. Psychological Methods, 7, 41–63.CrossRefGoogle ScholarPubMed
Davidian, M., & Giltinan, D. M. (1995). Nonlinear models for repeated measurement data, London: Chapman & HallGoogle Scholar
du Toit, S. H. C., & Cudeck, R. (2009). Estimation of the nonlinear random coefficient model when some random effects are separable. Psychometrika, 74, 65–82.CrossRefGoogle Scholar
Everitt, B. S., & Hand, D. J. (1981). Finite mixture distributions, London: Chapman & HallCrossRefGoogle Scholar
Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2011). Applied longitudinal analysis, 2New York, USA: WileyCrossRefGoogle Scholar
Frühwirth-Schnatter, S. (2006). Finite mixture and Markov switching models, New York, USA: SpringerGoogle Scholar
Grimm, K. J., & Ram, N. (2009). Nonlinear growth models in M plus and SAS. Structural Equation Modeling: A Multidisciplinary Journal, 16, 676–701.CrossRefGoogle Scholar
Gruen, B., & Leisch, F. (2007). Fitting finite mixtures of generalized linear regressions in R. Computational Statistics & Data Analysis, 51, 5247–5252.CrossRefGoogle Scholar
Hall, D. B., & Wang, L. (2005). Two-component mixtures of generalized linear mixed effects models for cluster correlated data. Statistical Modelling, 5, 21–37.CrossRefGoogle Scholar
Hancock, G. R., & Samuelsen, K. M. (2008). Advances in latent variable mixture models, Charlotte, NC: Information Age Publishing Inc.Google Scholar
Harring, J. R. (2009). A nonlinear mixed effects model for latent variables. Journal of Educational and Behavioral Statistics, 34, 293–318.CrossRefGoogle Scholar
Harring, J. R., Harring, J. R., & Hancock, G. R. (2012). Finite mixtures of nonlinear mixed effects models. Advances in longitudinal methods in the social and behavioral sciences, Charlotte, NC: Information Age Publishing Inc 159–192.Google Scholar
Harring, J. R., Cudeck, R., & du Toit, S. H. C. (2006). Fitting partially nonlinear random coefficient models as SEMs. Multivariate Behavioral Research, 41, 579–596.CrossRefGoogle ScholarPubMed
Hartford, A., & Davidian, M. (2000). Consequences of misspecifying assumptions in nonlinear mixed effects models. Computational Statistics & Data Analysis, 34, 139–164.CrossRefGoogle Scholar
Harwell, M. R. (1997). Analyzing the results of Monte Carlo studies in item response theory. Educational and Psychological Measurement, 57, 266–278.CrossRefGoogle Scholar
Harwell, M. R., Stone, C. A., Hsu, T., & Kirisci, L. (1996). Monte Carlo studies in item response theory. Applied Psychological Measurement, 20, 101–125.CrossRefGoogle Scholar
Hipp, J. R., & Bauer, D. J. (2006). Local solutions in the estimation of growth mixture models. Psychological Methods, 11, 36–53.CrossRefGoogle ScholarPubMed
Jennrich, R. I., & Schluchter, M. D. (1986). Unbalanced repeated measures models with structured covariance matrices. Biometrics, 42, 805–820.CrossRefGoogle ScholarPubMed
Kohli, N., & Harring, J. R. (2013). Modeling growth in latent variables using a piecewise function. Multivariate Behavioral Research, 48, 370–397.CrossRefGoogle ScholarPubMed
Kohli, N., Harring, J. R., & Hancock, G. R. (2013). Piecewise linear–linear latent growth mixture models with unknown knots. Educational and Psychological Measurement, 73, 935–955.CrossRefGoogle Scholar
Laird, N. M., & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38, 963–974.CrossRefGoogle ScholarPubMed
Liu, M., & Hancock, G. R. (2014). Unrestricted mixture models for class identification in growth mixture modeling. Educational and Psychological Measurement, 74, 557–584.CrossRefGoogle Scholar
Luo, X., Boos, D. D., & Tamura, R. N. (2004). Score tests for dose effect in the presence of non-responders. Statistics in Medicine, 23, 3581–3591.CrossRefGoogle ScholarPubMed
McLachlan, G., & Peel, D. (2000). Finite mixture models, New York, USA: WileyCrossRefGoogle Scholar
Muthén, B. O., Marcoulides, G. A., & Schumacker, R. E. (2001). Latent variable mixture modeling. New developments and techniques in structural equation modeling, Mahwah, NJ: Lawrence Erlbaum Associates 1–33.Google Scholar
Muthén, B. O. (2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29, 81–117.CrossRefGoogle Scholar
Muthén, B. O., & Kaplan, D. (2004). Latent variable analysis: Growth mixture modeling and related techniques. Handbook of quantitative methodology for the social sciences, Newbury Park, CA: Sage 345–368.Google Scholar
Muthén, B. O., & Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463–469.CrossRefGoogle ScholarPubMed
Nagin, D. S. (2005). Group-based modeling of development, Cambridge, MA: Harvard University PressCrossRefGoogle Scholar
Nylund, K. L., Asparouhov, T., & Muthén, B. (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study. Structural Equation Modeling: A Multidisciplinary Journal, 14, 535–569.CrossRefGoogle Scholar
Pavlic, M., Brand, R. J., & Cumming, S. R. (2001). Estimating probability of nonresponse to treatment using mixture distributions. Statistics in Medicine, 20, 1739–1753.CrossRefGoogle ScholarPubMed
Pinheiro, J. C., & Bates, D. M. (2000). Mixed-effects models in S and S-PLUS, New York, USA: Springer-VerlagCrossRefGoogle Scholar
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models, 2Thousand Oaks: Sage PublicationsGoogle Scholar
R Core Team (2013). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.Google Scholar
Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Seber, G. A. F., & Wild, C. J. (1989). Nonlinear regression, New York, USA: WileyCrossRefGoogle Scholar
Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence, New York, USA: Oxford University PressCrossRefGoogle Scholar
Smith, J. L., & Klebe, K. J. (1997, June). Design issues in learning experiments: A comparison between repeated measures ANOVA and a latent growth curve. Paper presented at the Annual Meeting of the Psychometric Society, Gatlinburg, TN.Google Scholar
Sokal, R. R., & Rohlf, F. J. (1995). Biometry: The principles and practice of statistics in biological research, 3New York, USA: W.H. FreemanGoogle Scholar
Titterington, D., Smith, A., & Makov, U. (1985). Statistical analysis of finite mixture distributions, Chichester: WileyGoogle Scholar
Tueller, S. J., Drotar, S., & Lubke, G. H. (2011). Addressing the problem of switched class labels in latent variable mixture model simulation studies. Structural Equation Modeling, 18, 110–131.CrossRefGoogle Scholar
Verbeke, G., & Lesaffre, E. (1996). A linear mixed-effects model with heterogeneity in the random effects population. Journal of the American Statistical Association, 91, 217–221.CrossRefGoogle Scholar
Verbeke, G., & Molenberghs, G. (2000). Linear mixed models for longitudinal data, New York, NY: Springer-VerlagGoogle Scholar
Vonesh, E. F. (2012). Generalized linear and nonlinear models for correlated data: Theory and applications using SAS, Cary, NC: SAS InstituteGoogle Scholar
Vonesh, E. F., & Carter, R. L. (1992). Mixed effects nonlinear regression for unbalanced repeated measures. Biometrics, 48, 1–17.CrossRefGoogle ScholarPubMed
Wang, L., & McArdle, J. J. (2008). A Simulation study comparison of bayesian estimation with conventional methods for estimating unknown change points. Structural Equation Modeling: A Multidisciplinary Journal, 15, 52–74.CrossRefGoogle Scholar
Wedel, M., & DeSarbo, W. S. (1995). A mixture likelihood approach for generalized linear models. Journal of Classification, 12, 21–55.CrossRefGoogle Scholar
Xu, W., & Hedeker, D. (2001). A random-effects mixture model for classifying treatment response in longitudinal clinical trials. Journal of Biopharmaceutical Statistics, 11, 253–273.CrossRefGoogle ScholarPubMed
Zopluoglu, C., Harring, J. R., & Kohli, N. (2014). FitPMM: An R routine to fit finite mixtures of piecewise mixed-effects models with unknown random knots. Applied Psychological Measurement, 38, 583–584.CrossRefGoogle Scholar