Nonlinear Change Models in Populations with Unobserved Heterogeneity
Abstract
When unobserved heterogeneity exists in populations where the phenomenon of interest is governed by a functional form of change linear in its parameters, the growth mixture model (GMM) is useful for modeling change conditional on latent class. However, when the functional form of interest is nonlinear in its parameters, the GMM is not very useful because it is based on a system of equations linear in its parameters. The nonlinear change mixture model (NCMM) is proposed, which explicitly addresses unobserved heterogeneity in situations where change follows a nonlinear functional form. Due to the integration of nonlinear multilevel models and finite mixture models, neither of which generally have closed form solutions, analytic solutions do not generally exist for the NCMM. Five methods of parameter estimation are developed and evaluated with a comprehensive Monte Carlo simulation study. The simulation showed that the parameters of the NCMM can be accurately estimated with several of the proposed methods, and that the method of choice depends on the precise question of interest.
References
1937). Personality: A psychological interpretation. New York: Henry Hold and Company.
(1984). An introduction to multivariate statistical analysis (2nd ed.). New York: Wiley, Inc.
(1988). Nonlinear regression analysis and its applications. New York: Wiley.
(1996). An alternative two stage least squares (2SLS) estimator for latent variable equations. Psychometrika, 61, 109–121.
(2001). Two-stage least squares and latent variable models: Simultaneous estimation and robustness to misspecifications. In , Structural equation modeling: Present and future. A festschrift in honor of Karl Joreskog (pp. 119–138). Lincolnwood, IL: Scientific Software International, Inc.
(1991). Models for learning data. In , Best methods for the analysis of change: Recent advances, unanswered questions, future directions. Washington, DC: American Psychological Association.
(1991). Estimating individual developmental functions: Methods and their assumptions. Child Development, 62, 23–43.
(2004). A two-stage regression model for epidemiological studies with multivariate disease classification data. Journal of the American Statistical Association, 99, 127–138.
(1996). Mixed-effects models in the study of individual differences with repeated measures data. Multivariate Behavioral Research, 31, 371–403.
(2007). Analysis of nonlinear patterns of change with random coefficient models. Annual Review of Psychology, 58, 615–637.
(1995). Nonlinear models for repeated measurement data. New York: Chapman & Hall.
(2003). Nonlinear models for repeated measurement data: An overview and update. Journal of Agricultural, Biological, and Environmental Statistics, 8, 387–419.
(2001). Cluster analysis as a method of recovering types of intraindividual growth trajectories: A Monte Carlo study. Multivariate Behavioral Research, 36, 501–5522.
(1994). Toward the combined use of nomothetic and idiographic methodologies in sport psychology: An empirical example. The Sport Psychologist, 8, 376–392.
(2004). MCLUST: Software for model based clustering (Version 2.1–5) [computer software and manual]. Retrieved from www.cran.r-project.org/.
(1990). Nonlinear mixed effects models for repeated measures data. Biometrics, 46, 673–687.
(1987). Statistical analysis with missing data. New York: Wiley, Inc.
(1988). Individual and variable-based approaches to longitudinal research on early risk factors. In , Studies of psychosocial risk: The power of longitudinal data (pp. 45–61). New York: Cambridge University Press.
(1973). Developmental changes in mental performance. Monographs of the Society for Research in Child Development, 38, 1–83.
(1988). Mixture models: Inference and applications to clustering. New York: Marcel Dekker.
(2001). Finite mixture models. New York: Wiley, Inc.
(1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, 557–585.
(2001a). Latent variable mixture modeling. In , New developments and techniques in structural equation modeling (pp. 1–33). Hillsdale, NJ: Erlbaum.
(2001b). Second-generation structural equation modeling with a combination of categorical and continuous latent variables: New opportunities for latent class/latent growth modeling. In , New methods for the analysis of change (pp. 291–322). Washington, DC: American Psychological Association.
(2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29, 81–117.
(2002). General growth mixture modeling for randomized preventive interventions. Biostatistics, 3, 459–475.
(2000). Integrating person-centered and variables-centered analyses: Growth mixture modeling with latent trajectory classes. Alcoholism: Clinical and Experimental Research, 24, 882–891.
(1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463–469.
(1999). Analyzing developmental trajectories: A semi-parametric, group-based approach. Psychological Methods, 4, 137–157.
(Eds.). Longitudinal research in the study of behavior and development. New York: Academic Press.
(2000). A mixture model for longitudinal data with application to assessment of noncompliance. Biometrics, 56, 464–472.
(2000). Mixed-effects models in S and S-Plus. New York: Springer.
(2004). NLME: Linear and nonlinear mixed effects models (version 3.3–48) [computer software and manual]. Retrieved from www.cran.r-project.org/.
(2005). R: A language and environment for statistical computing [computer software and manual], R foundation for statistical computing. Retrieved from www.r-project.org.
. (1983). Nonlinear regression modeling: A unified practical approach. New York: Marcel Dekker, Inc.
(2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks, CA: Sage.
(1959). A flexible growth function for empirical use. Journal of Experimental Botany, 10, 290–300.
(1966). Foundations of the theory of prediction. Homewood, IL: The Dorsey Press.
(2002). Missing data: Our view of the state of the art. Psychological Methods, 7, 147–177.
(1996). Adolescent risk factors for binge drinking during the transition to young adulthood: Variable and pattern-centered approaches to change. Developmental Psychology, 32, 659–673.
(2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York: Oxford University Press.
(1997). Two-stage method of estimation for general linear growth curve models. Biometrics, 53, 720–728.
(1990). Linear and nonlinear curve fitting. In , Statistical methods in longitudinal research, Volume II: Time series and categorical longitudinal data (pp. 289–318). Orlando, FL: Academic Press.
(2002). Attrition in longitudinal studies: How to deal with missing data. Journal of Clinical Epidemiology, 55, 329–337.
(1991). A dynamic systems model of cognitive and language growth. Psychological Review, 98, 3–52.
(1997). Linear and nonlinear models for the analysis of repeated measurements. New York: Marcel Dekker.
(1982). Maximum likelihood of misspecified models. Econometrica, 50, 1–25.
(2007). Multilevel covariance structure analysis by fitting multiple single-level models. Sociological Methodology, 37, 53–82.
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