Abstract
Latent growth curve models as structural equation models are extensively discussed in various research fields. Further methodological and statistical developments regarding mixture models are able to consider unobserved heterogeneity in developmental processes. Muthén (2001a, b) extended the classic structural equation approach by mixture components in terms of categorical latent classes resulting into the concept of growth mixture models. The paper discusses applications of growth mixture models with data on delinquent behavior of adolescents from the German panel study Crime in the modern City (CrimoC). Special attention is given to the distributions of the time-variant outcome variable (delinquency) as a count variable. The application of a mixture model assumes a negative binomial distributed count variable and discuss seven classes regarding different developments of delinquency. The mixture model is extended by a multinomial regression model containing five different exogenous variables explaining the latent class distributions. The concept of three-step latent class modeling (Vermunt 2010) is introduced and advantages of the conditional growth mixture model is discussed by an empirical example.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Agresti, A. 2002. Categorical data analysis, 2nd Ed. New York: Wiley.
Akaike, H. 1987. Factor analysis and the AIC. Psychometrika 52:317–332.
Asparouhov, T., and B. Muthén. 2013. Auxiliary variables in mixture modeling: A 3-step approach using Mplus. Mplus Webnotes 15. Version 6, February 7, 2013.
Bentrup, C. 2019. Untersuchungsdesign und Stichproben der Duisburger Kriminalitätsbefragung. In Delinquenz im Altersverlauf. Erkenntnisse der Langzeitstudie Kriminalität in der modernen Stadt, Eds. K. Boers and J. Reinecke, 95–120. Münster: Waxmann.
Boers, K., J. Reinecke, L. Mariotti, and D. Seddig. 2010. Explaining the development of adolescent violent delinquency. European Journal of Criminology 7:499–520.
Hilbe, J.M. 2011. Negative binomial regression, 2nd Ed. Cambridge: Cambridge University Press.
Krohn, M.D., C.L. Gibson, and T.P. Thornberry. 2013. Under the protective bud the bloom awaits: A review of theory and research on adult-onset and late-blooming offenders. In Handbook of life-course criminology: Emerging trends and directions for future research, Eds. M.D. Gibson and M.D. Krohn, 183–200. New York: Springer.
Lambert, D. 1992. Zero-inflated Poisson regression with an application to defects in manufacturing. Technometric 34:1–13.
Little, R.J.A., and D.B. Rubin. 2002. Statistical analysis with missing data, 2nd Ed. New York: Wiley.
Lo, Y., N.R. Mendell, and D.B. Rubin. 2001. Testing the number of components in a normal mixture. Biometrika 88:767–778.
Mariotti, L., and J. Reinecke. 2010. Delinquenzverläufe im Jugendalter: Wachstums- und Mischverteilungsmodelle unter Berücksichtigung unbeobachteter Heterogenität. In Sozialwissenschaftliche Forschungsdokumentationen 21. Münster: Institut für sozialwissenschaftliche Forschung e. V.
McArdle, J.J. 1988. Dynamic but structural equation modeling of repeated measures data. In Handbook of multivariate experimental psychology, Eds. J.R. Nesselroade and R.B. Cattell, 561–614. New York: Plenum.
McArdle, J.J., and D. Epstein. 1987. Latent growth curves within developmental structural equation models. Child Development 58:110–133.
McLachlan, G., and D. Peel. 2000. Finite mixture models. New York: Wiley.
Meredith, W., and J. Tisak. 1990. Latent curve analysis. Psychometrika 55:107–122.
Muthén, B. 1991. Analysis of longitudinal data using latent variable models with varying parameters. In Best methods for the analysis of change, Eds. L. Collins and J. Horn, 1–17. Washington DC: American Psychological Association.
Muthén, B. 1997. Latent variable modeling with longitudinal and multilevel data. In Sociological methodology, Ed. A. Raftery, 453–480. Boston: Blackwell Publishers.
Muthén, B. 2001a. Latent variable mixture modeling. In New developments and techniques in structural equation modeling, Eds. G.A. Marcoulides and R.E. Schumacker, 1–33. Lawrence Erlbaum Associates.
Muthén, B. 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, Eds. L.M. Collins and A. Sayer, 291–322. Washington, D.C.: APA.
Muthén, B.O. 2003. Statistical and substantive checking in growth mixture modeling: Comment on Bauer and Curran (2003). Psychological Methods 8:369–377.
Muthén, B.O. 2004. Latent variable analysis: Growth mixture modeling and related techniques for longitudinal data. In The Sage handbook of quantitative methodology for the social sciences, Ed. D. Kaplan, 345–368. Thousand Oaks: Sage.
Muthén, B.O. 2008. Latent variable hybrids: Overview of old and new models. In Advances in latent variable mixture models, Eds. G.R. Hancock and K.M. Samuelsen, 1–24. Charlotte: Information Age.
Muthén, B.O., and P.J. Curran. 1997. General longitudinal modeling of individual differences in experimental designs: A latent variable framework for analysis and power estimation. Psychological Methods 2:371–402.
Muthén, B., and K. Shedden. 1999. Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics 55:463–469.
Muthén, L., and B.O. Muthén. 1998–2017. Mplus user’s guide, 8th. Ed. Los Angeles: Muthén & Muthén.
Nagin, D.S. 1999. Analyzing developmental trajectories: A semi-parametric, group-based approach. Psychological Methods 4:139–157.
Nagin, D.S. 2005. Group-based modeling of development. Cambridge: Harvard University Press.
Nagin, D.S., and K.C. Land. 1993. Age, criminal careers, and population heterogeneity: Specification and estimation of a nonparametric, mixed Poisson model. Criminology 31:327–362.
Nyland, K.L., T. Asparouhov, and B.O. Muthén. 2007. Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study. Structural Equation Modeling 14:535–569.
Ramaswamy, V., W.S. DeSarbo, D.J. Reibstein, and W.T. Robinson. 1993. An empirical pooling approach for estimating marketing mix elasticities with PIMS data. Marketing Science 12:103–124.
Rao, C.R. 1958. Some statistical methods for comparison of growth curves. Biometrics 14:1–17.
Reinecke, J. 2006a. Delinquenzverläufe im Jugendalter: Empirische Überprüfung von Wachstums- und Mischverteilungsmodellen. In Sozialwissenschaftliche Forschungsdokumentationen 20. Münster: Institut für sozialwissenschaftliche Forschung e. V.
Reinecke, J. 2006b. Longitudinal analysis of adolescents deviant and delinquent behaviour. Applications of latent class growth curves and growth mixture models. Methodology 2:100–112.
Reinecke, J. 2013. Growth curve models and panel dropouts: Applications with criminological panel data. Netherlands Journal of Psychology 47:122–131.
Reinecke, J., and D. Seddig. 2011. Growth mixture models in longitudinal research. Advances in Statistical Analysis 95:415–434.
Roeder, K., K.G. Lynch, and D.S. Nagin. 1999. Modeling uncertainty in latent class membership: A case study in criminology. Journal of the American Statistical Association 94:766–776.
Schwarz, G. 1978. Estimating the dimension of a model. The Annals of Statistics 6:461–464.
Seddig, D., and J. Reinecke. 2017. Exploration and explanation of adolescent self-reported delinquency trajectories in the Crimoc study. In The International handbook of life-course criminology, Eds. V. van der Geest and A. Blokland, 159–178. London: Taylor & Francis.
Tucker, L.R. 1958. Determination of parameters of a functional relation by factor analysis. Psychometrika 23:19–23.
Vermunt, J.K. 2010. Latent class modeling with covariates: Two improved three-step approaches. Political Analysis 18:450–469.
Willett, J.B., and A.G. Sayer. 1994. Using covariance structure analysis to detect correlates and predictors of individual change over time. Psychological Bulletin 116:363–381.
Wickrama, K.A.S., T.K. Lee, C.W. O’ Neal, and F.O. Lorenz. 2016. Higher-order growth curves and mixture modeling with Mplus: A practical guide. New York: Routledge.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Fachmedien Wiesbaden GmbH, ein Teil von Springer Nature
About this chapter
Cite this chapter
Reinecke, J. (2020). Mixture Models in Longitudinal Research Designs. In: Mays, A., et al. Grundlagen - Methoden - Anwendungen in den Sozialwissenschaften. Springer VS, Wiesbaden. https://doi.org/10.1007/978-3-658-15629-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-658-15629-9_3
Published:
Publisher Name: Springer VS, Wiesbaden
Print ISBN: 978-3-658-15628-2
Online ISBN: 978-3-658-15629-9
eBook Packages: Social Science and Law (German Language)