Abstract
Latent growth curve models as structural equation models are extensively discussed in various research fields (Curran and Muthén in Am. J. Community Psychol. 27:567–595, 1999; Duncan et al. in An introduction to latent variable growth curve modeling. Concepts, issues and applications, 2nd edn., Lawrence Earlbaum, Mahwah, 2006; Muthén and Muthén in Alcohol. Clin. Exp. Res. 24(6):882–891, 2000a; in J. Stud. Alcohol. 61:290–300, 2000b). Recent methodological and statistical extension are focused on the consideration of unobserved heterogeneity in empirical data. Muthén extended the classic structural equation approach by mixture components, i.e. categorical latent classes (Muthén in Marcouldies, G.A., Sckumacker, R.E. (eds.), New developments and techniques in structural equation modeling, pp. 1–33, Lawrance Erlbaum, Mahwah, 2001a; in Behaviometrika 29(1):81–117, 2002; in Kaplan, D. (ed.), The SAGE handbook of quantitative methodology for the social sciences, pp. 345–368, Sage, Thousand Oaks, 2004). 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) (Boers et al. in Eur. J. Criminol. 7:499–520, 2010; Reinecke in Delinquenzverläufe im Jugendalter: Empirische Überprüfung von Wachstums- und Mischverteilungsmodellen, Institut für sozialwissenschaftliche Forschung e.V., Münster, 2006a; in Methodology 2:100–112, 2006b; in van Montfort, K., Oud, J., Satorra, A. (eds.), Longitudinal models in the behavioral and related sciences, pp. 239–266, Lawrence Erlbaum, Mahwah, 2007). Observed as well as unobserved heterogeneity will be considered with growth mixture models. Special attention is given to the distribution of the outcome variables as counts. Poisson and negative binomial distributions with zero inflation are considered in the proposed growth mixture models variables. Different model specifications will be emphasized with respect to their particular parameterizations.
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Boers, K., Reinecke, J., Seddig, D., Mariotti, L.: Explaining the development of adolescent violent delinquency. Eur. J. Criminol. 7, 499–520 (2010)
Celeux, G., Soromenho, G.: An entropy criterion for assessing the number of clusters in a mixture model. J. Classif. 13, 195–212 (1996)
Curran, P.J., Muthén, B.: The application of latent curve analysis to testing developmental theories in intervention research. Am. J. Community Psychol. 27, 567–595 (1999)
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. B 39(1), 1–38 (1977)
Duncan, T.E., Duncan, S.C., Strycker, L.A.: An Introduction to Latent Variable Growth Curve Modeling. Concepts, Issues and Applications, 2nd edn. Lawrence Earlbaum, Mahwah (2006)
Farrington, D.P., Loeber, R., Elliott, D.S.: Advancing knowledge about the onset of delinquency and crime. In: Lahey, B., Kazdin, A. (eds.) Advances in Clinical Child Psychology, vol. 13, pp. 283–342. Plenum Press, New York (1990)
Greene, W.: Functional forms for the negative binomial model for count data. Econ. Lett. 99, 585–590 (2008)
Hilbe, J.M.: Negative binomial regression. Cambridge University Press, Cambridge (2007)
Jeffries, N.O.: A note on testing the number of components in a normal mixture. Biometrika 90, 991–994 (2003)
Kreuter, F., Muthén, B.: Analyzing criminal trajectory profiles: bridging multilevel and group-based approaches using growth mixture modeling. J. Quant. Criminol. 24, 1–31 (2008a)
Kreuter, F., Muthén, B.: Longitudinal modeling of population heterogeneity: methodological challenges to the analysis of empirically derived criminal trajectory profiles. In: Hancock, G.R., Samuelsen, K.M. (eds.) Advances in Latent Variable Mixture, Modeling, pp. 53–75. Information Age Publishing, Charlotte (2008b)
Lambert, D.: Zero-inflated Poisson regression with an application to defects in manufacturing. Technometrics 34, 1–13 (1992)
Lo, Y., Mendell, N.R., Rubin, D.B.: Testing the number of components in a normal mixture. Biometrika 88(3), 767–778 (2001)
Mariotti, L., Reinecke, J.: Wachstums- und Mischverteilungsmodelle unter Berücksichtigung unbeobachteter Heterogenit: Empirische Analysen zum delinquenten Verhalten Jugendlicher in Duisburg. Institut für sozialwissenschaftliche Forschung e.V., Münster (2010)
McArdle, J.J.: Dynamic but structural equation modeling of repeated measures data. In: Nesselroade, J.R., Cattell, R.B. (eds.) Handbook of Multivariate Experimental Psychology, 2nd edn., pp. 561–614. Plenum, New York (1988)
McArdle, J.J., Epstein, D.: Latent growth curves within developmental structural equation models. Child Dev. 58, 110–133 (1987)
McLachlan, G., Peel, D.: Finite Mixture Models. Wiley, New York (2000)
Meredith, M., Tisak, J.: Latent curve analysis. Psychometrika 55(1), 107–122 (1990)
Muthén, B.: Analysis of longitudinal data using latent variable models with varying parameters. In: Collins, L.M., Horn, J.L. (eds.) Best Methods for the Analysis of Change: Recent Advances, Unanswered Questions, Future Directions, pp. 1–17. APA, Washington (1991)
Muthén, B.: Latent variable modeling with longitudinal and multilevel data. Sociol. Method. 27, 453–480 (1997)
Muthén, B.: Latent variable mixture modeling. In: Marcouldies, G.A., Schumacker, R.E. (eds.) New Developments and Techniques in Structural Equation Modeling, pp. 1–33. Lawrance Erlbaum, Mahaw (2001a)
Muthén, B.: Second-generation structural equation modeling with a combination of categorical and continuous latent variables: new opportunities for latent class/latent growth modeling. In: Collins, L., Sayer, A. (eds.) New Methods for the Analysis of Change, pp. 291–322. APA, Washington (2001b)
Muthén, B.: Beyond SEM: general latent variable modeling. Behaviormetrika 29(1), 81–117 (2002)
Muthén, B.: Statistical and substantive checking in growth mixture modeling: comment on Bauer and Curran (2003). Psychol. Methods 8, 369–377 (2003)
Muthén, B.: Latent variable analysis: growth mixture modeling and related techniques for longitudinal data. In: Kaplan, D. (ed.) The SAGE Handbook of Quantitative Methodology for the Social Sciences, pp. 345–368. Sage, Thousand Oaks (2004)
Muthén, B.: Latent variable hybrids: overview of old and new models. In: Hancock, G.R., Samuelsen, K.M. (eds.) Advances in Latent Variable Mixture Models, pp. 1–24. Information Age Publishing, Charlotte (2008)
Muthén, B., Asparouhov, T.: Growth mixture modeling: analysis with non-Gaussian random effects. In: Fitzmaurice, G., Davidian, M., Verbeke, G., Molenberghs, G. (eds.) Longitudinal Data Analysis, pp. 143–165. Chapman & Hall/CRC Press, Boca Raton (2008)
Muthén, B., Muthén, L.: Integrating person-centered and variable-centered analysis: growth mixture modeling with latent trajectory classes. Alcohol. Clin. Exp. Res. 24(6), 882–891 (2000a)
Muthén, B., Muthén, L.: The development of heavy drinking and alcohol-related problems from ages 18 to 37 in a U.S. national sample. Journal of Studies on Alcohol 61, 290–300 (2000b)
Muthén, L., Muthén, B.: Mplus User’s Guide, 4th edn. Muthén & Muthén, Los Angeles (2006)
Muthén, L., Muthén, B.: Mplus User’s Guide, 6th edn. Muthén & Muthén, Los Angeles (2010)
Muthén, B., Shedden, K.: Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics 55(2), 463–469 (1999)
Nagin, D.S.: Analyzing developmental trajectories: a semi-parametric, group-based approach. Psychol. Methods 4, 139–157 (1999)
Nagin, D.S.: Group-Based Modeling of Development. Harvard University Press, Cambridge (2005)
Nagin, D.S., Land, K.C.: Age criminal careers, and population heterogeneity: specification and estimation of a nonparametric, mixed Poisson model. Criminology 31(3), 327–362 (1993)
Nylund, K., Asparouhov, T., Muthén, B.O.: Deciding on the number of classes in latent class analysis and growth mixture modeling. A Monte Carlo simulation study. Struct. Eq. Model. 14, 535–569 (2007)
Odgers, C.L., Caspi, A., Poulton, R., Harrington, H., Thompson, M., Broadbent, J.M., Dickson, N., Sears, M.R., Hancox, B., Moffit, T.E.: Prediction of adult health burden by conduct problem subtypes in males. Arch. Gen. Psychiatry 64, 476–484 (2007)
Pöge, A.: Persönliche Codes bei Längsschnittstudien: Ein Erfahrungsbericht. ZA-Inf. 56, 50–69 (2005)
Rao, C.R.: Some statistical methods for comparison of growth curves. Biometrics 14, 1–17 (1958)
Reinecke, J.: Delinquenzverläufe im Jugendalter: Empirische Überprüfung von Wachstums- und Mischverteilungsmodellen. Institut für sozialwissenschaftliche Forschung e.V., Münster (2006a)
Reinecke, J.: Longitudinal analysis of adolescents’ deviant and delinquent behavior. Applications of latent class growth curves and growth mixture models. Methodology 2, 100–112 (2006b)
Reinecke, J.: The development of deviant and delinquent behavior of adolescents. Applications of latent class growth curves and growth mixture models. In: van Montfort, K., Oud, J., Satorra, A. (eds.) Longitudinal Models in the Behavioral and Related Sciences, pp. 239–266. Lawrence Erlbaum, Mahwah (2007)
Reinecke, J.: Klassifikation von Delinquenzverläufen: Eine Anwendung von Mischverteilungsmodellen. In: Reinecke, J., Tarnai, C. (eds.) Klassifikationsanalysen in Theorie und Praxis, pp. 189–218. Waxmann, Münster (2008)
Roeder, K., Lynch, K.G., Nagin, D.S.: Modeling uncertainty in latent class membership: a case study in criminology. J. Am. Stat. Assoc. 94, 766–776 (1999)
Ross, S.M.: Introduction to Probability Models, 5th edn. Academic Press, New York (1993)
Rost, J., Langeheine, R.: Applications of Latent Trait and Latent Class Models in the Social Sciences. Waxmann, Münster (1997)
Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)
Tucker, L.R.: Determination of parameters of a functional relation by factor analysis. Psychometrika 23, 19–23 (1958)
Willet, J.B., Sayer, A.G.: Using covariance structure analysis to detect correlates and predictors of individual change over time. Psychol. Bull. 116(2), 363–381 (1994)
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Reinecke, J., Seddig, D. Growth mixture models in longitudinal research. AStA Adv Stat Anal 95, 415–434 (2011). https://doi.org/10.1007/s10182-011-0171-4
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DOI: https://doi.org/10.1007/s10182-011-0171-4