Abstract
The development offinite mixture models in statistical research tries to cover the problem of unobserved heterogeneity within the general linear model (e.g., Titterington, Smith, and Makov, 1985). The term finite mixtures refers to the assumption that a sample of observations arises from a mixture of unknown proportions with a specific form of distribution in each population. Examples include mixtures of normal, exponential, and Bernoulli istributions. The conditional specification of a finite mixture model are discussed and applied in the social science literature. This model allows the probabilistic classification of observations intocomponents or classes and a simultaneous estimation of regression parameters for each mixture component (Wedel and DeSarbo, 1994, 1995). Probabilistic classifications are well known from latent class analysis, originally proposed by Lazarsfeld and Henry (1968).
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References
Arminger, Gerhard & Petra Stein 1997. Finite Mixtures of Covariance Structure Models with Regressors. 26: 148–182.
Arminger, Gerhard, Petra Stein, & Jörg Wittenberg. 1999. Mixtures of Conditional Mean and Covariance Structure Models. 64: 475–494.
Bauer, Daniel J. & Patrick J. Curran. 2003. Distributional Assumptions of Growth Mixture Models: Implications for Overextraction of Latent Trajectory Classes. 8: 338–363.
Boers, Klaus & Jost Reinecke. Hrsg. 2007. Delinquenz im Jugendalter. Erkenntnisse einer Münsteraner Längsschnittstudie. Kriminologie und Kriminalsoziologie, Bd. 3. Münster: Waxmann.
Boers, Klaus, Jost Reinecke, Daniel Seddig, & Luca Mariotti. 2010. Explaining the Development of Adolescent Violent Delinquency. 7: 499–520.
Cameron, A. Colin & Pravin K. Trivedi. 2007. Regression Analysis of Count Data. Cambridge: Cambridge University Press.
D’Unger, Amy V., Kenneth C. Land, Patricia L. McCall & Daniel S. Nagin. 1998. How Many Latent Classes of Delinquent/Criminal careers? Results from Mixed Poisson Regression Analyses of the London, Philadelphia and Racine Cohort Studies. 103: 1593–1630.
Farrington, David P. & West, Donald J. 1990. The Cambridge Study in Delinquent Development: A Long-Term Follow-up of 411 London Males. In Kriminalität: Persönlichkeit, Lebensgeschichte und Verhalten, eds. Hans Jürgen Kerner & Günther Kaiser, 115–138. Berlin: Springer.
Jedidi, Kamal, Harsharanjeet S. Jagpal, & Wayne S. DeSarbo. 1997a. Finite-Mixture Structural Equation Models for Response-Based Segmentation and Unobserved Heterogeneity. 16: 39–59.
Jedidi, Kamal, Harsharanjeet S. Jagpal, & Wayne S. DeSarbo, W. S. 1997b. STEMM: A General Finite Mixture Structural Equation Model. 14: 23–50.
Jedidi, Kamal, Venkat Ramaswamy, Wayne DeSarbo & Michel Wedel. 1996. On Estimating Finite Mixtures of Multivariate Regression and Simultaneous Equation Models. 3: 266–289.
Jones, Bobby L., Daniel S. Nagin & Kathryn Roeder. 2001. A SAS procedure based on mixture models for estimating developmental trajectories. 29: 374–393.
Lambert, Diane. 1992. Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. 34: 1–13.
Land, Kenneth C., Patricia L. McCall & Daniel S. Nagin. 1996. A Comparison of Poisson, Negative Binomial, and Semiparametric Mixed Poisson Regression Models with Empirical Applications to Criminal Careers Data. 24: 387–442.
Lazarsfeld, Paul F. & Neil W. Henry. 1968. Latent Structure Analysis. Boston: Houghton Mifflin Co.
McArdle, John J. 1988. Dynamic but Structural Equation Modeling of Repeated Measures Data. In Handbook of Multivariate Experimental Psychology, eds. John R. Nesselroade & Raymond B. Cattell, 561–614. New York: Plenum.
McArdle, John J. & David Epstein. 1987. Latent Growth Curves within Developmental Structural Equation Models. 58: 110–133.
Mariotti, Luca & Jost Reinecke. 2010. Delinquenzverläufe im Jugendalter: Wachstumsund Mischverteilungsmodelle unter Berücksichtigung unbeobachteter Heterogenität. In Sozialwissenschaftliche Forschungsdokumentationen 21. Münster: Institut für sozialwissenschaftliche Forschung e. V.
Meredith, William and John Tisak. 1990. Latent Curve Analysis. 55: 107–122.
Moffitt, Terrie. 1993. Adolescence-Limited and Life-Course-Persistent Antisocial Behavior: A Developmental Taxonomy. 100: 674–701.
Muthén, Bengt O. 1991. Analysis of Longitudinal Data using Latent Variable Models with Varying Parameters. In Best Methods for the Analysis of Change, eds. Linda M. Collins & John L. Horn, 1–17. Washington DC: American Psychological Association.
Muthén, Bengt O. 1997. Latent Variable Modeling with Longitudinal and Multilevel Data. In Sociological Methodology, ed. Adrian Raftery, 453–480. Boston: Blackwell Publishers.
Muthén, Bengt O. 2001a. Latent Variable Mixture Modeling. In New Developments and Techniques in Structural Equation Modeling, eds. George. A. Marcoulides and Randall E. Schumacker, 1–33. Lawrence Erlbaum Associates.
Muthén, Bengt O. 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. Linda M. Collins & Aline G. Sayer, 291–322. Washington, DC: American Psychological Association.
Muthén, Bengt O. 2003. Statistical and Substantive Checking in Growth Mixture Modeling: Comment on Bauer and Curran (2003). 8: 369–377.
Muthén, Bengt 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. David W. Kaplan, 345–368. Thousand Oaks: Sage.
Muthén, Bengt O. 2008. Latent Variable Hybrids: Overview of Old and New Models. In Advances in Latent Variable Mixture Models, eds. Gregory R. Hancock & Karen M. Samuelsen, 1–26. Charlotte, NC: Information Age Publishing.
Muthén, Bengt O. & Patrick J. Curran. 1997. General Longitudinal Modeling of Individual Differences in Experimental Designs: A Latent Variable Framework for Analysis and Power Estimation. 2: 371–402.
Muthén, Bengt O. & Kerby Shedden. 1999. Finite Mixture Modeling with Mixture Outcomes using the EM Algorithm55: 463–469.
Muthén, Linda K. & Bengt O. Muthén. 2007. (5th ed.). Los Angeles: Muthén & Muthén.
Nagin, Daniel S. & Kenneth C. Land. 1993. Age, Criminal Careers, and Population Heterogeneity: Specification and Estimation of a Nonparametric, Mixed Poisson Model. 31: 327–362.
Nagin, Daniel S. (1999). Analyzing Developmental Trajectories: A Semi-parametric, Group-based Approach. 4: 139–157.
Nagin, Daniel S. (2005). Group-Based Modeling of Development. Cambridge: Harvard University Press.
Piquero Alex R. 2008. Taking Stock of Developmental Trajectories of Criminal Activity over the Life Course. In The Long View of Crime. A Synthesis of Longitudinal Research, ed. Akiva M. Liberman, 23–78. New York: Springer.
Rao, C. Radhakrishna. 1958. Some Statistical Methods for Comparison of Growth Curves. 14: 1–17.
Reinecke, Jost. 2006a. Delinquenzverläufe im Jugendalter: Empirische Überprüfung von Wachstums- und Mischverteilungsmodellen. In Sozialwissenschaftliche Forschungsdokumentationen 20. Münster: Institut für sozialwissenschaftliche Forschung e. V.
Reinecke, Jost. 2006b. Longitudinal Analysis of Adolescents’ Deviant and Delinquent Behaviour. Applications of Latent Class Growth Curves and Growth Mixture Models. 2: 100–112.
Roeder, Kathryn, Kevin G. Lynch & Daniel S. Nagin. 1999. Modeling Uncertainty in Latent Class Membership: A Case Study in Criminology. 94: 766–776.
Ross, Sheldon M. 1993. (5th ed.). New York: Academic Press.
Titterington, D. M., A. F. M. Smith & U. E. Makov. 1985. Statistical Analysis of Finite Mixture Distributions. Chichester: Wiley.
Tracy, Paul E., Marvin E. Wolfgang & Robert M. Figlio. 1990. Delinquency in Two Birth Cohorts. New York: Plenum.
Tremblay, Richard E., Lyse Desmarais-Gervais, Claude Gagnon & Patrice M. Charlebois. 1987. The Preschool Behaviour Questionnaire: Stability of its Factor Structure between Cultures, Sexes, Ages and Socioeconomic Classes. 10: 467–484.
Tucker, Ledyard R. 1958. Determination of Parameters of a Functional Relation by Factor Analysis. 23: 19–23.
van Dulmen, Manfred H. M., Elizabeth A. Goncy, Andrea Vest & Daniel J. Flannery. 2009. Group-Based Trajectory Modeling of Externalizing Behavior Problems from Childhood through Adulthood: Exploring Discrepancies in the Empirical Findings. In The Development of Persistent Criminality,ed. Joanne Savage, 288–314. Oxford: Oxford University Press.
Wedel, Michel and Wayne S. de Sarbo. 1994. A Review of Recent Developments in Latent Class Regression Models. In Advanced Methods of Marketing Research, ed. Richard P. Bagozzi, 352–388. Cambridge: Blackwell.
Wedel, Michel and Wayne S. de Sarbo. 1995. A Mixture Likelihood Approach for Generalized Linear Models. Journal of Classification 12: 21–56.
Yung, Yiu-Fai. 1997. Finite Mixture in Confirmatory Factor-Analysis Models. 62: 297–330.
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Reinecke, J. (2012). Mixture Models for Longitudinal Analysis: Applications of Adolescents’ Development of Delinquency. In: Salzborn, S., Davidov, E., Reinecke, J. (eds) Methods, Theories, and Empirical Applications in the Social Sciences. VS Verlag für Sozialwissenschaften. https://doi.org/10.1007/978-3-531-18898-0_10
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DOI: https://doi.org/10.1007/978-3-531-18898-0_10
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