Studies of human development require longitudinal data analysis methods that describe within- and between-individual variation in developmental and behavioral trajectories. This article reviews life-course data analysis methods for modeling these trajectories, as well as their application in studies of antisocial behavior and of crime in childhood, in adolescence, and throughout life. We set the stage by introducing growth curve (hierarchical linear) models. We focus our review on finite mixture models for life-course data, known as group-based trajectory and growth mixture models. We then discuss how these models are applied within criminology and developmental psychology, recent controversies over their substantive use and interpretation, and important issues of statistical practice and the challenges they raise. Building on the critical literature, we offer several recommendations for the applied users of the models. Finally, we present the most recent method of examining behavioral trajectories in criminology, the unimodal curve registration (UCR) approach. We briefly contrast the UCR model with growth curve and finite mixture models for life-course data analysis.


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Literature Cited

  1. Akaike H. 1973. Information theory and an extension of the maximum likelihood principle. Second International Symposium on Information Theory BN Petrov, F Csaki 267–81 Budapest: Akad. Kiado [Google Scholar]
  2. Bandeen-Roche K, Miglioretti DL, Zeger SL, Rathouz PJ. 1997. Latent variable regression for multiple discrete outcomes. J. Am. Stat. Assoc. 92:4401375–86 [Google Scholar]
  3. Barnett A, Blumstein A, Farrington DP. 1987. Probabilistic models of youthful criminal careers. Criminology 25:183–108 [Google Scholar]
  4. Bartholomew DJ. 1995. What is statistics?. J. R. Stat. Soc. A 158:1–20 [Google Scholar]
  5. Bauer DJ. 2007. Observations on the use of growth mixture models in psychological research. Multivar. Behav. Res. 42:4757–86 [Google Scholar]
  6. Bauer DJ, Curran PJ. 2003a. Distributional assumptions of growth mixture models: Implications for overextraction of latent trajectory classes. Psychol. Methods 8:3338–63 [Google Scholar]
  7. Bauer DJ, Curran PJ. 2003b. Overextraction of latent trajectory classes: Much ado about nothing? Reply to Rindskopf (2003), Muthén (2003), and Cudeck and Henly (2003). Psychol. Methods 8:384–92 [Google Scholar]
  8. Bauer DJ, Curran PJ. 2004. The integration of continuous and discrete latent variable models: Potential problems and promising opportunities. Psychol. Methods 9:13–29 [Google Scholar]
  9. Blokland AA, Nagin D, Nieuwbeerta P. 2005. Life span offending trajectories of a Dutch conviction cohort. Criminology 43:4919–54 [Google Scholar]
  10. Blumstein A, Cohen J. 1979. Estimation of individual crime rates from arrest records. J. Crim. Law Criminol. 70:4561–85 [Google Scholar]
  11. Blumstein A, Cohen J. 1987. Characterizing criminal careers. Science 237:4818985–91 [Google Scholar]
  12. Blumstein A, Farrington DP, Moitra S. 1985. Delinquency careers: Innocents, desisters, and persisters. Crime Justice 6:187–219 [Google Scholar]
  13. Blumstein A, Moitra S. 1980. The identification of “career criminals” from “chronic offenders” in a cohort. Law Policy 2:3321–34 [Google Scholar]
  14. Boscardin CK, Muthén B, Francis DJ, Baker EL. 2008. Early identification of reading difficulties using heterogeneous developmental trajectories. J. Educ. Psychol. 100:1192 [Google Scholar]
  15. Bowers AJ, Sprott R. 2012. Examining the multiple trajectories associated with dropping out of high school: a growth mixture model analysis. J. Educ. Res. 105:3176–95 [Google Scholar]
  16. Brame R, Mulvey EP, Piquero AR. 2001. On the development of different kinds of criminal activity. Sociol. Methods Res. 29:3319–41 [Google Scholar]
  17. Brame R, Nagin DS, Wasserman L. 2006. Exploring some analytical characteristics of finite mixture models. J. Quant. Criminol. 22:131–59 [Google Scholar]
  18. Brame R, Paternoster R, Piquero AR. 2012. Thoughts on the analysis of group-based developmental trajectories in criminology. Justice Q. 29:4469–90 [Google Scholar]
  19. Britt CL. 1992. Constancy and change in the US age distribution of crime: a test of the “invariance hypothesis.”. J. Quant. Criminol. 8:2175–87 [Google Scholar]
  20. Brumback LC, Lindstrom MJ. 2004. Self modeling with flexible, random time transformations. Biometrics 60:2461–70 [Google Scholar]
  21. Bushway S, Sweeten G, Nieuwbeerta P. 2009. Measuring long term individual trajectories of offending using multiple methods. J. Quant. Criminol. 25:3259–86 [Google Scholar]
  22. Bushway SD, Thornberry TP, Krohn MD. 2003. Desistance as a developmental process: a comparison of static and dynamic approaches. J. Quant. Criminol. 19:2129–53 [Google Scholar]
  23. Chassin L, Flora DB, King KM. 2004. Trajectories of alcohol and drug use and dependence from adolescence to adulthood: the effects of familial alcoholism and personality. J. Abnorm. Psychol. 113:4483 [Google Scholar]
  24. Chib S. 1992. Bayesian inference in the Tobit censored regression model. J. Econom. 51:79–99 [Google Scholar]
  25. Colder CR, Mehta P, Balanda K, Campbell RT, Mayhew K. et al. 2001. Identifying trajectories of adolescent smoking: an application of latent growth mixture modeling. Health Psychol. 20:2127–35 [Google Scholar]
  26. Connell AM, Frye AA. 2006. Growth mixture modelling in developmental psychology: overview and demonstration of heterogeneity in developmental trajectories of adolescent antisocial behaviour. Infant Child Dev. 15:6609–21 [Google Scholar]
  27. Connor JT. 2006. Multivariate Mixture Models to Describe Longitudinal Patterns of Frailty in American Seniors PhD Thesis, Dep. Stat., H. John Heinz III Sch. Public Policy Manag. Carnegie Mellon Univ., Pittsburgh, PA [Google Scholar]
  28. DeBoor C. 1978. A Practical Guide to Splines Berlin: Springer-Verlag [Google Scholar]
  29. D'Unger AV, Land KC, McCall PL. 2002. Sex differences in age patterns of delinquent/criminal careers: results from Poisson latent class analyses of the Philadelphia cohort study. J. Quant. Criminol. 18:4349–75 [Google Scholar]
  30. Eggleston EP, Laub JH, Sampson RJ. 2004. Methodological sensitivities to latent class analysis of long-term criminal trajectories. J. Quant. Criminol. 20:11–26 [Google Scholar]
  31. Elliott MR, Gallo JJ, Ten Have TR, Bogner HR, Katz IR. 2005. Using a Bayesian latent growth curve model to identify trajectories of positive affect and negative events following myocardial infarction. Biostatistics 6:1119–43 [Google Scholar]
  32. Esbensen F-A, Huizinga D. 1990. Community structure and drug use: from a social disorganization perspective. Justice Q. 7:691–709 [Google Scholar]
  33. Geisser S. 1993. Predictive Inference: An Introduction London: Chapman & Hall [Google Scholar]
  34. Gervini D, Gasser T. 2004. Self-modelling warping functions. J. R. Stat. Soc. B 66:4959–71 [Google Scholar]
  35. Gibbons R, Hedeker D, DuToit S. 2010. Advances in analysis of longitudinal data. Annu. Rev. Clin. Psychol. 19:79–107 [Google Scholar]
  36. Goldstein H. 2011. Multilevel Statistical Models Chichester, UK: Wiley [Google Scholar]
  37. Greenberg DF. 1991. Modeling criminal careers. Criminology 29:117–46 [Google Scholar]
  38. Greenwood PW, Turner S. 1987. Selective incapacitation revisited. Why High-Rate Offenders Are Hard to Predict. Santa Monica, CA: Rand [Google Scholar]
  39. Hamil-Luker J, Land KC, Blau J. 2004. Diverse trajectories of cocaine use through early adulthood among rebellious and socially conforming youth. Soc. Sci. Res. 33:2300–21 [Google Scholar]
  40. Haviland AM, Nagin DS. 2005. Causal inferences with group based trajectory models. Psychometrika 70:3557–78 [Google Scholar]
  41. Heckman J, Singer B. 1984. A method for minimizing the impact of distributional assumptions in econometric models for duration data. Econometrica 52:2271–320 [Google Scholar]
  42. Hirschi T, Gottfredson MR. 1983. Age and the explanation of crime. Am. J. Sociol. 89:3552–48 [Google Scholar]
  43. Hoeksma JB, Kelderman H. 2006. On growth curves and mixture models. Infant Child Dev. 15:6627–34 [Google Scholar]
  44. Hynes K, Clarkberg M. 2005. Women's employment patterns during early parenthood: a group-based trajectory analysis. J. Marriage Fam. 67:1222–39 [Google Scholar]
  45. Jennings WG, Reingle JM. 2012. On the number and shape of developmental/life-course violence, aggression, and delinquency trajectories: a state-of-the-art review. J. Crim. Justice 40:6472–89 [Google Scholar]
  46. Jones BL, Nagin DS. 2007. Advances in group-based trajectory modeling and an SAS procedure for estimating them. Sociol. Methods Res. 35:4542–71 [Google Scholar]
  47. Jones BL, Nagin DS, Roeder K. 2001. A SAS procedure based on mixture models for estimating developmental trajectories. Sociol. Methods Res. 29:3374–93 [Google Scholar]
  48. Kass RE, Raftery AE. 1995. Bayes factors. J. Am. Stat. Assoc. 90:430773–95 [Google Scholar]
  49. Kass RE, Wasserman L. 1995. A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. J. Am. Stat. Assoc. 90:431928–34 [Google Scholar]
  50. Kreuter F, Muthén B. 2008. Analyzing criminal trajectory profiles: bridging multilevel and group-based approaches using growth mixture modeling. J. Quant. Criminol. 24:11–31 [Google Scholar]
  51. Lacourse E, Coté S, Nagin DS, Vitaro F, Brendgen M, Tremblay RE. 2002. A longitudinal-experimental approach to testing theories of antisocial behavior development. Dev. Psychopathol. 14:4909–24 [Google Scholar]
  52. Laird NM, Ware JH. 1982. Random-effects models for longitudinal data. Biometrics 38:963–74 [Google Scholar]
  53. Land KC, Nagin DS, McCall PL. 2001. Discrete-time hazard regression models with hidden heterogeneity: the semiparametric mixed Poisson regression approach. Sociol. Methods Res. 29:3342–73 [Google Scholar]
  54. Laub JH, Sampson RJ. 2001. Understanding desistance from crime. Crime Justice 28:1–69 [Google Scholar]
  55. Leiby BE, Sammel MD, Ten Have TR, Lynch KG. 2009. Identification of multivariate responders and non-responders by using Bayesian growth curve latent class models. J. R. Stat. Soc. C 58:4505–24 [Google Scholar]
  56. Leoutsakos J-MS, Muthén BO, Breitner J, Lyketsos CG. 2012. Effects of non-steroidal anti-inflammatory drug treatments on cognitive decline vary by phase of pre-clinical Alzheimer disease: findings from the randomized controlled Alzheimer's disease Anti-inflammatory Prevention Trial. Int. J. Geriatr. Psychiatry 27:4364–74 [Google Scholar]
  57. Li F, Duncan TE, Hops H. 2001. Examining developmental trajectories in adolescent alcohol use using piecewise growth mixture modeling analysis. J. Stud. Alcohol Drugs 62:2199–210 [Google Scholar]
  58. Lu ZL, Zhang Z, Lubke G. 2011. Bayesian inference for growth mixture models with latent class dependent missing data. Multivar. Behav. Res. 46:4567–97 [Google Scholar]
  59. Manrique-Vallier D. 2010. Longitudinal Mixed Membership Models with Applications to Disability Survey Data PhD Thesis, Carnegie Mellon Univ., Pittsburgh, PA [Google Scholar]
  60. McLachlan GJ, Peel D. 2000. Finite Mixture Models New York: Wiley [Google Scholar]
  61. Moffitt TE. 1993. Adolescence-limited and life-course-persistent antisocial behavior: a developmental taxonomy. Psychol. Rev. 100:674–701 [Google Scholar]
  62. Muthén B. 2004. Latent variable analysis. The Sage Handbook of Quantitative Methodology for the Social Sciences D Kaplan 345–68 Thousand Oaks, CA: Sage [Google Scholar]
  63. Muthén B, Asparouhov T. 2008. Growth mixture modeling: analysis with non-Gaussian random effects. Advances in Longitudinal Data Analysis G Fitzmaurice, M Davidian, G Verbeke, G Molenberghs 143–65 Boca Raton, FL: Chapman & Hall/CRC Press [Google Scholar]
  64. Muthén B, Brown CH, Masyn K, Jo B, Khoo S-T. et al. 2002. General growth mixture modeling for randomized preventive interventions. Biostatistics 3:4459–75 [Google Scholar]
  65. Muthén B, Brown HC. 2009. Estimating drug effects in the presence of placebo response: causal inference using growth mixture modeling. Stat. Med. 28:273363–85 [Google Scholar]
  66. Muthén B, Shedden K. 1999. Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics 55:2463–69 [Google Scholar]
  67. Muthén LK, Muthén B. 2005. Growth modeling with latent variables using Mplus Multilevel Model. in Mplus, Univ. Calif. Los Angel. Stat. Consult. Group, Los Angeles, CA. http://www.ats.ucla.edu/stat/mplus/seminars/mlmMplus_JH/Hopkins_Day1_fixed_May20051.pdf [Google Scholar]
  68. Muthén LK, Muthén BO. 2012. Mplus: Statistical Analysis with Latent Variables: User's Guide Los Angeles, CA: Muthén & Muthén, 7th ed.. [Google Scholar]
  69. Nagin DS. 2005. Group-Based Modeling of Development Cambridge, MA: Harvard Univ. Press [Google Scholar]
  70. Nagin DS, Land KC. 1993. Age, criminal careers, and population heterogeneity: specification and estimation of a nonparametric, mixed Poisson model. Criminology 31:3327–62 [Google Scholar]
  71. Nagin DS, Odgers CL. 2010a. Group-based trajectory modeling in clinical research. Annu. Rev. Clin. Psychol. 6:109–38 [Google Scholar]
  72. Nagin DS, Odgers CL. 2010b. Group-based trajectory modeling (nearly) two decades later. J. Quant. Criminol. 26:4445–53 [Google Scholar]
  73. Nagin DS, Tremblay RE. 1999. Trajectories of boys' physical aggression, opposition, and hyperactivity on the path to physically violent and nonviolent juvenile delinquency. Child Dev. 70:51181–96 [Google Scholar]
  74. Nagin DS, Tremblay RE. 2001. Analyzing developmental trajectories of distinct but related behaviors: a group-based method. Psychol. Methods 6:118–34 [Google Scholar]
  75. Nagin DS, Tremblay RE. 2005a. Developmental trajectory groups: fact or a useful statistical fiction. Criminology 43:4873–904 [Google Scholar]
  76. Nagin DS, Tremblay RE. 2005b. What has been learned from group-based trajectory modeling? Examples from physical aggression and other problem behaviors. Ann. Am. Acad. Polit. Soc. Sci. 602:82–117 [Google Scholar]
  77. Neelon B, Swamy GK, Burgette LF, Miranda ML. 2011. A Bayesian growth mixture model to examine maternal hypertension and birth outcomes. Stat. Med. 30:222721–35 [Google Scholar]
  78. Odgers CL, Moffitt TE, Broadbent JM, Dickson N, Hancox RJ. et al. 2008. Female and male antisocial trajectories: from childhood origins to adult outcomes. Dev. Psychopathol. 20:02673–716 [Google Scholar]
  79. Paternoster R, Brame R, Farrington DP. 2001. On the relationship between adolescent and adult conviction frequencies. J. Quant. Criminol. 17:3201–5 [Google Scholar]
  80. Pettit L. 1990. The conditional predictive ordinate for the normal distribution. J. R. Stat. Soc. B 52:175–84 [Google Scholar]
  81. Pickles A, Croudace T. 2010. Latent mixture models for multivariate and longitudinal outcomes. Stat. Methods Med. Res. 19:3271–89 [Google Scholar]
  82. Pinheiro JC, Bates DM. 2000. Mixed-Effects Models in S and S-Plus New York: Springer-Verlag [Google Scholar]
  83. Piquero AR. 2008. Taking stock of developmental trajectories of criminal activity over the life course. The Long View of Crime: A Synthesis of Longitudinal Research AM Liberman 23–78 New York: Springer [Google Scholar]
  84. Piquero AR, Blumstein A, Brame R, Haapanen R, Mulvey EP, Nagin DS. 2001. Assessing the impact of exposure time and incapacitation on longitudinal trajectories of criminal offending. J. Adolesc. Res. 16:154–74 [Google Scholar]
  85. Piquero AR, Brame R, Mazerolle P, Haapanen R. 2002. Crime in emerging adulthood. Criminology 40:1137–70 [Google Scholar]
  86. Qureshi I, Fang Y. 2011. Socialization in open source software projects: a growth mixture modeling approach. Organ. Res. Methods 14:1208–38 [Google Scholar]
  87. Ramsay JO, Li X. 1998. Curve registration. J. R. Stat. Soc. B 60:351–63 [Google Scholar]
  88. Raudenbush SW. 2001. Comparing personal trajectories and drawing causal inferences from longitudinal data. Annu. Rev. Psychol. 52:501–25 [Google Scholar]
  89. Raudenbush SW. 2005. How do we study “What happens next”?. Ann. Am. Acad. Polit. Soc. Sci. 602:131–44 [Google Scholar]
  90. Raudenbush SW, Bryk AS. 2002. Hierarchical Linear Models: Applications and Data Analysis Methods Newbury Park, CA: Sage [Google Scholar]
  91. Reinecke J, Seddig D. 2011. Growth mixture models in longitudinal research. Adv. Stat. Anal. 95:4415–34 [Google Scholar]
  92. Roeder K, Lynch KG, Nagin DS. 1999. Modeling uncertainty in latent class membership: a case study in criminology. J. Am. Stat. Assoc. 94:766–76 [Google Scholar]
  93. Rolfe MI, Mengersen KL, Vearncombe KJ, Andrew B, Beadle GF. 2011. Bayesian estimation of extent of recovery for aspects of verbal memory in women undergoing adjuvant chemotherapy treatment for breast cancer. J. R. Stat. Soc. C 60:5655–74 [Google Scholar]
  94. Rousseau J, Mengersen K. 2011. Asymptotic behaviour of the posterior distribution in overfitted mixture models. J. R. Stat. Soc. B 73:5689–710 [Google Scholar]
  95. Sampson RJ, Laub JH. 1993. Crime in the Making: Pathways and Turning Points Through the Life Course Cambridge, MA: Harvard Univ. Press [Google Scholar]
  96. Sampson RJ, Laub JH. 2003. Life-course desisters? Trajectories of crime among delinquent boys followed to age 70. Criminology 41:3555–92 [Google Scholar]
  97. Sampson RJ, Laub JH. 2005. A life-course view of the development of crime. Ann. Am. Acad. Polit. Soc. Sci. 602:112–45 [Google Scholar]
  98. Saunders JM. 2010. Understanding random effects in group-based trajectory modeling: an application of Moffitt's developmental taxonomy. J. Drug Issues 40:1195–220 [Google Scholar]
  99. Schwarz G. 1978. Estimating the dimension of a model. Ann. Stat. 6:2461–64 [Google Scholar]
  100. Skardhamar T. 2010. Distinguishing facts and artifacts in group-based modeling. Criminology 48:1295–320 [Google Scholar]
  101. Slaughter JC, Herring AH, Thorp JM. 2009. A Bayesian latent variable mixture model for longitudinal fetal growth. Biometrics 65:41233–42 [Google Scholar]
  102. Sobel ME, Muthén B. 2012. Compliance mixture modelling with a zero-effect complier class and missing data. Biometrics 68:41037–45 [Google Scholar]
  103. Sterba SK, Baldasaro RE, Bauer DJ. 2012. Factors affecting the adequacy and preferability of semiparametric groups-based approximations of continuous growth trajectories. Multivar. Behav. Res. 47:4590–634 [Google Scholar]
  104. Tarpey T. 2007. Linear transformations and the k-means clustering algorithm: applications to clustering curves. Am. Stat. 61:134–40 [Google Scholar]
  105. Tarpey T, Petkova E, Lu Y, Govindarajulu U. 2010. Optimal partitioning for linear mixed effects models: applications to identifying placebo responders. J. Am. Stat. Assoc. 105:491968–77 [Google Scholar]
  106. Telesca D, Erosheva E, Kreager DA, Matsueda RL. 2012. Modeling criminal careers as departures from a unimodal age-crime curve: the case of marijuana use. J. Am. Stat. Assoc. 18:2159–77 [Google Scholar]
  107. Telesca D, Inoue LYT. 2008. Bayesian hierarchical curve registration. J. Am. Stat. Assoc. 103:481328–39 [Google Scholar]
  108. Tremblay RE, Vitaro F, Nagin D, Pagani L, Séguin JR. 2003. The Montréal Longitudinal and Experimental Study: rediscovering the power of descriptions. Taking Stock of Delinquency: An Overview of Findings from Contemporary Longitudinal Studies TP Thornberry, MD Krohn 205–54 New York: Kluwer Academic/Plenum [Google Scholar]
  109. Underwood MK, Beron KJ, Rosen LH. 2009. Continuity and change in social and physical aggression from middle childhood through early adolescence. Aggress. Behav. 35:5357–75 [Google Scholar]
  110. Wang C-P, Hendricks Brown C, Bandeen-Roche K. 2005. Residual diagnostics for growth mixture models: Examining the impact of a preventive intervention on multiple trajectories of aggressive behavior. J. Am. Stat. Assoc. 100:4711054–76 [Google Scholar]
  111. Wiesner M, Capaldi DM. 2003. Relations of childhood and adolescent factors to offending trajectories of young men. J. Res. Crime Delinq. 40:3231–62 [Google Scholar]
  112. Wiesner M, Windle M. 2004. Assessing covariates of adolescent delinquency trajectories: A latent growth mixture modeling approach. J. Youth Adolesc. 33:5431–42 [Google Scholar]
  113. Wolfgang M, Figlio R, Sellin T. 1972. Delinquency in a Birth Cohort Chicago: Univ. Chicago Press [Google Scholar]

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