Model Selection Criteria for Latent Growth Models Using Bayesian Methods

  • Zhenqiu (Laura) LuEmail author
  • Zhiyong Zhang
  • Allan Cohen
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 89)


Research in applied areas, such as statistical, psychological, behavioral, and educational areas, often involves the selection of the best available model from among a large set of candidate models. Considering that there is no well-defined model selection criterion in a Bayesian context and that latent growth mixture models are becoming popular in many areas, the goal of this study is to investigate the performance of a series of model selection criteria in the framework of latent growth mixture models with missing data and outliers in a Bayesian context. This study conducted five simulation studies to cover different cases, including latent growth curve models with missing data, latent growth curve models with missing data and outliers, growth mixture models with missing data and outliers, extended growth mixture models with missing data and outliers, and latent growth models with different classes. Simulation results show that almost all the proposed criteria can effectively identify the true models. This study also illustrated the application of these model selection criteria in real data analysis. The results will help inform the selection of growth models by researchers seeking to provide states with accurate estimates of the growth of their students.


Growth Curve Model Model Selection Criterion Growth Mixture Model Latent Growth Curve Model Latent Growth Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank the reviewer Dr. Daniel Bolt for his very helpful comments and suggestions, which greatly improved the quality of this article.


  1. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Automat Contr 1919(6):716–723CrossRefMathSciNetGoogle Scholar
  2. Anderson TW, Bahadur RR (1962) Classification into two multivariate normal distributions with different covariance matrices. Ann Math Stat 33:420–431CrossRefzbMATHMathSciNetGoogle Scholar
  3. Bartholomew DJ, Knott M (1999) Latent variable models and factor analysis: Kendall’s library of statistics, vol 7, 2nd edn. Edward Arnold, New YorkGoogle Scholar
  4. Bozdogan H (1987) Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika 52:345–370CrossRefzbMATHMathSciNetGoogle Scholar
  5. Bureau of Labor Statistics, U.S. Department of Labor (1997) National longitudinal survey of youth 1997 cohort, 1997–2003 (rounds 1–7). [computer file]. Produced by the National Opinion Research Center, the University of Chicago and distributed by the Center for Human Resource Research, The Ohio State University. Columbus, OH: 2005. Retrieved from
  6. Casella G, George EI (1992) Explaining the Gibbs sampler. Am Stat 46(3):167–174MathSciNetGoogle Scholar
  7. Celeux G, Forbes F, Robert C, Titterington D (2006). Deviance information criteria for missing data models. Bayesian Anal 4:651–674CrossRefMathSciNetGoogle Scholar
  8. Dunson DB (2000) Bayesian latent variable models for clustered mixed outcomes. J R Stat Soc B 62:355–366CrossRefMathSciNetGoogle Scholar
  9. Geweke J (1992) Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In: Bernado JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 4. Clarendon Press, Oxford, pp 169–193Google Scholar
  10. Glynn RJ, Laird NM, Rubin DB (1986) In: Wainer H (ed) Drawing inferences from self-selected samples. Springer, New York, pp 115–142CrossRefGoogle Scholar
  11. Little RJA (1993) Pattern-mixture models for multivariate incomplete data. J Am Stat Assoc 88:125–134zbMATHGoogle Scholar
  12. Little RJA (1995) Modelling the drop-out mechanism in repeated-measures studies. J Am Stat Assoc 90:1112–1121CrossRefzbMATHMathSciNetGoogle Scholar
  13. Little RJA, Rubin DB (2002) Statistical analysis with missing data, 2nd edn. Wiley-Interscience, New YorkCrossRefzbMATHGoogle Scholar
  14. Lu Z, Zhang Z (2014) Robust growth mixture models with non-ignorable missingness data: models, estimation, selection, and application. Comput Stat Data Anal 71:220–240CrossRefGoogle Scholar
  15. Lu Z, Zhang Z, Lubke G (2011) Bayesian inference for growth mixture models with latent-class-dependent missing data. Multivariate Behav Res 46:567–597CrossRefGoogle Scholar
  16. Lu Z, Zhang Z, Cohen A (2013a) Bayesian inference for latent growth curve models with non-ignorable missing data. Struct Equ Modeling (manuscript submitted for publication)Google Scholar
  17. Lu ZL, Zhang Z, Cohen A (2013b) In: Millsap RE, van der Ark LA, Bolt DM, Woods CM (eds) New developments in quantitative psychology, vol 66. Springer, New York, pp 275–304CrossRefGoogle Scholar
  18. Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics: theory and methods. Wiley, New YorkGoogle Scholar
  19. McLachlan GJ, Peel D (2000) Finite mixture models. Wiley, New YorkzbMATHGoogle Scholar
  20. Muthén B, Shedden K (1999) Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics 55(2):463–469CrossRefzbMATHGoogle Scholar
  21. Oldmeadow C, Keith JM (2011) Model selection in Bayesian segmentation of multiple DNA alignments. Bioinformatics 27:604–610CrossRefGoogle Scholar
  22. Schwarz GE (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464CrossRefzbMATHGoogle Scholar
  23. Sclove LS (1987) Application of mode-selection criteria to some problems in multivariate analysis. Psychometrics 52:333–343CrossRefGoogle Scholar
  24. Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc Series B Stat Methodol 64(4):583–639CrossRefzbMATHGoogle Scholar
  25. Spiegelhalter DJ, Thomas A, Best N, Lunn D (2003) WinBUGS manual Version 1.4 (MRC Biostatistics Unit, Institute of Public Health, Robinson Way, Cambridge CB2 2SR, UK,
  26. Yuan K-H, Lu Z (2008) SEM with missing data and unknown population using two-stage ML: theory and its application. Multivariate Behav Res 43:621–652CrossRefGoogle Scholar
  27. Zhang Z, Lai K, Lu Z, Tong X (2013) Bayesian inference and application of robust growth curve models using student’s t distribution. Struct Equ Modeling 20(1):47–78CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Zhenqiu (Laura) Lu
    • 1
    Email author
  • Zhiyong Zhang
    • 2
  • Allan Cohen
    • 1
  1. 1.University of GeorgiaAthensUSA
  2. 2.University of Notre DameNotre DameUSA

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