Detecting Multiple Random Changepoints in Bayesian Piecewise Growth Mixture Models

Psychometrika volume 83, pages733–750(2018)Cite this article

Abstract

Piecewise growth mixture models are a flexible and useful class of methods for analyzing segmented trends in individual growth trajectory over time, where the individuals come from a mixture of two or more latent classes. These models allow each segment of the overall developmental process within each class to have a different functional form; examples include two linear phases of growth, or a quadratic phase followed by a linear phase. The changepoint (knot) is the time of transition from one developmental phase (segment) to another. Inferring the location of the changepoint(s) is often of practical interest, along with inference for other model parameters. A random changepoint allows for individual differences in the transition time within each class. The primary objectives of our study are as follows: (1) to develop a PGMM using a Bayesian inference approach that allows the estimation of multiple random changepoints within each class; (2) to develop a procedure to empirically detect the number of random changepoints within each class; and (3) to empirically investigate the bias and precision of the estimation of the model parameters, including the random changepoints, via a simulation study. We have developed the user-friendly package BayesianPGMM for R to facilitate the adoption of this methodology in practice, which is available at https://github.com/lockEF/BayesianPGMM. We describe an application to mouse-tracking data for a visual recognition task.

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References

  1. Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6), 716–723.

    Article  Google Scholar 

  2. Bradway, K. P., & Thompson, C. W. (1962). Intelligence at adulthood: A twenty-five year follow-up. Journal of Educational Psychology, 53(1), 1.

    Article  Google Scholar 

  3. Brooks, S. P., & Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7(4), 434–455.

    Google Scholar 

  4. Carlin, B. P., Gelfand, A. E., & Smith, A. F. (1992). Hierarchical Bayesian analysis of changepoint problems. Applied Statistics, 41(2), 389–405.

    Article  Google Scholar 

  5. Casella, G., & George, E. I. (1992). Explaining the Gibbs sampler. The American Statistician, 46(3), 167–174.

    Google Scholar 

  6. Chib, S. (1998). Estimation and comparison of multiple change-point models. Journal of Econometrics, 86(2), 221–241.

    Article  Google Scholar 

  7. Cudeck, R., & Klebe, K. J. (2002). Multiphase mixed-effects models for repeated measures data. Psychological Methods, 7(1), 41.

    Article  PubMed  Google Scholar 

  8. Daniels, M. J., & Kass, R. E. (1999). Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models. Journal of the American Statistical Association, 94(448), 1254–1263.

    Article  Google Scholar 

  9. Delattre, M., Lavielle, M., Poursat, M.-A., et al. (2014). A note on BIC in mixed-effects models. Electronic Journal of Statistics, 8(1), 456–475.

    Article  Google Scholar 

  10. Dominicus, A., Ripatti, S., Pedersen, N. L., & Palmgren, J. (2008). A random change point model for assessing variability in repeated measures of cognitive function. Statistics in Medicine, 27(27), 5786–5798.

    Article  PubMed  PubMed Central  Google Scholar 

  11. Fearnhead, P. (2006). Exact and efficient Bayesian inference for multiple changepoint problems. Statistics and Computing, 16(2), 203–213.

    Article  Google Scholar 

  12. Fearnhead, P., & Liu, Z. (2011). Efficient Bayesian analysis of multiple changepoint models with dependence across segments. Statistics and Computing, 21(2), 217–229.

    Article  Google Scholar 

  13. Gallistel, C. R., Fairhurst, S., & Balsam, P. (2004). The learning curve: Implications of a quantitative analysis. Proceedings of the National Academy of Sciences of the United States of America, 101(36), 13124–13131.

    Article  PubMed  PubMed Central  Google Scholar 

  14. Gelman, A., et al. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Analysis, 1(3), 515–534.

    Article  Google Scholar 

  15. Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4), 457–472.

    Article  Google Scholar 

  16. Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.

    Article  PubMed  Google Scholar 

  17. Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–732.

    Article  Google Scholar 

  18. Hall, C. B., Lipton, R. B., Sliwinski, M., & Stewart, W. F. (2000). A change point model for estimating the onset of cognitive decline in preclinical Alzheimer’s disease. Statistics in Medicine, 19(11–12), 1555–1566.

    Article  PubMed  Google Scholar 

  19. Hall, C. B., Ying, J., Kuo, L., Sliwinski, M., Buschke, H., Katz, M., et al. (2001). Estimation of bivariate measurements having different change points, with application to cognitive ageing. Statistics in Medicine, 20(24), 3695–3714.

    Article  PubMed  Google Scholar 

  20. Incera, S., & McLennan, C . T. (2016). Mouse tracking reveals that bilinguals behave like experts. Bilingualism: Language and Cognition, 19(03), 610–620.

    Article  Google Scholar 

  21. Karlberg, J. (1987). On the modelling of human growth. Statistics in Medicine, 6(2), 185–192.

    Article  PubMed  Google Scholar 

  22. Karlberg, J., Jalil, F., Lam, B., Low, L., & Yeung, C. (1994). Linear growth retardation in relation to the three phases of growth. European Journal of Clinical Nutrition, 48, S25–43.

    PubMed  Google Scholar 

  23. Kieffer, M. J. (2012). Before and after third grade: Longitudinal evidence for the shifting role of socioeconomic status in reading growth. Reading and Writing, 25(7), 1725–1746.

    Article  Google Scholar 

  24. Kohli, N., Harring, J. R., & Hancock, G. R. (2013). Piecewise linear-linear latent growth mixture models with unknown knots. Educational and Psychological Measurement, 73(6), 935–955.

    Article  Google Scholar 

  25. Kohli, N., Harring, J., & Zopluoglu, C. (2016). A finite mixture of nonlinear random coefficient models for continuous repeated measures data. Psychometrika, 81(3), 851–880.

    Article  PubMed  Google Scholar 

  26. Kohli, N., Hughes, J., Wang, C., Zopluoglu, C., & Davison, M. (2015a). Fitting a linear-linear piecewise growth mixture model with unknown knots: A comparison of two common approaches to inference. Psychological Methods, 20(2), 259–275.

    Article  PubMed  Google Scholar 

  27. Kohli, N., Sullivan, A. L., Sadeh, S., & Zopluoglu, C. (2015b). Longitudinal mathematics development of students with learning disabilities and students without disabilities: A comparison of linear, quadratic, and piecewise linear mixed effects models. Journal of School Psychology, 53(2), 105–120.

    Article  PubMed  Google Scholar 

  28. Kreisman, M. B. (2003). Evaluating academic outcomes of head start: An application of general growth mixture modeling. Early Childhood Research Quarterly, 18(2), 238–254.

    Article  Google Scholar 

  29. Lai, Y., & Albert, P. S. (2014). Identifying multiple change points in a linear mixed effects model. Statistics in Medicine, 33(6), 1015–1028.

    Article  PubMed  Google Scholar 

  30. Morrell, C. H., Pearson, J. D., Carter, H. B., & Brant, L. J. (1995). Estimating unknown transition times using a piecewise nonlinear mixed-effects model in men with prostate cancer. Journal of the American Statistical Association, 90(429), 45–53.

    Article  Google Scholar 

  31. Muniz Terrera, G., Van den Hout, A., & Matthews, F. (2011). Random change point models: investigating cognitive decline in the presence of missing data. Journal of Applied Statistics, 38(4), 705–716.

    Article  Google Scholar 

  32. Myung, I. J. (2000). The importance of complexity in model selection. Journal of Mathematical Psychology, 44(1), 190–204.

    Article  PubMed  Google Scholar 

  33. Naumova, E. N., Must, A., & Laird, N. M. (2001). Tutorial in biostatistics: Evaluating the impact of critical periods in longitudinal studies of growth using piecewise mixed effects models. International Journal of Epidemiology, 30(6), 1332–1341.

    Article  PubMed  Google Scholar 

  34. Papastamoulis, P. (2014). Handling the label switching problem in latent class models via the ECR algorithm. Communications in Statistics-Simulation and Computation, 43(4), 913–927.

    Article  Google Scholar 

  35. Papastamoulis, P. (2015). label. switching: An R package for dealing with the label switching problem in MCMC outputs. arXiv preprint arXiv:1503.02271.

  36. Plummer, M. (2016). rjags: Bayesian Graphical Models using MCMC. R package version 4-6.

  37. Schwarz, G., et al. (1978). Estimating the dimension of a model. The annals of statistics, 6(2), 461–464.

    Article  Google Scholar 

  38. Sinharay, S. (2004). Experiences with Markov chain Monte Carlo convergence assessment in two psychometric examples. Journal of Educational and Behavioral Statistics, 29(4), 461–488.

    Article  Google Scholar 

  39. Spiegelhalter, D., Thomas, A., Best, N., & Gilks, W. (1996). BUGS 0.5: Bayesian inference using Gibbs sampling manual (pp. 1–59). Cambridge: MRC Biostatistics Unit, Institute of Public Health.

    Google Scholar 

  40. Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4), 583–639.

    Article  Google Scholar 

  41. Sullivan, A. L., Kohli, N., Farnsworth, E. M., Sadeh, S., & Jones, L. (2017). Longitudinal models of reading achievement of students with learning disabilities and without disabilities. School Psychology Quarterly, 32(3), 336–349.

  42. Vaida, F., & Blanchard, S. (2005). Conditional Akaike information for mixed-effects models. Biometrika, 92(2), 351–370.

    Article  Google Scholar 

  43. Vehtari, A., Gelman, A., & Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5), 1413–1432.

    Article  Google Scholar 

  44. Wang, L., & McArdle, J. J. (2008). A simulation study comparison of Bayesian estimation with conventional methods for estimating unknown change points. Structural Equation Modeling, 15(1), 52–74.

    Article  Google Scholar 

  45. Wang, Y., & Fang, Y. (2009). Least square and empirical Bayes approaches for estimating random change points. Journal of Data Science, 7(1), 111–127.

    Google Scholar 

  46. Zhao, L., & Banerjee, M. (2012). Bayesian piecewise mixture model for racial disparity in prostate cancer progression. Computational Statistics & Data Analysis, 56(2), 362–369.

    Article  Google Scholar 

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Acknowledgements

We would like to thank Sara Incera and Conor T. McLennan of Cleveland State University for graciously providing the mouse-tracking data described in Sect. 4. This work was supported in part by NIH grant ULI RR033183/KL2 RR0333182 [to EFL].

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Affiliations

  1. Division of Biostatistics, School of Public Health, University of Minnesota, A460 Mayo Building, MMC 303 420 Delaware Street, S.E., Minneapolis, MN , 55455, USA

    Eric F. Lock & Maitreyee Bose

  2. Department of Educational Psychology, College of Education and Human Development, University of Minnesota, Minneapolis, MN, USA

    Nidhi Kohli

Authors
  1. Eric F. Lock
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  2. Nidhi Kohli
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  3. Maitreyee Bose
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Correspondence to Eric F. Lock.

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Lock, E.F., Kohli, N. & Bose, M. Detecting Multiple Random Changepoints in Bayesian Piecewise Growth Mixture Models. Psychometrika 83, 733–750 (2018). https://doi.org/10.1007/s11336-017-9594-5

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Keywords

  • Bayesian
  • longitudinal data
  • Markov chain Monte Carlo
  • mixture model
  • nonlinear random effects models
  • piecewise function