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Covariance pattern mixture models: Eliminating random effects to improve convergence and performance

  • Daniel McNeishEmail author
  • Jeffrey Harring
Article

Abstract

Growth mixture models (GMMs) are prevalent for modeling unknown population heterogeneity via distinct latent classes. However, GMMs are riddled with convergence issues, often requiring researchers to atheoretically alter the model with cross-class constraints simply to obtain convergence. We discuss how within-class random effects in GMMs exacerbate convergence issues, even though these random effects rarely help answer typical research questions. That is, latent classes provide a discretization of continuous random effects, so including additional random effects within latent classes can unnecessarily complicate the model. These random effects are commonly included in order to properly specify the marginal covariance; however, random effects are inefficient for patterning a covariance matrix, resulting in estimation issues. Such a goal can be achieved more simply through covariance pattern models, which we extend to the mixture model context in this article (covariance pattern mixture models, or CPMMs). We provide evidence from theory, simulation, and an empirical example showing that employing CPMMs (even if they are misspecified) instead of GMMs can circumvent the computational difficulties that can plague GMMs, without sacrificing the ability to answer the types of questions commonly asked in empirical studies. Our results show the advantages of CPMMs with respect to improved class enumeration and less biased class-specific growth trajectories, in addition to their vastly improved convergence rates. The results also show that constraining the covariance parameters across classes in order to bypass convergence issues with GMMs leads to poor results. An extensive software appendix is included to assist researchers in running CPMMs in Mplus.

Keywords

Finite mixture modeling Convergence Growth mixture modeling Constraints Latent class analysis 

Notes

References

  1. Azevedo, C. L., Fox, J. P., & Andrade, D. F. (2016). Bayesian longitudinal item response modeling with restricted covariance pattern structures. Statistics and Computing, 26, 443–460.CrossRefGoogle Scholar
  2. Bauer, D. J., & Curran, P. J. (2003). Distributional assumptions of growth mixture models: Implications for overextraction of latent trajectory classes. Psychological Methods, 8, 338–363.  https://doi.org/10.1037/1082-989X.8.3.338 CrossRefPubMedGoogle Scholar
  3. Bergman, L. R., & Magnusson, D. (1997). A person-oriented approach in research on developmental psychopathology. Development and Psychopathology, 9, 291–319.  https://doi.org/10.1017/S095457949700206X CrossRefPubMedGoogle Scholar
  4. Bonanno, G. A. (2004). Loss, trauma, and human resilience: Have we underestimated the human capacity to thrive after extremely aversive events? American Psychologist, 59, 20–28.  https://doi.org/10.1037/0003-066X.59.1.20 CrossRefPubMedGoogle Scholar
  5. Burton, P., Gurrin, L., & Sly, P. (1998). Tutorial in biostatistics: Extending the simple linear regression model to account for correlated responses: An introduction to generalized estimating equations and multi-level mixed modeling. Statistics in Medicine, 17, 1261–1291.  https://doi.org/10.1002/0470023724.ch1a CrossRefPubMedGoogle Scholar
  6. Codd, C. L., & Cudeck, R. (2014). Nonlinear random-effects mixture models for repeated measures. Psychometrika, 79, 60–83.  https://doi.org/10.1007/s11336-013-9358-9 CrossRefPubMedGoogle Scholar
  7. Cole, V. T., & Bauer, D. J. (2016). A note on the use of mixture models for individual prediction. Structural Equation Modeling, 23, 615–631.  https://doi.org/10.1080/10705511.2016.1168266 CrossRefPubMedPubMedCentralGoogle Scholar
  8. Cudeck, R., & Codd, C. L. (2012). A template for describing individual differences in longitudinal data with application to the connection between learning and ability. In J. R. Harring & G. R. Hancock (Eds.), Advances in longitudinal methods in the social and behavioral sciences (pp. 3–24). Charlotte, NC: Information AgeGoogle Scholar
  9. Davidian, M., & Giltinan, D. M. (1995). Nonlinear models for repeated measurement data. New York, NY: Chapman & Hall.Google Scholar
  10. Depaoli, S., van de Schoot, R., van Loey, N., & Sijbrandij, M. (2015). Using Bayesian statistics for modeling PTSD through Latent Growth Mixture Modeling: Implementation and discussion. European Journal of Psychotraumatology, 6, 27516.  https://doi.org/10.3402/ejpt.v6.27516 CrossRefPubMedGoogle Scholar
  11. Diallo, T. M., Morin, A. J., & Lu, H. (2016). Impact of misspecifications of the latent variance–covariance and residual matrices on the class enumeration accuracy of growth mixture models. Structural Equation Modeling, 23, 507–531.  https://doi.org/10.1080/10705511.2016.1169188 CrossRefGoogle Scholar
  12. Diallo, T. M., Morin, A. J., & Lu, H. (2017). The impact of total and partial inclusion or exclusion of active and inactive time invariant covariates in growth mixture models. Psychological Methods, 22, 166–190.CrossRefPubMedGoogle Scholar
  13. Diggle, P. J., Heagerty, P., Liang, K. Y., & Zeger, S. L. (2002). Analysis of longitudinal data (2nd). New York, NY: Oxford University Press.Google Scholar
  14. Dziak, J. J., Lanza, S. T., & Tan, X. (2014). Effect size, statistical power, and sample size requirements for the bootstrap likelihood ratio test in latent class analysis. Structural Equation Modeling, 21, 534–552.CrossRefPubMedPubMedCentralGoogle Scholar
  15. Enders, C. K., & Tofighi, D. (2008). The impact of misspecifying class-specific residual variances in growth mixture models. Structural Equation Modeling, 15, 75–95.  https://doi.org/10.1080/10705510701758281 CrossRefGoogle Scholar
  16. Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2011) Applied longitudinal analysis (2nd). Philadelphia, PA: Wiley.CrossRefGoogle Scholar
  17. Flora, D. B., & Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9, 466–491.  https://doi.org/10.1037/1082-989X.9.4.466 CrossRefPubMedPubMedCentralGoogle Scholar
  18. Grimm, K. J., & Stegmann, G. (2019). Modeling change trajectories with count and zero-inflated outcomes: Challenges and recommendations. Addictive Behaviors, 94, 4–15.  https://doi.org/10.1016/j.addbeh.2018.09.016 CrossRefPubMedGoogle Scholar
  19. Grimm, K. J., & Widaman, K. F. (2010). Residual structures in latent growth curve modeling. Structural Equation Modeling, 17, 424–442.  https://doi.org/10.1080/10705511.2010.489006 CrossRefGoogle Scholar
  20. Harring, J. R., & Blozis, S. A. (2014). Fitting correlated residual error structures in nonlinear mixed-effects models using SAS PROC NLMIXED. Behavior Research Methods, 46, 372–384.CrossRefPubMedGoogle Scholar
  21. Harring, J. R., & Blozis, S. A. (2016). A note on recurring misconceptions when fitting nonlinear mixed models. Multivariate Behavioral Research, 51, 805–817.  https://doi.org/10.1080/00273171.2016.1239522 CrossRefPubMedGoogle Scholar
  22. Harring, J. R., & Hodis, F. A. (2016). Mixture modeling: Applications in educational psychology. Educational Psychologist, 51, 354–367.  https://doi.org/10.1080/00461520.2016.1207176 CrossRefGoogle Scholar
  23. Hox, J. (2010). Multilevel analyses: Techniques and applications. Mahwah, NJ: ErlbaumCrossRefGoogle Scholar
  24. Infurna, F. J., & Grimm, K. J. (2017). The use of growth mixture modeling for studying resilience to major life stressors in adulthood and old age: Lessons for class size and identification and model selection. Journals of Gerontology, 73B, 148–159.  https://doi.org/10.1093/geronb/gbx019 Google Scholar
  25. Infurna, F. J., & Luthar, S. S. (2016). Resilience to major life stressors is not as common as thought. Perspectives on Psychological Science, 11, 175–194.  https://doi.org/10.1177/1745691615621271 CrossRefPubMedPubMedCentralGoogle Scholar
  26. Jennrich, R. I., & Schluchter, M. D. (1986). Unbalanced repeated-measures models with structured covariance matrices. Biometrics, 42, 805–820.  https://doi.org/10.2307/2530695 CrossRefPubMedGoogle Scholar
  27. Jung, T., & Wickrama, K. A. S. (2007). An introduction to latent class growth analysis and growth mixture modeling. Social and Personality Psychology Compass, 2, 302–317.CrossRefGoogle Scholar
  28. Kreuter, F., & Muthén, B. (2008). Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling. Journal of Quantitative Criminology, 24, 1–31.  https://doi.org/10.1007/s10940-007-9036-0 CrossRefGoogle Scholar
  29. Laursen, B., & Hoff, E. (2006). Person-centered and variable-centered approaches to longitudinal data. Merrill-Palmer Quarterly, 52, 377–389.CrossRefGoogle Scholar
  30. Li, M., Harring, J. R., & Macready, G. B. (2014). Investigating the feasibility of using Mplus in the estimation of growth mixture models. Journal of Modern Applied Statistical Methods, 13, 31.  https://doi.org/10.22237/jmasm/1398918600 CrossRefGoogle Scholar
  31. Liang, K. Y., & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13–22.  https://doi.org/10.1093/biomet/73.1.13 CrossRefGoogle Scholar
  32. Liu, M., & Hancock, G. R. (2014). Unrestricted mixture models for class identification in growth mixture modeling. Educational and Psychological Measurement, 74, 557–584.  https://doi.org/10.1177/0013164413519798 CrossRefGoogle Scholar
  33. Liu, S., Rovine, M. J., & Molenaar, P. (2012a). Selecting a linear mixed model for longitudinal data: Repeated measures analysis of variance, covariance pattern model, and growth curve approaches. Psychological Methods, 17, 15–30.  https://doi.org/10.1037/a0026971 CrossRefPubMedGoogle Scholar
  34. Liu, S., Rovine, M. J., & Molenaar, P. C. (2012b). Using fit indexes to select a covariance model for longitudinal data. Structural Equation Modeling, 19, 633–650.CrossRefGoogle Scholar
  35. Lix, L., & Sajobi, T. (2010). Discriminant analysis for repeated measures data: A review. Frontiers in Psychology, 1, 146.CrossRefPubMedPubMedCentralGoogle Scholar
  36. McLachlan, G. J. (1987). On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Applied Statistics, 36, 318–324.CrossRefGoogle Scholar
  37. McNeish, D., & Matta, T. (2018). Differentiating between mixed effects and latent curve approaches to growth modeling. Behavior Research Methods, 50, 1398–1414.  https://doi.org/10.3758/s13428-017-0976-5 CrossRefPubMedGoogle Scholar
  38. McNeish, D., Stapleton, L. M., & Silverman, R. D. (2017). On the unnecessary ubiquity of hierarchical linear modeling. Psychological Methods, 22, 114–140.  https://doi.org/10.1037/met0000078 CrossRefPubMedGoogle Scholar
  39. Mehta, P. D., & West, S. G. (2000). Putting the individual back into individual growth curves. Psychological Methods, 5, 23–43.  https://doi.org/10.1037/1082-989X.5.1.23 CrossRefPubMedGoogle Scholar
  40. Molenaar, P. C. (2004). A manifesto on psychology as idiographic science: Bringing the person back into scientific psychology, this time forever. Measurement, 2, 201–218.  https://doi.org/10.1207/s15366359mea0204_1 CrossRefGoogle Scholar
  41. Molenaar, P. C., & Campbell, C. G. (2009). The new person-specific paradigm in psychology. Current Directions in Psychological Science, 18, 112–117.CrossRefGoogle Scholar
  42. Morin, A. J., Maïano, C., Nagengast, B., Marsh, H. W., Morizot, J., & Janosz, M. (2011). General growth mixture analysis of adolescents’ developmental trajectories of anxiety: The impact of untested invariance assumptions on substantive interpretations. Structural Equation Modeling, 18, 613–648.  https://doi.org/10.1111/j.1467-8721.2009.01619.x CrossRefGoogle Scholar
  43. Musu-Gillette, L. E., Wigfield, A., Harring, J. R., & Eccles, J. S. (2015). Trajectories of change in students’ self-concepts of ability and values in math and college major choice. Educational Research and Evaluation, 21, 343–370.  https://doi.org/10.1080/13803611.2015.1057161 CrossRefGoogle Scholar
  44. Muthén, B., & Muthén, L. K. (2000). Integrating person-centered and variable-centered analyses: Growth mixture modeling with latent trajectory classes. Alcoholism: Clinical and experimental research, 24, 882–891.  https://doi.org/10.1111/j.1530-0277.2000.tb02070.x CrossRefGoogle Scholar
  45. Muthén, B., & Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463–469.  https://doi.org/10.1111/j.0006-341X.1999.00463.x CrossRefPubMedGoogle Scholar
  46. Muthén, B. O. (2001). 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. M., & Sayer, A. (Eds.), New methods for the analysis of change (pp. 291–322). Washington, DC: American Psychological Association.CrossRefGoogle Scholar
  47. Muthén, B. O., & Curran, P. J. (1997). General longitudinal modeling of individual differences in experimental designs: A latent variable framework for analysis and power estimation. Psychological Methods, 2, 371–402.  https://doi.org/10.1037/1082-989X.2.4.371 CrossRefGoogle Scholar
  48. Nagin, D. S. (1999). Analyzing developmental trajectories: A semiparametric, group-based approach. Psychological Methods, 4, 139–157.  https://doi.org/10.1037/1082-989X.4.2.139 CrossRefGoogle Scholar
  49. Nagin, D. S. (2005). Group-based modeling of development. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar
  50. Nagin, D. S., & Tremblay, R. E. (2001). Analyzing developmental trajectories of distinct but related behaviors: A group-based method. Psychological Methods, 6, 18–34.  https://doi.org/10.1037/1082-989X.6.1.18 CrossRefPubMedGoogle Scholar
  51. Nylund, K. L., Asparouhov, T., & Muthén, B. O. (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study. Structural Equation Modeling, 14, 535–569.  https://doi.org/10.1080/10705510701575396 CrossRefGoogle Scholar
  52. Nylund-Gibson, K., Grimm, R., Quirk, M., & Furlong, M. (2014). A latent transition mixture model using the three-step specification. Structural Equation Modeling, 21, 439–454.CrossRefGoogle Scholar
  53. Nylund-Gibson, K., & Masyn, K. E. (2016). Covariates and mixture modeling: Results of a simulation study exploring the impact of misspecified effects on class enumeration. Structural Equation Modeling, 23, 782–797.  https://doi.org/10.1080/10705511.2016.1221313 CrossRefGoogle Scholar
  54. Petras, H., & Masyn, K. (2010). General growth mixture analysis with antecedents and consequences of change. In A. Piquero & D. Weisburd (Eds.), Handbook of quantitative criminology (pp. 69–100). New York, NY: Springer.CrossRefGoogle Scholar
  55. Sclove, S. L. (1987). Application of model-selection criteria to some problems in multivariate analysis. Psychometrika, 52, 333–343.CrossRefGoogle Scholar
  56. Sher, K. J., Jackson, K. M., & Steinley, D. (2011). Alcohol use trajectories and the ubiquitous cat’s cradle: Cause for concern? Journal of Abnormal Psychology, 120, 322–335.  https://doi.org/10.1037/a0021813 CrossRefPubMedPubMedCentralGoogle Scholar
  57. Sterba, S. K., & Bauer, D. J. (2010). Matching method with theory in person-oriented developmental psychopathology research. Development and Psychopathology, 22, 239–254.  https://doi.org/10.1017/S0954579410000015 CrossRefPubMedGoogle Scholar
  58. Sterba, S. K., & Bauer, D. J. (2014). Predictions of individual change recovered with latent class or random coefficient growth models. Structural Equation Modeling, 21, 342–360.  https://doi.org/10.1080/10705511.2014.915189 CrossRefGoogle Scholar
  59. Tofighi, D., & Enders, C. K. (2007). Identifying the correct number of classes in a growth mixture models. In G. R. Hancock & K. M. Samuelsen (Eds.), Advances in latent variable mixture models (pp. 317–341). Greenwich, CT: Information Age.Google Scholar
  60. Van De Schoot, R., Sijbrandij, M., Winter, S. D., Depaoli, S., & Vermunt, J. K. (2017). The GRoLTS checklist: guidelines for reporting on latent trajectory studies. Structural Equation Modeling, 24, 451–467.Google Scholar
  61. van de Schoot, R., Sijbrandij, M., Depaoli, S., Winter, S. D., Olff, M., & Van Loey, N. E. (2018). Bayesian PTSD-trajectory analysis with informed priors based on a systematic literature search and expert elicitation. Multivariate Behavioral Research, 53, 267–291.  https://doi.org/10.1080/00273171.2017.1412293 CrossRefPubMedGoogle Scholar
  62. Verbeke, G., & Lesaffre, E. (1996). A linear mixed-effects model with heterogeneity in the random-effects population. Journal of the American Statistical Association, 91, 217–221.CrossRefGoogle Scholar
  63. Verbeke, G., & Molenberghs, G. (2000). Linear mixed models for longitudinal data. New York, NY: Springer.Google Scholar
  64. Vonesh, E. F. (2013). Generalized linear and nonlinear models for correlated data: Theory and applications using SAS. New York, NY: Springer.Google Scholar
  65. Wilkinson, L., & the Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594–604.  https://doi.org/10.1037/0003-066X.54.8.594 CrossRefGoogle Scholar
  66. Yang, C. C. (2006). Evaluating latent class analysis models in qualitative phenotype identification. Computational Statistics and Data Analysis, 50, 1090–1104.CrossRefGoogle Scholar

Copyright information

© The Psychonomic Society, Inc. 2019

Authors and Affiliations

  1. 1.Arizona State UniversityTempeUSA
  2. 2.University of MarylandCollege ParkUSA

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