Fitting a linear-linear piecewise growth mixture model with unknown knots: A comparison of two common approaches to inference

Psychological Methods

Volumn 20, Issue 2, 2015, Pages 259-275

  Kohli, Nidhi ^a   Hughes, John ^a   Wang, Chun ^a   Zopluoglu, Cengiz ^b   Davison, Mark L ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

^b UNIVERSITY OF MIAMI   (United States)

Author keywords

Bayesian; Finite mixture; Longitudinal data; Maximum likelihood; Piecewise function

Indexed keywords

ALGORITHM; BAYES THEOREM; COMPARATIVE STUDY; LONGITUDINAL STUDY; MARKOV CHAIN; MONTE CARLO METHOD; STATISTICAL MODEL;

ALGORITHMS; BAYES THEOREM; LIKELIHOOD FUNCTIONS; LINEAR MODELS; LONGITUDINAL STUDIES; MARKOV CHAINS; MODELS, STATISTICAL; MONTE CARLO METHOD;

EID: 84929509342     PISSN: 1082989X     EISSN: None     Source Type: Journal    
DOI: 10.1037/met0000034     Document Type: Article

References (56)

1. Barnard, J., McCulloch, R., & Meng, X.-L. (2000). Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. Statistica Sinica, 10, 1281-1312.
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. http://dx.doi.org/10.1037/1082-989X.8.3.338
3. Bradway, K., & Thompson, C. (1962). Intelligence at adulthood: A twentyfive year followup. Journal of Educational Psychology, 53, 1-4. http://dx.doi.org/10.1037/h0045764
4. Brooks, S., Gelman, A., Jones, G. L., & Meng, X.-L. (Eds.). (2011). Handbook of Markov chain Monte Carlo. London: Chapman and Hall/CRC. http://dx.doi.org/10.1201/b10905
5. Chou, C.-P., Yang, D., Pentz, M. A., & Hser, Y.-I. (2004). Piecewise growth curve modeling approach for longitudinal prevention study. Computational Statistics & Data Analysis, 46, 213-225. http://dx.doi .org/10.1016/S0167-9473(03)00149-X
6. Congdon, P. (2001). Bayesian statistical modeling. New York, NY: Wiley.
7. Cudeck, R. (1996). Mixed-effects models in the study of individual differences with repeated measures data. Multivariate Behavioral Research, 31, 371-403. http://dx.doi.org/10.1207/s15327906mbr3103_6
8. Cudeck, R., & Browne, M. W. (1983). Cross-validation of covariance structures. Multivariate Behavioral Research, 18, 147-167. http://dx.doi .org/10.1207/s15327906mbr1802_2
9. Cudeck, R., & Codd, C. (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 Age Publishing.
10. Cudeck, R., & Harring, J. R. (2010). Developing a random coefficient model for nonlinear repeated measures data. In S.-M. Chow, E. Ferrer, & F. Hsieh (Eds.), Statistical methods for modeling human dynamics: An interdisciplinary dialogue (pp. 289-318). New York, NY: Routledge.
11. Cudeck, R., & Klebe, K. J. (2002). Multiphase mixed-effects models for repeated measures data. Psychological Methods, 7, 41-63. http://dx.doi .org/10.1037/1082-989X.7.1.41
12. Depaoli, S. (2013). Mixture class recovery in GMM under varying degrees of class separation: Frequentist versus Bayesian estimation. Psychological Methods, 18, 186-219. http://dx.doi.org/10.1037/a0031609
13. 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, 5786-5798. http://dx.doi.org/10.1002/sim.3380
14. Flegal, J., Haran, M., & Jones, G. (2008). Markov chain Monte Carlo: Can we trust the third significant figure? Statistical Science, 23, 250-260. http://dx.doi.org/10.1214/08-STS257
15. Fox, P. W., Hershberger, S. L., & Bouchard, T. J., Jr. (1996). Genetic and environmental contributions to the acquisition of a motor skill. Nature, 384, 356-358. http://dx.doi.org/10.1038/384356a0
16. Fraser, D. A. S. (2011). Is Bayes posterior just quick and dirty confidence? Statistical Science, 26, 299-316. http://dx.doi.org/10.1214/11-STS352
17. Frensch, P. A., Lindenberger, U., & Kray, J. (1999). Imposing structure on an unstructured environment: Ontogenetic changes in the ability to form rules of behavior under conditions of low environmental predictability. In A. D. Friederici & R. Menzel (Eds.), Learning: Rule extraction and representation (pp. 139-162). Berlin: De Gruyter.
18. Fruhwirth-Schnatter, S. (2006). Finite mixture and Markov switching models. New York, NY: Springer.
19. Grimm, K. J. (2008). Longitudinal associations between reading and mathematics achievement. Developmental Neuropsychology, 33, 410-426. http://dx.doi.org/10.1080/87565640801982486
20. 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, 1555-1566. http://dx.doi.org/10.1002/(SICI)1097-0258(20000615/30)19:11/12_1555::AID-SIM445_3.0.CO;2-3
21. Hall, C. B., Ying, J., Kuo, L., Sliwinski, M., Buschke, H., Katz, M., & Lipton, R. B. (2001). Estimation of bivariate measurements having different change points, with application to cognitive ageing. Statistics in Medicine, 20, 3695-3714. http://dx.doi.org/10.1002/sim.1113
22. Hamilton, J. (2009). An investigation of growth mixture models when data are collected with unequal selection probabilities: A Monte Carlo study. Unpublished doctoral dissertation, University of Maryland, College Park.
23. Haran, M., & Hughes, J. (2012). Batchmeans: Consistent batch means estimation of Monte Carlo standard errors. Minneapolis, MN. R package version 1.0 -1.
24. Harring, J. R., Cudeck, R., & du Toit, S. H. C. (2006). Fitting partially nonlinear random coefficient models as SEMs. Multivariate Behavioral Research, 41, 579-596. http://dx.doi.org/10.1207/s15327906mbr4104_7
25. Haughton, D. M. A., Oud, J. H. L., & Jansen, R. A. R. G. (1997). Information and other criteria in structural equation model selection. Communications in Statistics Simulation and Computation, 26, 1477-1516. http://dx.doi.org/10.1080/03610919708813451
26. Hipp, J. R., & Bauer, D. J. (2006). Local solutions in the estimation of growth mixture models. Psychological Methods, 11, 36-53. http://dx .doi.org/10.1037/1082-989X.11.1.36
27. Jasra, A., Holmes, C., & Stephens, D. (2005). Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Statistical Science, 20, 50 - 67. http://dx.doi.org/10.1214/088342305000000016
28. Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematics difficulties versus children with comorbid mathematics and reading difficulties. Child Development, 74, 834-850. http://dx.doi.org/10.1111/1467-8624.00571
29. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773-795. http://dx.doi.org/10.1080/01621459.1995.10476572
30. Kohli, N., Harring, J. R., & Hancock, G. R. (2013). Piecewise linear-linear latent growth mixture models with unknown knots. Educational and Psychological Measurement, 73, 935-955. http://dx.doi.org/10.1177/0013164413496812
31. Kreisman, M. B. (2003). Evaluating academic outcomes of Head Start: An application of general growth mixture modeling. Early Childhood Research Quarterly, 18, 238 -254. http://dx.doi.org/10.1016/S0885-2006(03)00027-9
32. Li, F., Duncan, T., Duncan, S., & Hops, H. (2001). Piecewise growth mixture modeling of adolescent alcohol use data. Structural Equation Modeling, 8, 175-204. http://dx.doi.org/10.1207/S15328007SEM0802_2
33. Lubke, G., & Muthén, B. (2007). Performance of factor mixture models as a function of model size, covariate effects, and class-specific parameters. Structural Equation Modeling, 14, 26-47. http://dx.doi.org/10.1207/s15328007sem1401_2
34. McCrady, B. S., Epstein, E. E., Cook, S., Jensen, N., & Hildebrandt, T. (2009). A randomized trial of individual and couple behavioral alcohol treatment for women. Journal of Consulting and Clinical Psychology, 77, 243-256. http://dx.doi.org/10.1037/a0014686
35. Meredith, W., & Tisak, J. (1990). Latent curve analysis. Psychometrika, 55, 107-122. http://dx.doi.org/10.1007/BF02294746
36. Miles, T. R., & Miles, E. (1992). Dyslexia and mathematics. London:Routledge.
37. Muthén, B. (2001). Latent variable mixture modeling. In G. A. Marcoulides & R. E. Schumacker (Eds.), New developments and techniques in structural equation modeling (pp. 1-33). Mahwah, NJ: Erlbaum.
38. Muthén, B. (2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29, 81-117. http://dx.doi.org/10.2333/bhmk.29.81
39. 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. http://dx.doi.org/10.1111/j.1530-0277.2000.tb02070.x
40. Muthén, B., & Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463-469. http://dx .doi.org/10.1111/j.0006-341X.1999.00463.x
41. Muthén, L. K., & Muthén, B. O. (1998 -2010). Mplus user's guide (6th ed.). Los Angeles, CA: Muthén & Muthén.
42. Nylund, K. L., Asparouhov, T., & Muthén, B. (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. http://dx.doi.org/10.1080/10705510701575396
43. Oehlert, G. W. (1992). A note on the delta method. The American Statistician, 46, 27-29.
44. Paris, S. G. (2005). Reinterpreting the development of reading skills. Reading Research Quarterly, 40, 184-202. http://dx.doi.org/10.1598/RRQ.40.2.3
45. Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd International Workshop on Distributed Statistical Computing. March 20-22, 2003, Technische Universität Wien, Vienna, Austria.
46. Preacher, K. J., & Hancock, G. R. (in press). Meaningful aspects of change as novel random coefficients: A general method for reparameterizing longitudinal models. Psychological Methods.
47. R Development Core Team. (2009). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.
48. Robert, C. P. (2001). The Bayesian choice (2nd ed.). New York, NY: Springer.
49. Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461-464. http://dx.doi.org/10.1214/aos/1176344136
50. Shin, T., Davison, M. L., Long, J., Chan, C.-K., & Heistad, D. (2013). Exploring gains in reading and mathematics among regular and exceptional students using growth curve modeling. Learning and Individual Differences, 23, 92-100. http://dx.doi.org/10.1016/j.lindif.2012.10.002
51. Silverman, R. D., Speece, D. L., Harring, J. R., & Ritchey, K. D. (2012). Fluency has a role in the simple view of reading. Scientific Studies of Reading, 16, 1-26.
52. Steiger, J. H., & Lind, J. C. (May, 1980). Statistically based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society; Iowa City, IA.
53. Tolvanen, A. (2008). Latent growth mixture modeling: A simulation study. Unpublished doctoral dissertation, University of Jyvaskyla.
54. Tueller, S. J., Drotar, S., & Lubke, G. H. (2011). Addressing the problem of switched class labels in latent variable mixture model simulation studies. Structural Equation Modeling, 18, 110-131. http://dx.doi.org/10.1080/10705511.2011.534695
55. 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, 52-74. http://dx.doi .org/10.1080/10705510701758265
56. Yao, W., & Lindsay, B. G. (2009). Bayesian mixture labeling by highest posterior density. Journal of the American Statistical Association, 104, 758-767. http://dx.doi.org/10.1198/jasa.2009.0237