Growth mixture modeling (GMM) is a method for identifying multiple unobserved sub-populations, describing longitudinal change within each unobserved sub-population, and examining differences in change among unobserved sub-populations. We provide a practical primer that may be useful for researchers beginning to incorporate GMM analysis into their research. We briefly review basic elements of the standard latent basis growth curve model, introduce GMM as an extension of multiple-group growth modeling, and describe a four-step approach to conducting a GMM analysis. Example data from a cortisol stress-response paradigm are used to illustrate the suggested procedures.
Arbuckle, J.L. ( 1995-2008). AMOS User’s Guide. Spring House, PA: Amos Development Corp.
2.
Bauer, D.J., & Curran, P.J. ( 2003a). Distributional assumptions of growth mixture models: Implications for overextraction of latent trajectory classes. Psychological Methods, 8, 338-363.
3.
Bauer, D.J., & Curran, P.J. ( 2003b). Overextraction of latent trajectory classes: Much ado about nothing? Reply to Risdskopf (2003), Muthén (2003), and Cudeck and Henly (2003). Psychological Methods, 8, 384-393.
4.
Bauer, D.J., & Curran, P.J. ( 2004). The integration of continuous and discrete latent variable models: Potential problems and promising opportunities. Psychological Methods, 9, 3-29.
5.
Bollen, K.A., & Curran, P.J. ( 2006). Latent curve models: A structural equation perspective . New York: Wiley.
6.
Bryk, A.S., & Raudenbush, S.W. (1987). Application of hierarchical linear models to assessing change. Psychological Bulletin , 101, 147-158.
7.
Bryk, A.S., & Raudenbush, S.W. (1992). Hierarchical linear models: Applications and data analysis. Newbury Park, CA: Sage Publications.
8.
Burchinal, M., Nelson, L., & Poe, M. ( 2006). Growth curve analysis: An introduction to various methods for analyzing longitudinal data. Monographs of the Society for Research in Child Development, 71, 65-87.
9.
Connell, A.M., & Frye, A.A. ( 2006). Growth mixture modeling in developmental psychology: Overview and demonstration of heterogeneity in developmental trajectories of adolescent antisocial behaviour. Infant and Child Development , 15, 609-621.
10.
Duncan, T., Duncan, S., Stryker, L., Li, F., & Alpert, A. ( 2006). An introduction to latent variable growth curve modeling . Mahwah, NJ: Erlbaum.
11.
Greenbaum, P.E., Del Boca, F.K., Darkes, J., Wang, C., & Goldman, M.S. ( 2005). Variation in the drinking trajectories of freshmen college students. Journal of Consulting and Clinical Psychology, 73, 229-238.
12.
Grimm, K.J., & Ram, N. (in press). Non-linear growth models in Mplus and SAS. Structural Equation Modeling: A Multidisciplinary Journal.
13.
Grimm, K.J., & Ram, N. ( 2009). A second order growth mixture model for developmental research. Research in Human Development, 6, 121-143.
14.
Hair, J.F., & Black, W.C. ( 2000). Cluster analysis. In L.G. Grimm & P.R. Yarnold (Eds.), Reading and understanding more multivariate statistics (pp. 147-205). Washington, DC: APA.
15.
Harring, J. ( 2005). Nonlinear mixed effects mixture model: A model for clustering nonlinear longitudinal profile. Unpublished doctoral dissertation. University of Minnesota.
16.
Heim, C., Elhert, U., & Hellhammer, D. (2000). The potential role of hypocortisolism in the pathophysiology of stress-related bodily disorders. Psychoneuroendocrinology, 25, 1-35.
17.
Hipp, J.R., & Bauer, D.J. ( 2006). Local solutions in the estimation of growth mixture models . Psychological Methods, 11, 36-53.
18.
Jedidi, K., Ramaswamy, V., & Desarbo, W.S. ( 1993). A maximum likelihood method for latent class regression involving a censored dependent variable. Psychometrika, 358, 375-394.
19.
Jones, B.L., & Nagin, D.S. ( 2007). Advances in group-based trajectory modeling and an SAS procedure for estimating them. Sociological Methods & Research, 35, 542-571.
20.
Jones, B.L., Nagin, D.S., & Roeder, K. ( 2001). A SAS procedure based on mixture models for estimating developmental trajectories. Sociological Methods & Research , 29, 374-393.
21.
Jung, T., & Wickrama, K.A.S. (2008). An introduction to latent class growth analysis and growth mixture modeling. Social and Personality Psychology Compass, 2, 302-317.
22.
Kelley, K. ( 2005). Estimating nonlinear change models in heterogeneous populations when class membership is unknown: Defining and developing the latent classification differential change model. Unpublished doctoral dissertation. University of Notre Dame, IN.
Li, F., Duncan, T.E., Duncan, S.C., & Acock, A. ( 2001). Latent growth modeling of longitudinal data: A finite growth mixture modeling approach. Structural Equation Modeling , 8, 493-530.
25.
Little, R.J.A., & Rubin, D.B. ( 1987) Statistical analysis with missing data. New York: John Wiley & Sons.
26.
Lo, Y., Mendell, N.R., & Rubin, D.B. ( 2001). Testing the number of components in a normal mixture . Biometrika, 88, 767-778.
27.
Masyn, K. ( 2009). Discrete-time survival factor mixture analysis for low-frequency recurrent event histories. Research in Human Development, 6, 165-194.
28.
McArdle, J.J. ( 2009). Latent variable modeling of differences and changes with longitudinal data. Annual Review of Psychology, 60, 577-605.
McArdle J.J., & Nesselroade, J.R. (2003). Growth curve analysis in contemporary psychological research. In J. Shinka & W. Velicer (Eds.), Comprehensive handbook of psychology, Vol. 2: Research methods in psychology (pp. 447-480). New York: Wiley.
31.
McLachlan, G.J., & Peel, D. ( 2000). Finite mixture models. New York: John Wiley & Sons.
32.
Meredith, W., & Tisak, J. ( 1990). Latent curve analysis. Psychometrika , 55, 107-122.
33.
Miller, G.E., Chen, E., & Zhou, E.S. ( 2007). If it goes up, must it come down? Chronic stress and the hypothalamic-pituitary-adrenocortical axis in humans. Psychological Bulletin, 133, 25-45.
34.
Muthén, B. ( 2003). Statistical and substantive checking in growth mixture modeling: Comment on Bauer and Curran (2003). Psychological Methods, 8, 369-377.
35.
Muthén, B. ( 2004). Latent variable analysis. Growth mixture modeling and related techniques for longitudinal data. In D. Kaplan (Ed.), The Sage handbook of quantitative methodology for the social sciences (pp. 345-368). Thousand Oaks, CA: Sage Publications.
36.
Muthén, B.O., & Masyn, K. ( 2005). Discrete-time survival mixture analysis. Journal of Educational and Behavioral Statistics, 30, 27-58.
37.
Muthén, B.O., & Muthén, L.K. (2000). Integrating person-centered and variable-centered analysis: Growth mixture modeling with latent trajectory classes. Alcoholism: Clinical and Experimental Research, 24, 882-891.
38.
Muthén, B.O., & Shedden, K. ( 1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463-469.
Nagin, D.S. ( 2005). Group-based modeling of development. Cambridge, MA: Harvard University Press.
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.
Ram, N., & Grimm, K. ( 2007). Using simple and complex growth models to articulate developmental change: Matching theory to method. International Journal of Behavioral Development, 31, 303-316.
45.
Rindskopf, D. ( 2003). Mixture or homogeneous? Comment on Bauer and Curran (2003) . Psychological Methods, 8, 364-368.
46.
Rogosa, D.R., & Willet, J.B. ( 1985). Understanding correlates of change by modeling individual differences in growth. Psychometrika, 50, 203-228.
47.
Seeman, T.E., Berkman, L.F., Gulanski, B., Robbins, R., Greenspan, S., Charpentier, P., & Rowe, J. ( 1995). Self-esteem and neuroendocrine response to challenge: MacArthur Successful Aging Studies. Psychosomatic Research , 39, 69-84.
48.
Seeman, T.E., Singer, B., & Charpentier, P. (1995). Gender differences in pattern of HPA axis response to challenge: MacArthur Studies of Successful Aging. Psychoneuroendicrinology, 20, 711-725.
49.
Singer, J.D., & Willet, J.B. ( 2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York: Oxford University Press.
50.
Stanger, C. ( 2006). Latent growth mixture models: An important new tool for developmental researchers. Infant and Child Development, 15, 635-637.
51.
Vuong, Q.H. ( 1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57, 307-333.
52.
Wang, M., & Bodner, T.E. ( 2007). Growth mixture modeling: Identifying and predicting unobserved subpopulations with longitudinal data. Organizational Research Methods, 10, 635-656.
53.
Wohlwill, J.F. ( 1973). The study of behavioral development. San Diego, CA: Academic Press.
