In this article, the authors illustrate how random coefficient modeling can be used to develop growth models for the analysis of longitudinal data. In contrast to previous discussions of random coefficient models, this article provides step-by-step guidance using a model comparison framework. By approaching the modeling this way, the authors are able to build off a regression foundation and progressively estimate and evaluate more complex models. In the model comparison framework, the article illustrates the value of using likelihood tests to contrast alternative models (rather than the typical reliance on tests of significance involving individual parameters), and it provides code in the open-source language R to allow readers to replicate the results. The article concludes with practical guidelines for estimating growth models.
Get full access to this article
View all access options for this article.
References
1.
Abelson, R. P. (1995). Statistics as a principled argument. Hillsdale, NJ: Lawrence Erlbaum.
2.
Bliese, P. D. (2000). Within-group agreement, non-independence, and reliability: Implications for data aggregation and analysis. In K. J. Klein & S. W. Kozlowski (Eds.), Multilevel theory, research, and methods in organizations (pp. 349-381). San Francisco: Jossey-Bass.
3.
Bliese, P. D. (2002a). Multilevel modeling in R: A brief introduction to R, the multilevel package, and the NLME package. Washington, DC: Walter Reed Army Institute of Research.
4.
Bliese, P. D. (2002b). Multilevel random coefficient modeling in organizational research: Examples using SAS and S-PLUS. In F. Drasgow & N. Schmitt (Eds.), Measuring and analyzing behavior in organizations: Advances in measurement and data analysis (pp. 401-445). San Francisco: Jossey-Bass.
5.
Bryk, A. S., & Raudenbush, S. W. (1987). Application of hierarchical linear models to assessing change. Psychological Bulletin, 101, 147-158.
6.
Bryk, A. S., & Raudenbush, S. W. (1992). Hierarchical linear models. Newbury Park, CA: Sage.
7.
Bryk, A. S., Raudenbush, S. W., & Congdon, R. T. (1994). Hierarchical linear modeling with the HLM/2L and HLM/3L programs. Chicago: Scientific Software, International.
8.
Chan, D. (1998). The conceptualization and analysis of change over time: An integrative approach incorporating longitudinal means and covariance structures analysis (LMACS) and multiple indicator latent growth modeling (MLGM). Organizational Research Methods, 1, 421-483.
9.
Chan, D. (2002). Latent growth modeling. In F. Drasgow & N. Schmitt (Eds.), Measuring and analyzing behavior in organizations: Advances in measurement and data analysis (pp. 302-349). San Francisco: Jossey-Bass.
10.
Chan, D., Ramey, S., Ramey, C., & Schmitt, N. (2000). Modeling intraindividual changes in children’s social skills at home and at school: A multivariate latent growth approach to understanding between-settings differences in children’s social skills development. Multivariate Behavioral Research, 35, 365-396.
11.
Chan, D., & Schmitt, N. (2000). Interindividual differences in intraindividual changes in proactivty during organizational entry: A latent growth modeling approach to understanding newcomer adaptation. Journal of Applied Psychology, 85, 190-210.
12.
Chou, C., Bentler, P. M., & Pentz, M. A. (1998). Comparisons of two statistical approaches to study growth curves: The multilevel model and the latent curve analysis. Structural Equation Modeling, 5, 247-266.
13.
Cohen, J., & Cohen, P. (1983). Applied multiple regression/correlation analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
14.
Deadrick, D. L., Bennett, N., & Russell, C. J. (1997). Using hierarchical linear modeling to examine dynamic performance criteria over time. Journal of Management, 23, 745-757.
15.
DeShon, R. P., Ployhart, R. E., & Sacco, J. M. (1998). The estimation of reliability in longitudinal models. International Journal of Behavior and Development, 22, 493-515.
16.
Garst, H., Frese, M., & Molenaar, P. C. M. (2000). The temporal factor of change in stressor-strain relationships: A growth curve model on a longitudinal study in East Germany. Journal of Applied Psychology, 85, 417-438.
17.
Gonzales, R., & Griffin, D. (2001). Testing parameters in structural equation modeling: Every “one” matters. Psychological Methods, 6, 258-269.
18.
Gottman, J. M. (1995). The analysis of change. Mahwah, NJ: Lawrence Erlbaum.
19.
Hand, D., & Crowder, M. (1996). Practical longitudinal data analysis. London: Chapman & Hall.
20.
Hofmann, D. A. (1997). An introduction to the logic and rationale of hierarchical linear models. Journal of Management, 23, 723-744.
21.
Hofmann, D. A., & Gavin, M. (1998). Centering decisions in hierarchical linear models: Theoretical and methodological implications for research in organizations. Journal of Management, 24, 623-641.
22.
Hofmann, D. A., Griffin, M. A., & Gavin, M. B. (2000). The application of hierarchical linear modeling to organizational research. In K. J. Klein & S. W. Kozlowski (Eds.), Multilevel theory, research, and methods in organizations (pp. 467-511). San Francisco: Jossey-Bass.
23.
Hofmann, D. A., Jacobs, R., & Baratta, J. E. (1993). Dynamic criteria and the measurement of change. Journal of Applied Psychology, 78, 194-204.
24.
Kenny, D. A., & Judd, C. M. (1986). Consequences of violating the independence assumption in analysis of variance. Psychological Bulletin, 99, 422-431.
25.
Kreft, I., & de Leeuw, J. (1998). Introducing multilevel modeling. London: Sage.
26.
Lance, C. E., Meade, A. W., & Williamson, G. M. (2000). We should measure change—and here’s how. In G. M. Williamson & D. R. Shaffer (Eds.), Physical illness and depression in older adults: A handbook of theory, research, and practice (pp. 201-235). New York: Kluwer Academic/Plenum.
27.
Lance, C. E., Vandenberg, R. J., & Self, R. M. (2000). Latent growth models of individual change: The case of newcomer adjustment. Organizational Behavior & Human Decision Processes, 83, 107-140.
28.
Littell, R. C., Milliken, G. A., Stroup, W. W., & Wolfinger, R. D. (1996). SAS system for mixed models. Cary, NC: SAS Institute.
29.
Little, R. J. A., & Schlenker, N. (1995). Missing data. In G. Arminger, C. C. Clogg, & M. E. Sobel (Eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 39-75). New York: Plenum.
30.
Muthen, L. K., & Muthen, B. O. (1998). Mplus user’s guide. Los Angeles: Muthen & Muthen.
31.
Pinheiro, J. C., & Bates, D. M. (2000). Mixed-effects models in S and S-PLUS. New York: Springer-Verlag.
32.
Ployhart, R. E., & Hakel, M. D. (1998). The substantive nature of performance variability: Predicting interindividual differences in intraindividual performance. Personnel Psychology, 51, 859-901.
33.
Ployhart, R. E., Holtz, B. C., & Bliese, P. D. (2002). Longitudinal data analysis: Applications of random coefficient modeling to leadership research. Leadership Quarterly, 13, 455-486.
34.
Raudenbush, S. W. (2001). Comparing personal trajectories and drawing causal inferences from longitudinal data. Annual Review of Psychology, 52, 501-525.
35.
Schnabel, K. U., Little, T. D., & Baumert, J. (2000). Modeling longitudinal and multilevel data. In T. D. Little, K. U. Schnabel, & J. Baumert (Eds.), Modeling longitudinal and multilevel data: Practical issues, applied approaches, and specific examples (pp. 9-13). Mahwah, NJ: Lawrence Erlbaum.
36.
Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 24, 323-355.
37.
Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 2, 4-69.
38.
Willett, J. B., & Sayer, A. G. (1994). Using covariance structural analysis to detect correlates and predictors of individual change over time. Psychological Bulletin, 116, 363-381.
