Sample Sizes for Two-Group Second-Order Latent Growth Curve Models

Second-order latent growth curve models (S. C. Duncan &; Duncan, 1996 Duncan, S. C. and Duncan, T. E. 1996. A multivariate growth curve analysis of adolescent substance use.. Structural Equation Modeling, 3: 323–347. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]; McArdle, 1988 McArdle, J. J. 1988. “Dynamic but structural equation modeling of repeated measures data.”. In Handbook of multivariate experimental psychology, , 2nd ed. Edited by: Cattell, R. B. and Nesselroade, J. 564–614. New York: Plenum.. [Crossref] , [Google Scholar]) can be used to study group differences in change in latent constructs. We give exact formulas for the covariance matrix of the parameter estimates and an algebraic expression for the estimation of slope differences. Formulas for calculations of the required sample size are presented, illustrated, and discussed. They are checked by Monte Carlo simulations in Mplus and also by Satorra and Saris's (1985) Satorra, A. and Saris, W. E. 1985. The power of the likelihood ratio test in covariance structure analysis.. Psychometrika, 50: 83–90. [Crossref], [Web of Science ®] , [Google Scholar] power approximation techniques for small and medium effect sizes (Cohen, 1988 Cohen, J. 1988. Statistical power analysis for the behavioral sciences. Hillsdale, NJ: Erlbaum..  [Google Scholar]). Results are similar across methods. Not surprisingly, sample sizes decrease with effect sizes, indicator reliabilities, number of indicators, frequency of observation, and duration of study. The relative importance of these factors is also discussed, alone and in combination. The use of the sample size formula is illustrated using a modification of empirical results from Stoel, Peetsma, and Roeleveld (2003) Stoel, R. D., Peetsma, T. T. and Roeleveld, J. 2003. Relations between the development of school investment, self-confidence, and language achievement in elementary education: A multivariate latent growth curve approach.. Learning and Individual Differences, 13: 313–333. [Crossref], [Web of Science ®] , [Google Scholar].