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

Abstract

Second-order latent growth curve models (S. C. Duncan & Duncan, 1996; McArdle, 1988) 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) power approximation techniques for small and medium effect sizes (Cohen, 1988). 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).

Notes

¹See, for example, Diggle et al. (2002) for descriptions of longitudinal models.

²For descriptions of multilevel models, see, for example, Bryk and Raudenbush (1992), Goldstein (2003), and Longford (1993).

³We focus on the second-order model in which indicators of latent variables are observed at each occasion.

⁴See, for example, Jöreskog and Sörbom (1979) for a general SEM model with a measurement and a structural part.

⁵This is based on the general latent variable multiple population formulation (see, e.g., Bollen, 1989; Jöreskog & Sörbom, 1979; B. O. Muthén & Curran, 1997).

⁶Previous studies have also focused on reliability at Time 1 (see, e.g., Hertzog et al., 2006).

⁷This formula follows from R _k = λ² _k Var(η₁)/(λ² _k Var(η₁) + Var(ε _k )), which is related to the definition of reliability from classical test theory (e.g., McDonald, 1999).

⁸Only fairly large sample sizes were used. Had sample sizes been smaller (e.g., n = 10), biases might have been larger.

^aThe simulations for d = .5 resulted in covariance matrices that were not positive definite for some replications. The largest number of nonpositive definite matrices (.0033%) came from the R = .6, k = 3 model.

⁹They used children from a longitudinal cohort study in The Netherlands and fitted second-order models to study development in self-confidence, school investment, and language ability. They had some attrition and used all available data to fit separate growth models for each concept as well as a multivariate model to study relations between the three concepts as well as intelligence. For the purpose of this illustration, we focus on the self-concept model and assume no attrition. They presented reliability estimates of the indicators and latent variables and error variances were deduced from those.

¹⁰The sample sizes do not relate to each other by a factor of 6.25 here because we used a rounded value for α₃.

¹¹See, for example, B. O. Muthén and Curran (1997) and Curran and Muthén (1999) for interaction interpretations and power computations for first order models.