Harry Garst posted on Thursday, May 30, 2002 - 10:42 am
Dear Linda and Bengt,
I'm preparing an article using growth curves. I did include the measurement model and I already demonstrated measurement invariance. I interpret the variances of the random intercepts as individual differences at T1, the variances of the slopes as developmental variance and the residual variance as state variance. Using the difference chi-square test I can demonstrate that a linear growth curve model fits significantly better than a random intercept model. Now I want to demonstrate the relevance of this finding by providing an estimate of the effect size of developmental changes. How much do people systematically (and linearly) change over time, compared with how much people already varied at the beginning of the period? Do people change very much over time or are these changes relatively minor in comparison to the between variation? It's is no problem to decompose the variance in a random intercept model. This will be the ICC and I guess it is the same as Cronbach’s alpha. However, in the linear growth curve model the intercept factor typically covaries with the slope factor. This implies that in the calculation of the total variance for all the waves there appears a covariance term. I can calculate the part attributable to state variance by taken the ratio of the sum of the residual variances and the total variance. I can calculate the part attributable to individual differences at T1 by dividing the variance of the intercept factor by the total variance. Finally, I can estimate the part attributable to individual linear changes by dividing the variance of the slope factor by the total variance. However, this procedure must be wrong, because the parts don’t add up to the total variance, due to the existence of the covariance term. How can I decompose the variances of two factors in case these factors covary? Is there a way to show the readers how much linear change (developmental) there is relative to total variances of the latent factors aggregated over time? Is there any literature about this subject? Thank you very much in advance.
As you indicate, there is no simple decomposition of the variance due to the growth factors when the growth factors covary. Off hand, I don't remember having seen this addressed in the literature. You can try to define the growth factors to be uncorrelated by changing the centering. The writings by David Rogosa, e.g. his Myths article, would seem to be relevant.