Growth curve reliability
Message/Author
 Peggy Clements posted on Monday, October 09, 2006 - 6:06 pm
There is an article in the most recent issue of Psychological Methods that demonstrates the importance of "growth curve reliability" for having adequate power to detect slope-to-slope effects. The GCR (as I'm sure you know) is the ratio of the variance of the growth curve to the total variance. I understand where to find the variance for the growth curve, but am not sure where to find the total variance (the total variance of x at time t).

Thanks.
 Linda K. Muthen posted on Tuesday, October 10, 2006 - 9:46 am
I am not familiar with this but do you mean the total variance of y at time t? You can obtain these numbers from the RESIDUAL output.
 Peggy Clements posted on Tuesday, October 10, 2006 - 10:42 am
[Post 1 of 2]
I'm not sure if this is what they mean. The article I am referring to is by Hertzog, Lindenberger, Ghisletta, and von Oertzen, titled: on the power of multivariate latent growth curve models to detect correlated change. They write:

"A second issue is the effect of growth curve reliability (GCR) on the power to detect covariance between the slopes. Define the total variance of x at time t as [sigma-squared, subcripts x and t]. This variance can be decomposed into two components,
(a) that due to individual differences in latent intercepts and slopes, namely
{[sigma-squared, subscript Ix]
+ [(Beta-squared, subscript 2t)*(sigma-squared, subscript Sx)]} (where sigma-squared Ix and sigma-squared Sx are the variances of the intercepts and slopes for x, respectively), and
(b) that due to error (sigma-squared subscript ex). Then GCR is defined as ... [(sigma-squared, sub xt) - (sigma-squared, sub ex)]/(sigma-squared, sub xt), that is, the ratio of variance is determined by the latent growth curve to ttal variance. The same expressions apply variable y."
 Peggy Clements posted on Tuesday, October 10, 2006 - 10:42 am
[post 2 of 2]
As you suggest, maybe they are referring to the matrix of the variances of all of the x and all of the times (t). If that's the case, I'm still not sure how to end up with a single statistic that is between 0 and 1. [They note that the power to detect slope-to-slope associations is greatly reduced if GCR is <.9]. In other words, I see that the RESIDUAL ouptut provides both the estimated covariance matrix and the residual covariance matrix, but I'm not clear how to transform the information in those matrices into this GCR statistic.

I can send you the quotation I am refering to (with the actual greek symbols instead of my translation of the equations into text), if that would be helpful.

This seems like a topic that could be of interest to people who conduct LGM, so I'm hoping that this extended message is worth the time it takes to read and answer. Thanks much.
 Bengt O. Muthen posted on Tuesday, October 10, 2006 - 5:32 pm
I looked at the Psych Methods article and as far as I understand this variance ratio is simply the R-squared of the outcome x at a certain time point as a function of the growth factors. This is printed in the Mplus output when Standardized is requested in the Output command. A high R-square will naturally increase the power to detect a correlation between slopes of two parallel processes. The total variance is obtained when requesting Residual in the Output command.
 Jon Heron posted on Friday, December 11, 2015 - 1:32 am
We've been attempting to estimate a growth model where growth-curve reliability (GCR) is constrained with reference to the paper mentioned above.

In doing so we have discovered two things

[1] The quoted formula for the variance explained by a linear growth model is lacking the 2ab*cov(I,S) term

[2] When one estimated the reliability this does indeed agree with R-squared as Bengt says. However both of these measures are based on the model-implied variances and covariances rather than the samples statistics of the raw data. A poor-fitting mean-structure can then lead to model-implied var/covs which do not align with those in the data.

This is little more than a half-formed thought at the minute, but it feels to me that when considering GCR, once should ensure a good fitting mean curve as a preliminary step. Perhaps that is good practice for LGM in general.
 Jon Heron posted on Friday, December 11, 2015 - 1:37 am
An apology to Hertzog et al.
their paper states

"... covariances between intercepts and slopes were fixed to 0."
 Bengt O. Muthen posted on Saturday, December 12, 2015 - 4:50 pm
Yes, the growth model imposes restrictions on both the means and the variances, so the reliability-correcting approach needs to pay attention to the fit of observed to estimated variances.
 Jon Heron posted on Monday, December 14, 2015 - 5:05 am
thanks Bengt