High Correlation Between S and Q
Message/Author
 Scott D. Martin posted on Tuesday, May 24, 2011 - 12:48 pm
Dear Mplus Team,

We wanted to ask you for some help understanding the causes leading to a correlation greater than 1 between the slope and quadratic factors.

Our linear model (unconditional) fit was modest and there was evidence to look at the quadratic form. We ran the quadratic model and results came back with a correlation between S and Q > 1. In searching through other posts on the Mplus site and several references in articles and books (including Bollen and Curran, “Latent Curve Models”) this problem appears somewhat common, but it was not clear in these references why the correlations are so high.

We thought that since the time points (0 and 1) would be the same when squared in the quadratic factor as in the linear factor that this might cause the high correlation, so following guidelines in the M+ manual and elsewhere, we tried a different centering and different parameter values for time, but the correlation between S and Q for both of these models still came back as greater than 1.

We did run a model allowing the time scores to be estimated and the model fit much better, but we still wanted to understand what causes the high correlations. Could you offer some input?
 Linda K. Muthen posted on Tuesday, May 24, 2011 - 4:41 pm
I think the main reason is that the time scores are the same for the time points with time scores of zero and one. You can try centering the times scores at the middle time point, that is, have the time score of zero there.
 Scott D. Martin posted on Monday, May 30, 2011 - 11:58 am
Hi Linda,

Thank you for the quick reply. That thought did occur to us previously, so following guidelines in the M+ manual (and as you have just suggested), we tried a different centering:

MODEL: i s q | yaplgal1@-1 yaplgal2@0 yaplgal3@1.2 yaplgal4@3.6;

but the correlation between S and Q still came back as greater than 1. (times 1.2 and 3.6 were based on our sample data collection times which were collected at unequal intervals).
 Linda K. Muthen posted on Monday, May 30, 2011 - 3:07 pm
What were your original time scores? And what were the measurement occasions.
 Scott D. Martin posted on Tuesday, May 31, 2011 - 6:41 pm
Hi Linda,
Our original time scores were:
MODEL: i s q | yaplgal1@0 yaplgal2@1 yaplgal3@2.2 yaprlgal4@4.6;

Measurement occasions were baseline, 10 month, 22 month, and 46 month follow up, so we based the time scores on the number of months from baseline.
 Linda K. Muthen posted on Wednesday, June 01, 2011 - 10:38 am
You can try adding residual covariances across time, for example,

yaplgal1 WITH yaplfal2;

etc.