Currently, I am working on a simple cross-lagged path analysis (i.e., not latent variables) with two variables measured at three time points. My question concerns the standardized parameter estimates. Specifically, when constraining the cross-lagged paths to be equal across time, the standardized estimates (STDYX - I'm using continuous control variables) are not equal. For example, the standardized estimate for Time 1 Variable X --> Time 2 Variable Y is not equal to Time 2 Variable X --> Time 3 Variable Y when that path is constrained to be equal in the code.
Is there a way to force these estimates to be equal? How would this be done and what would the potential implications for model fit be?
The estimates are standardized using the standard deviations of the variables involved. If these are not the same, the standardized estimates will not be the same even if the raw estimates are held equal.
You can constrain X variable variances to be equal across time but you still have the Y variances to contend with. But I would argue that you should focus on the equality of unstandardized coefficients. I would not confound them with (uninteresting) changes in variances and residual variances as you would if you focus on standardized coeffs.
A follow-up question regarding modeling cross-lagged path analysis (involving three time points). It is my understanding from a few published works that one should constrain the stability effects to be equal, as well as the cross-lagged paths. If this is done, as you noted above, the unstandardized estimates will be the same but the standardized estimates are likely to be different for paths constrained to be equal. In this scenario, how does one test for mediation effects, given that one would use the same labels to constrain the relevant paths to be equal. Testing for mediation using path labels under model constraint (e.g., Med1 = a*b*c, where b is the label for two cross-lagged paths or two stability paths constrained to be equal) would be problematic. Thank you.
Below is a Sample syntax involving crossed lagged effects of two team processes each measured at three time points (T2, T3, T4), and subsequent effect on team performance (measured once at time 5):
TmCd_T3r on TmCd_T2r (b1); TmCd_T4r on TmCd_T3r (b1);
Gcoh_T3 on Gcoh_T2 (b2); Gcoh_T4 on Gcoh_T3 (b2);
Gcoh_T2 with TmCd_T2r; GCoh_T3 with TmCd_T3r; GCoh_T4 with TmCd_T4r;
TmCd_T3r on Gcoh_T2 (b3); TmCd_T4r on Gcoh_T3 (b3);
Gcoh_T3 on TmCd_T2r (b4); Gcoh_T4 on TmCd_T3r (b4);
TeamPerf on Gcoh_T4 (q) TeamPerf on TmCd_T4r (s);
One possible indirect path from Gcoh_T2 to TeamPerf is: Gcoh_T2 --> Gcoh_T3 --> Gcoh_T4 --> Team perf. However, given that (Gcoh_T2 --> Gcoh_T3) is constrained to be equal to (Gcoh_T3 --> Gcoh_T4) as denoted by the label b2, the mediation effect would be b2*b2*q. Is it a problem in this case because b2 (unstandardized) is the same value? In calculating the indirect effect, it seems to me that mplus will only actually multiple b2*q rather than b2*b2*q. If these paths were not constrained to be equal, this would not be a problem because the paths would have different labels.
Please let me know if i am thinking about this correctly. Thanks