Modification index
Message/Author
 Sally Czaja posted on Monday, October 16, 2006 - 12:41 pm
What does it mean for an MI to suggest freeing "B on A" when the model already includes "A on B"? or freeing "C with B" when the model includes "B on C"? How is that not redundant? thanks.
 Linda K. Muthen posted on Tuesday, October 17, 2006 - 7:48 am
These parameters are not the same so if they would be identified if they are free, a modificaton index will be given for them. For example, a reciprocal interaction model would have both a ON b and b ON a.
 Sanjoy Bhattacharjee posted on Monday, December 10, 2007 - 6:10 pm
Dear Prof. Muthen/s,

I have the following situation (Model 1)

Y1 = f(X1, Y2, Y3, Y4)
Y2 = f(X2)
Y3 = f(X3)
Y4 = f(X4)

Y2, Y3 and Y4 are correlated, and our rationale suggests that Y2 -Y4 better be used as indicators of an underlying latent construct rather than being separately used in the equation 1. We run the following (model 2)

Y1 = f(X1, eta)
eta BY Y2, Y3, Y4
eta ON X5

where Y1 is binary and Y2 -Y4 are ordered categorical. X’s are vector of strictly exogenous variables.

Below is the comparison of model fit indices

model 1
CFI = 0.324
TLI = -0.081
RMSEA = 0.136
WRMR = 2.192

model 2
CFI = 1.00
TLI = 1.024
RMSEA = 0.00
WRMR = 0.595

As expected, model 2 does a better fit. However, following MI, I added (Y2 with Y3; Y3 with Y4; Y 3 with Y4) in model 1. I got the following fit results for model 1

Modified model 1 (after incorporating MI's suggestion)

CFI = 1.00
TLI = 1.045
RMSEA = 0.00
WRMR = 0.462

Q1. Why do the fit indices change so significantly for the model 1 once we estimate the parameters of the “Psi matrix”

Q2. Which of the model should I consider? Model 2 or the modified model 1?

Thanks and regards
Sanjoy
 Bengt O. Muthen posted on Tuesday, December 11, 2007 - 11:29 am
Model 2 uses x5 which is not part of Model 1, so the models are difficult to compare.

In Model 2 y2-y4 are correlated even conditional on the covariates. This is because you have a residual variance in eta. The default for Model 1 should give residual correlations for the y's as well so I don't understand how you get non-zero MI's for those correlations. But if your Model 1 indeed has zero residual corr's for the y's, Model 1 is therefore quite different from Model 2.
 Sanjoy Bhattacharjee posted on Wednesday, December 12, 2007 - 8:50 am
Thank you very much professor Muthen.