Testing the threshold assumption in L...
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 Jeffrey Long posted on Wednesday, November 17, 2004 - 2:22 pm
I am running a latent growth curve analysis with five ordered categorical items at seven time points. The items are assumed to be unidimensional. I understand the thresholds must be held invariant in order to specify the mean structure. However, I would like to test the assumption of invariant thresholds across time. Is this possible in MPLUS? NOte that I cannot free all thresholds as I run out of df (even when the loadings are invariant). Perhaps using an overly elaborate mean structure, I could free the thresholds of each item one at a time and compute the scaled nested chi-square based on WLSM? I am unsure of this strategy as the mean structure is partly defined by the thresholds. Any advice would be much appreciated.
 bmuthen posted on Wednesday, November 17, 2004 - 3:28 pm
You can use the WLSMV DIFFTEST option for chi-square difference testing of the model with full measurement invariance (thresholds and loadings) and no measurement invariance. In the former model the thresholds and loadings are invariant and - using the default Delta parameterization - you fix deltas to 1 for all 5 items at the first time point and free delta for the items of the other time points. The factor mean is fixed at zero for the first time point and free for the other. In the latter model deltas are all fixed at 1 and all factor means are fixed at 0. See also the growth example in Mplus Web Note #4. You probably end up finding that you have partial measurement invariance, where some of the items do not show invariance. As you say, partial invariance can be studied stepwise with invariance imposed for each item at a time.
 Jeffrey Long posted on Thursday, November 18, 2004 - 6:42 pm
Bengt, I wonder if you could clarify. I am using the theta parametization. The restricted model, H0, has invariant loadings and invariant thresholds, and the residual variances are fixed at unity for time 1 and free the remaining times. My othogonal cubic polynomial model is:
i BY f1-f7@1;
l BY f1@-3 f2@-2 f3@-1 f4@0 f5@1 f6@2 f7@3;
q BY f1@5 f2@0 f3@-3 f4@-4 f5@-3 f6@0 f7@5;
c BY f1@-1 f2@1 f3@1 f4@0 f5@-1 f6@-1 f7@1;
[f1-f7@0 i@0 l q c];

The less restictive model, H1, has varying loadings and thresholds. Because of the varying thresholds I must fix all the residual variances at unity. Do I use the same growth curve model and mean specification as with H0? When I run the model above, MPLUS says the model is not identified.

Another question, under H0, can I use the derivatives with respect to tau (TECH2) as modification indices to suggest what thresholds might be set free? Thanks.
 Linda K. Muthen posted on Friday, November 19, 2004 - 6:06 am
Send the full output to support@statmodel.com and I will take a look at it.

You can use derivatives but modification indices are now available for categorical outcomes.