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| Longitudinal invariance with categori... |
 
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My question concerns tests of longitudinal invariance when an instrument has categorical indicators. Assume that I have one-factor construct which is measured with 3 dichotomous items at 2 points in time.
1. In a *multiple group* approach, typically a basline model is estimated in which all parameters are freely estimated. Subseq. models then impose constraints on thresholds/loadings, latent (co)variances, and eventually latent means. HOWEVER, in the case of longitudinal data this is not possible, since freely estimating *all* thesholds will necessarily result in under-indentified latent means. That is, even after I fix the latent mean at time 1 to 0, the latent mean at time 2 is not identified (without constraints). However if I constrain thresholds (and factor loadings by association) to be equal over time, this baseline model seems much more stringent than is the multiple group case. Am I missing something or is that all I can do?
2. If I assume that the baseline model necessarily imposes constraints on thresholds/factor loadings across time, I would still like to compare this model against models with constraints on latent (co)variances and latent means. Imposing constraints on latent (co)variances is not a problem. However constraining latent means is. Specifically, initially the time 1 latent mean is fixed to 0 and the time 2 mean is interpreted as the change in mean level relative to time 1. In the model that I try to equate latent means in the customary way (e.g., [latent-x-t1 latent-x-t2] (##);) I get an error implying under-identification. If I continue to fix the latent mean at time 1 to 0, obviously there is no constraint being imposed and the latent mean at time 2 continues to be interpretted as relative change. Again, if I were doing a multiple group analysis, I could simply equate latent means separately using "model" lines (as Linda suggested Millsap in her 11/27/00 post to this board). However in the case of long. invariance, I can't seem to impose constraints on latent means over time and still have them be identified. Again, am I missing something?
I appreciate your thoughts - mikew@unc.edu |
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| bmuthen posted on Tuesday, December 19, 2000 - 2:02 pm
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A useful first model is to analyze both time points together with no across-time invariance. In this model you have the factor means fixed at zero at both time points since you allow the thresholds to be different.
Then you turn to the model with invariant thresholds and loadings, letting the delta scale factors be different across time (see Users Guide, Appendix). In this model the factor means are zero at time 1 and free at time 2. The factor covariance matrices are different across time. Following that you can use WLS chi-square difference testing and restrict factor means to zero at time 2 to test that they are different from time 1, and you can restrict the factor covariance matrices to test that. |
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