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?
bmuthen posted on Tuesday, December 19, 2000 - 2:02 pm
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.
You would compare the same models across time as you would normally compare across groups. The Topic 4 course handout shows this for continuous outcomes under Multiple Indicator Growth. The measurement invariance models we suggest for categorical outcomes are described in the new Chapter 14 at the end of the discussion of multiple group analysis.
I am analyzing longitudinal categorical data with 9 indicators of a single factor measured 3 times. The data are quite similar in structure to Example 6.15, the multiple indicator growth model with categorical indicators.
As a preliminary step to the growth model, I am doing a multi-wave CFA to test for measurement invariance (modeling the longitudinal data simultaneously, not as separate groups, to preserve the longitudinal structure). In your recommendations concerning testing for measurement invariance (Ch. 14, p. 433 of the User's Guide), you recommend that the more restrictive model for categorical data should hold factor loadings and thresholds to equality over time, to fix scale factors at 1 at one wave and free at other waves, and to fix the factor means at 0 in wave waves and free in the other waves.
With such data, Mplus sets all factor means to 0 by default (this is true in ex6.15, too). When I fix wave 1 factor mean at 0 and free factor means in the other groups (per the User's Guide suggestion), I get small and non-significant factor means at waves 2 and 3 and model fit improves to a nonsignificant degree (relative to model with all factor means at 0).
(continued from above because post size is limited)
So, my inclination is to keep all factor means fixed at 0 for parsimony. But I am not clear on the effect that this has on the interpretation and validity of factor loadings and factor covariances, which I'm interested in. My sense is one fixes the factor mean at 0 in one group to provide a scaling of the factor means relative to the other waves (roughly mean differences in theta, in IRT terms). So, the fact that the means are not really different when freed in waves 2 and 3 suggests that there is little mean level change in the construct (i.e., symptom severity is roughly equal across time in my case of a one-factor severity construct).
So, I think the specific questions are:
1) Is it valid to fix the means at zero at all waves or does this muddy the interpretation? 2) Is my interpretation of the nonsignificant factor mean differences valid (i.e., little mean level change in the construct)? 3) Are there any concerns of model nonidentification? (I know estimating all factor means freely will cause nonidentification...)
1) Yes, but I would not recommend it because it is like model trimming - report the non-significance instead.
3) Note that in the factor analysis case, the means are free for all but the reference time point. But for growth modeling the factor intercepts are fixed at zero for all time points because the factor mean changes are handled by the growth factors.
Yes, post size is limited - it is intentional: We only have time for short questions
Hi I have read a lot of posts about this but i still don't get it straight. I have a CFA model with four factors and categorical indicators that fit well in the two time points available separately. I would like to assess if the loadings are similar in the two time points I have specified two models as stated in the manual with both time points together: "1. Thresholds and factor loadings free across groups (time); scale factors fixed at one in all groups (time); factor means fixed at zero in all groups (time) 2. Thresholds and factor loadings constrained to be equal across groups (time); scale factors fixed at one in one group (time) and free in the others; factor means fixed at zero in one group (time) and free in the others" My question is; Can I compare this two models with difftest to assess the equality of loadings and thresholds over time? I don't see clearly that this are nested models because of the second one having some free parameters that the first one doesn't have.
I have an autoregressive path model with parameterization=theta (3 waves)
categorical are u1 u2 u3;
model: !autoregressive paths f3 on f2; f2 on f1; u3 on u2; u2 on u1; !cross-lagged paths f3 on u2; f2 on u1; u3 on f2; u2 on f1; !within-time covariances u1 with f1; u2 with f2; u3 with f3; !residual correlations u1 with u2 u3; u2 with u3; f1 with f2 f3; f2 with f3; !covariates u1 f1 on male age;
How would I set up an autoregressive path model model up to capture similarities in measurements across time in the u variables? I would like to assume the u variable has measurement invariance across time, and also, the measurement error of u is correlated across time. Is that possible? As for the residual correlations, The "u1 with u2 u3" presently statements show up in the psi matrix and not the theta matrix in TECH1, so it looks like they are simply correlations of the variables.
Starting from your last statement, the fact that u1 with u2 ends up in Psi instead of Theta should be ignored - it still refers to the residual covariance between the variables (the reason it ends up there is that a latent variable gets put behind the u's that have regressions; it's just an Mplus trick).
With a continuous outcome that is repeatedly measured over time, it is possible to identify and estimate a measurement error. But I don't think you can identify correlated errors over time of say y1 with y2 if you also have y2 on y1. Same for a categorical outcome.
Are the syntax associated with the "Item Response Modeling in Mplus: A Multi-Dimensional, Multi-Level, and Multi-Timepoint Example" paper going to be posted to statmodel? I had not seen anything uploaded and was uncertain. Thanks
Dear Professor Muthen, I am also interested in Mplus syntax/output associated with the Table 6 (and Table 3) from "Item Response Modeling in Mplus: A Multi-Dimensional, Multi-Level, and Multi-Timepoint Example" paper. I would be very grateful if you could send it to me also (firstname.lastname@example.org).