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 Mike Willoughby posted on Tuesday, December 19, 2000 - 10:22 am
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
 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.
 Randall MacIntosh posted on Tuesday, August 03, 2010 - 4:19 pm
I have data (n=1300) from a 2-wave panel on a single construct measured by 10 dichotomous items.

What is the most appropriate model to assess longitudinal measurement invariance?

Thanks.
 Linda K. Muthen posted on Tuesday, August 03, 2010 - 4:30 pm
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.
 Michael Hallquist posted on Thursday, March 10, 2011 - 12:46 pm
Linda and Bengt,

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).
 Michael Hallquist posted on Thursday, March 10, 2011 - 12:47 pm
(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...)

Thanks so much for your input,
Michael
 Bengt O. Muthen posted on Thursday, March 10, 2011 - 6:29 pm
1) Yes, but I would not recommend it because it is like model trimming - report the non-significance instead.

2) Yes.

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 :-)
 Jaume Aguado Carné posted on Friday, October 21, 2011 - 9:55 am
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.

Thank you
J.
 Linda K. Muthen posted on Friday, October 21, 2011 - 10:35 am
Yes, these two models are nested and can be tested with DIFFTEST.
 Joseph E. Glass posted on Sunday, June 03, 2012 - 5:39 pm
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.

Thank you!
 Bengt O. Muthen posted on Sunday, June 03, 2012 - 6:22 pm
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.
 Joseph E. Glass posted on Sunday, June 03, 2012 - 7:22 pm
Thank you for your quick response!

The model appears to be identified with these cross-time correlations and regressions specified, in that it has no problems with convergence and has good fit. Does that sound wrong?

I am suspicious though as to why the cross-time residual correlations are negative for the observed categorical and continuous variables. Does this sound like an incorrectly specified model?
 Bengt O. Muthen posted on Monday, June 04, 2012 - 10:00 am
Then we need to see the output - please send to Support.
 Bengt O. Muthen posted on Monday, June 04, 2012 - 9:26 pm
Looks like the model is identified by restrictions across the different processes. Don't know what to say about the negative cross-time residual correlations.
 Dustin Pardini posted on Thursday, March 14, 2013 - 2:57 pm
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
 Bengt O. Muthen posted on Thursday, March 14, 2013 - 4:13 pm
Haven't done this yet, but should. I can send you the ones you are interested in.
 Dustin Pardini posted on Thursday, March 14, 2013 - 7:22 pm
That would be great. I'm trying to establish longitudinal measurement invariance of a bifactor model over 9 waves with an accelarated cohort design and have been running into a few problems.

I believe the syntax related to your Table 6 dealing with longitudinal two-level analysis using bi-factor CFA and the Bayes estimator would be helpful, or any others you suggest. Thanks again.

dap38@pitt.edu
 Bengt O. Muthen posted on Friday, March 15, 2013 - 4:52 pm
Will send the Table 6 output.
 Pawel Grygiel posted on Wednesday, July 31, 2013 - 2:07 am
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 (grygiel@uniwersytet.rzeszow.pl).
 Bengt O. Muthen posted on Wednesday, July 31, 2013 - 10:32 am
Will have it posted today.
 Bengt O. Muthen posted on Thursday, August 01, 2013 - 8:09 am
The Table 6 run has been posted.
 RuoShui posted on Tuesday, April 01, 2014 - 4:20 am
Dear Drs. Muthen,

I am pretty new to longitudinal measurement invariance with categorical indicators. After reading the users' guide, I am not sure if I had this correct. Could you please kindly have a look at the syntax?
1. Testing configural invariance
USEVARIABLES ARE .....
CATEGORICAL=
a1-a3 b1-b3 c1-c3;
Model:
f1 BY a1-a3;
f2 BY b1-b3;
f3 BY c1-c3;
[f1-f3@0];
{a1-a3@1 b1-b3@1 c1-c3@1};
a1 WITH b1;
a1 WITh c1;
b1 WITH c1;
...
savedata: difftest is configural.dat

2. Testing metric and scalar invariance in tandem
Analysis; difftest is configural.dat
Model:
f1 BY a1
a2 (1)
a3 (2)
f2 BY b1
b2 (1)
b3 (2)
f3 BY c1
c2 (1)
c3 (2)
[a1$1 b1$1 c1$1](3);
[a1$2 b1$2 c1$2](4);
[a2$1 b2$1 c2$1](5);
[a2$2 b2$2 c2$2](6);
[a3$1 b3$1 c3$1](7);
[a3$2 b3$2 c3$2](8);
[a1@0];
{a1-a3@1};
a1 WITH b1;
a1 WITh c1;
b1 WITH c1;
...
Thank you so much for your help!
 Linda K. Muthen posted on Tuesday, April 01, 2014 - 6:01 am
See the Version 7.1 Language Addendum on the website with the user's guide. All models for testing for measurement invariance along with new features for testing for measurement invariance are described starting on page 4.
 RuoShui posted on Tuesday, April 01, 2014 - 4:22 pm
Thank you so much Dr. Muthen. That was really helpful.

I have a related question. From the user's guide, it seems that the factor loading and threshold should be constrained in tandem when testing measurement invariance for ordinal indicators. But the language addendum carried out metric and scalar invariance in separate steps. When I tested scalar invariance, the modification indices suggested relaxing the two thresholds for 1 item. My question is, do I also need to relax its factor loading at the same time I relax its two thresholds?

Another question is when I fit a linear growth curve modeling, there is this warning message. Could you please let me know what this means? Thank you very much!

THE MODEL CONTAINS A NON-ZERO CORRELATION BETWEEN DEPENDENT VARIABLES.
SUCH CORRELATIONS ARE IGNORED IN THE COMPUTATION OF THE FACTOR SCORES.
 Linda K. Muthen posted on Wednesday, April 02, 2014 - 11:20 am
The Version 7.1 is a more thorough description of measurement invariance models for more specific situations. As stated there, thresholds can be tested separate from factor loadings for ordinal items.

You must have a WITH statement between two dependent variables which means that factor scores cannot be computed.
 RuoShui posted on Wednesday, April 02, 2014 - 12:52 pm
Dear Dr. Muthen,

Thank you very much!

Yes, I did correlate the error variance between the same ordinal indicators at different time points using WITH statement. But i did not get the error message when I fitted similar models with continuous indicators.
Does this only happen for ordinal indicators?
And is it a concern that factor scores cannot be computed? In other words, does using WITH statement between two ordinal indicators invalidate the model?

Thank you again for your time.
 Linda K. Muthen posted on Wednesday, April 02, 2014 - 3:27 pm
This situation happens only with categorical items and weighted least squares estimation. No, the WITH statement does not invalidate the model. This is only a problem if you want factor scores based on a model with a WITH statement.
 RuoShui posted on Wednesday, April 02, 2014 - 3:43 pm
Thank you very much Dr. Muthen!
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