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 Liz Goncy posted on Tuesday, May 13, 2014 - 7:30 am
I am conducting measurement invariance analyses across two waves of 26 indicators on two factors. I have successfully run an unconstrained model and am comparing this to a model with constraints on loadings and thresholds. I get the following error message when I constrain the thresholds, but not the loadings.

THE CHI-SQUARE COMPUTATION COULD NOT BE COMPLETED BECAUSE OF A SINGULAR MATRIX.

THE STANDARD ERRORS OF THE MODEL PARAMETER ESTIMATES COULD NOT BE COMPUTED. THE MODEL MAY NOT BE IDENTIFIED. CHECK YOUR MODEL.
PROBLEM INVOLVING PARAMETER 200.

THE CONDITION NUMBER IS -0.137D-15.

I believe my model is identified and I can't seem to figure out what the problem is. When I ran the model treating the indicators as continuous, I did not have the same problem. I have a large sample (N = 3800). Any guidance is appreciated.
 Linda K. Muthen posted on Tuesday, May 13, 2014 - 9:12 am
Please send the output and your license number to support@statmodel.com.
 Lois Downey posted on Thursday, June 05, 2014 - 8:09 am
I'm investigating longitudinal measurement invariance for a single group over three time points, using clustered data. The hypothesized 2-factor model includes 8 categorical indicators. I have two questions related to constraining scale factors:

1) Example 6.15 in User's Guide Version 7, includes a statement constraining the scale factors for all indicators for the first time point to 1, and freely estimating the remaining scale factors. In response to a thread dealing with invariance between groups, Bengt indicated that scale factors are needed whenever one is testing either metric or scalar invariance. In my models of scalar invariance (either with or without metric invariance), the syntax provided in Example 6.15 works just fine. However, if I do not constrain the thresholds to invariance, the model terminates with an indication that the standard errors of model parameter estimates could not be computed (with a pointer to one of the indicators in the delta matrix). What might cause this, and is there a solution?

2) In the same response related to scale factors in tests for invariance between groups, Bengt indicates that "You fix scale factors in the configural case because in that case you are not comparing factors across groups." By "FIX," does he mean that one should constrain the scale factors for all indicators at all time points to 1?

Thank you!
 Bengt O. Muthen posted on Thursday, June 05, 2014 - 3:33 pm
1) Threshold invariance is needed to be able to identify factor mean differences across time.

2) Yes.
 Lois Downey posted on Thursday, June 05, 2014 - 3:43 pm
Thank you.

Does your response to my question #1 imply that one should never test factor loading invariance in the absence of threshold invariance?

Thanks.
 Bengt O. Muthen posted on Friday, June 06, 2014 - 7:53 am
You can do that if your categorical variables are polytomous and not binary, but it requires special modeling and there isn't much of a point doing so in my view. See our description of invariance testing in the Version 7.1 Language Addendum on our website under User's Guide.

Regarding the point of it, what would make loadings invariant and not thresholds? Together they form the response curves so I think one should deal with them in tandem.
 Lois Downey posted on Tuesday, June 24, 2014 - 10:13 am
Just one further question. In the 7.1 Language Addendum, in discussing the testing for metric invariance with polytomous indicators (p. 8), it says, "The METRIC setting is allowed for ordered categorical (ordinal) variables in some cases. Then it is allowed, the metric of a factor must be set by fixing a factor loading to one. The METRIC setting is not allowed ... when the metric of the factors is set by fixing the factor variance to one."

However, on page 10, in discussing the specifics of testing for metric invariance with polytomous indicators when using maximum likelihood estimation, this restriction appears to be relaxed. Is the earlier restriction true only for weighted least squares estimation?
 Bengt O. Muthen posted on Tuesday, June 24, 2014 - 2:56 pm
No, the restriction is not relaxed. No the restriction is not only for weighted least squares estimation.
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