 Delta and theta in multi-group MI CFA...    Message/Author  Melissa Harry posted on Wednesday, June 28, 2017 - 3:15 pm
Dear Drs. Muth�n,

I am running both multi-group measurement invariance (MI) models, as well as within-group longitudinal MI models with categorical data using WLSMV estimation in Mplus. The problem I have is that some of the multi-group MI models are testing partial invariance, where the two groups have different correlating residuals. This requires theta parameterization. However, my scalar longitudinal within-group MI models require delta parameterization, because I receive an error that theta cannot be used with scale factors. The example problem line is:
{a1-c1@1 a4-c4};

My questions are as follows:

1) Should I use the default delta parameterization for baseline CFAs and longitudinal within-group MI models, and theta parameterization for the multi-group invariance models, some of which test partial MI?

2) Will using different parameterizations confound my results?  Bengt O. Muthen posted on Wednesday, June 28, 2017 - 6:23 pm
The Delta parameterization can handle correlated residuals.

The Theta parameterization doesn't need scale factors because residual variances are instead free.  Melissa Harry posted on Wednesday, June 28, 2017 - 8:33 pm
Thank you for the quick reply, Dr. Muth�n. What I have found is that Delta works when the same residuals are correlated for both groups in the main model statement, but it doesn't work when only one group has correlated residuals. When I use Delta, I get this error message:

*** ERROR
The following MODEL statements are ignored:
* Statements in Group G1:
d1 WITH e1

Here is the basic syntax that works with Theta, but not Delta:

GROUPING = grouping_var (0 = G1 1 = G2);
Analysis:
Type = complex;
ESTIMATOR IS WLSMV;
Parameterization is Delta;
MODEL = CONFIGURAL METRIC SCALAR
MODEL: f1 BY a1 b1 c1 d1 e1;
MODEL G1: d1 WITH e1;

Given this error with Delta, should I just use Theta for all of my MI models, both partial multi-group and longitudinal MI? It sounds like I can since I don't need a scale factor in the longitudinal scalar models if I use Theta parameterization (according to your post above). Much thanks!  Melissa Harry posted on Thursday, June 29, 2017 - 5:41 am
I am also going by the explanation that you had posted on June 7, 2013, including that "Theta parameterization lets you access the residual variances of the factor indicators as parameters, whereas in the Delta parameterization, the delta parameters are functions of factor variances, factor loadings, and residual variances," which to me would be the reason why only Theta works with the above syntax. Would that be correct? Thank you!

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