

Delta and theta in multigroup MI CFA... 

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Dear Drs. Muthén, I am running both multigroup measurement invariance (MI) models, as well as withingroup longitudinal MI models with categorical data using WLSMV estimation in Mplus. The problem I have is that some of the multigroup MI models are testing partial invariance, where the two groups have different correlating residuals. This requires theta parameterization. However, my scalar longitudinal withingroup MI models require delta parameterization, because I receive an error that theta cannot be used with scale factors. The example problem line is: {a1c1@1 a4c4}; My questions are as follows: 1) Should I use the default delta parameterization for baseline CFAs and longitudinal withingroup MI models, and theta parameterization for the multigroup invariance models, some of which test partial MI? 2) Will using different parameterizations confound my results? 


The Delta parameterization can handle correlated residuals. The Theta parameterization doesn't need scale factors because residual variances are instead free. 


Thank you for the quick reply, Dr. Muthén. 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 multigroup 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! 


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! http://www.statmodel.com/discussion/messages/11/464.html?1489024424 

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