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In a multiple group problem that is a CFA of a set of dichotomous items, I would like to constrain the residual variances to invariance over groups. The manual discusses equality constraints in the multiple group context on pp.7980. I do not want to constrain the residual variances to equality within the groups; only crossgroup constraints are desired. Unfortunately, it is not clear from the manual how this would be done in the multigrp case. My idea is that, for y1, y2, y3 (say), you would specify in the general model command: MODEL f1 by y1y3; y1 (1) y2 (2) y3 (3); This would, I thought, introduce invariance without also producing withingroup equality. But it doesn't work; the program gives an error. So how is it done? 


There are two issues here. First, with categorical outcomes, residual variances (Theta diagonal elements) do not enter into the model. Their values are computed as "remainders", see Appendix, page 271. This is because correlation matrices are analyzed, so that the diagonal does not enter into the computations (does not carry information). So, these residual variances cannot be held equal over groups as in the continuous case. There are scale factor parameters than can be worked with, but they refer to the inverted y* s.d.'s and are therefore functions of several parameters. The second issue has to do with between and within group equalities. One note here is that only one equality () can be specified per line. MODEL: f1 BY y1y4; y1y4 (1); specifies that the residual variances of continuous outcomes y1y4 will be held equal both within and between groups. MODEL: f1 BY y1y4; y1 (1); y2 (2); y3 (3); y4 (4); specifies that the residual variances of continuous outcomes y1y4 are equal across groups but not within groups. 

Feng Tian posted on Thursday, April 10, 2008  3:18 pm



I have a question also related to constraining the residual variances. I tried the following approach to constrain the residual to be equal across both groups, MODEL: f1 BY y1y4; y1 (1); y2 (2); y3 (3); y4 (4); but when I checked the output I found that the residual variances are not equal across groups. For the first group, they are all 1s (one)s but for the second, they seem like free estimates. Why does this happen? 


It sounds like you have categorical outcomes and are using the Theta parameterization in which case residual variances must be fixed to one in one group. If this is not the case, please send your input, data, output, and license number to support@statmodel.com. 

anonymous posted on Thursday, April 10, 2008  3:52 pm



Yes, I have categorical data and use Theta parameterization. How do I constrain the residual variances to be equal in this case? 


The comparison is of a model with residual variance one in one group and free in the other groups to a model with the residual variances fixed to one in all groups. 

anonymous posted on Thursday, April 10, 2008  6:01 pm



so how to fix residual variances to one for all groups? I followed e.g 12.10 (equalities in a multiple group analysis) with model: y1y4 (1); which should hold residual variances of y1 to y4 equal to each other and equal across groups. but still the output shows the residual variances fixed to one only for one group not the other. Is this because that example is for continuous variables? and how to fix residual variances to one for all groups or constrain them to be equal across groups? 


y1y4@1; 


I am trying to test invariance of uniqueness across two groups in multiplegroup EFA. My indicators are continuous. When I tried to model, say: f1f3 BY y1y12 (*1); y1y12@1; I got a message saying the residuals cannot be fixed. What is the correct way to test invariance of residuals in EFA models? 


The correct way is to compare a model with the residual variances free across groups to a model with them held equal across groups. f1f3 BY y1y12 (*1); y1y12: versus f1f3 BY y1y12 (*1); y1 (2) y2 (3) y4 (4) y4 (5); 

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