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Message/Author
 Roger Millsap posted on Thursday, January 27, 2000 - 10:09 pm
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.79-80. I do not want to constrain the residual
variances to equality within the groups; only
cross-group 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 y1-y3;
y1 (1) y2 (2) y3 (3);

This would, I thought, introduce invariance
without also producing within-group equality.
But it doesn't work; the program gives an
error. So how is it done?
 Bengt O. Muthen posted on Friday, January 28, 2000 - 10:14 am
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 y1-y4;
y1-y4 (1);

specifies that the residual variances of continuous outcomes y1-y4 will be held equal both within and between groups.

MODEL: f1 BY y1-y4;
y1 (1);
y2 (2);
y3 (3);
y4 (4);

specifies that the residual variances of continuous outcomes y1-y4 are equal across groups but not within groups.
 Feng Tian posted on Thursday, April 10, 2008 - 9:18 am
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 y1-y4;
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?
 Linda K. Muthen posted on Thursday, April 10, 2008 - 9:24 am
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 - 9:52 am
Yes, I have categorical data and use Theta parameterization. How do I constrain the residual variances to be equal in this case?
 Linda K. Muthen posted on Thursday, April 10, 2008 - 10:19 am
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 - 12:01 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:
y1-y4 (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?
 Linda K. Muthen posted on Thursday, April 10, 2008 - 2:46 pm
y1-y4@1;
 Priscilla Lui posted on Thursday, September 05, 2013 - 5:30 pm
I am trying to test invariance of uniqueness across two groups in multiple-group EFA. My indicators are continuous.
When I tried to model, say:

f1-f3 BY y1-y12 (*1);
y1-y12@1;

I got a message saying the residuals cannot be fixed.
What is the correct way to test invariance of residuals in EFA models?
 Linda K. Muthen posted on Friday, September 06, 2013 - 1:34 pm
The correct way is to compare a model with the residual variances free across groups to a model with them held equal across groups.

f1-f3 BY y1-y12 (*1);
y1-y12:

versus

f1-f3 BY y1-y12 (*1);
y1 (2)
y2 (3)
y4 (4)
y4 (5);
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