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 emmanuel bofah posted on Monday, August 12, 2013 - 9:48 am
on page 2 Mplus LANGUAGE ADDENDUM 7.1 is said that the
The alignment optimization method consists of three steps:
1. Analysis of a configural model with the same number of factors and same pattern of zero factor loadings in all groups.
2. Alignment optimization of the measurement parameters, factor loadings and intercepts/thresholds according to a
simplicity criterion that favors few non-invariant measurement parameters.
3. Adjustment of the factor means and variances in line with the optimal alignment.
my question is why is that a model with ALIGNMENT = FREE (CONFIGURAL);
give the same estimates as ALIGNMENT = FREE; because i was thinking you need to be able to compare the configural and the scalar but their estimates are the same.
 Linda K. Muthen posted on Monday, August 12, 2013 - 9:51 am
The default is CONFIGURAL so the two specifications you show are the same.
 emmanuel bofah posted on Monday, August 12, 2013 - 10:49 am
why not possible to specify:
ALIGNMENT = FREE (METRIC);
ALIGNMENT = FREE (SCALAR);
How can i specific scalar model with the alignment.
 Linda K. Muthen posted on Monday, August 12, 2013 - 11:34 am
The alignment method avoids using a metric or scalar model. The definition of the alignment method is that it is based on the configural model.
 Peter Halpin posted on Friday, February 14, 2014 - 9:40 am
Hello,

Is alignment implemented for ordered categorical data?
 Bengt O. Muthen posted on Friday, February 14, 2014 - 11:16 am
No, not yet. Only binary.
 Tait Medina posted on Tuesday, February 25, 2014 - 6:19 am
Is it possible to test the equality of regression coefficients across groups under the alignment method? For example, I am interested in regressing the factor on age and testing whether the coefficient for age is different across groups?

Thank you!
 Bengt O. Muthen posted on Tuesday, February 25, 2014 - 12:15 pm
The alignment method currently doesn't handle covariates. But you can divide people into age groups and then you get a*b groups in the alignment run, where a is the number of age groups and b is the number of original groups.
 Tait Medina posted on Tuesday, February 25, 2014 - 3:11 pm
Thank you for your reply.

I have a question about the output from Alignment=Fixed. How are the estimates given under MODEL RESULTS connected to the numbers given under “Item Parameters In The Alignment Optimization Metric”? I have read Web Note 18, but am having a hard time mapping the output in these two sections onto the equations.

Also, the R2 measures given in Table 7 in Web Note 18 need to be calculated by hand using Eq 13 and 14. Correct?

Thank you.
 Tihomir Asparouhov posted on Wednesday, February 26, 2014 - 5:16 pm
The “Item Parameters In The Alignment Optimization Metric” section contains the alignment results in the metric in which the alignment optimization is performed, i.e., after all indicator variables are standardized and also under constraint (10) from web note 18, also including the factor mean fixed to 0 in the corresponding group. These parameters are scale reversed back to the original metric of the variables and also to factor variance fixed to 1 in the corresponding group if that is the requested parameterization. There is a way to get these parameters as your final model estimates. You will need to standardize your variables using the standardized option of the define command as well as the analysis option METRIC=PRODUCT;

R2 is computed with the upcoming Mplus 7.2 but you can compute it by hand as well.
 Tait Medina posted on Monday, April 07, 2014 - 1:21 pm
Thank you for your response. I have another question about the alignment approach.

I have noticed that when I use ALIGNMENT=FREE I receive a warning that I should switch to ALIGNMENT=FIXED and a reference group (or baseline group) is suggested. How is the suggestion for a baseline group determined? I have played around with using different baseline groups trying to get a feel for this new approach and have noticed that the choice of group impacts the results under the APPROXIMATE MEASUREMENT INVARIANCE (NONINVARIANCE) FOR GROUPS section. Could you provide a bit more insight into this? Thank you.
 Tihomir Asparouhov posted on Monday, April 07, 2014 - 2:50 pm
> I have noticed that when I use ALIGNMENT=FREE I receive a warning that I should switch to ALIGNMENT=FIXED and a reference group (or baseline group) is suggested. How is the suggestion for a baseline group determined?

It is the group with the smallest absolute factor mean value. Presumably fixing that parameter to 0 would lead to the smallest misspecification.


> I have played around with using different baseline groups trying to get a feel for this new approach and have noticed that the choice of group impacts the results under the APPROXIMATE MEASUREMENT INVARIANCE (NONINVARIANCE) FOR GROUPS section. Could you provide a bit more insight into this?

This is explained in Section 5.3 in
http://statmodel.com/examples/webnotes/webnote18.pdf
Fixing the factor mean to 0 in one group can lead to biased results if that mean is not 0.

You can also try using TOLERANCE=0.01. This option seems to yield more robust results and will be Mplus default in the upcoming Mplus 7.2
 Tait Medina posted on Tuesday, April 15, 2014 - 9:24 am
I am a little confused by this output (below). The p-values seem to suggest that the item intercept for CHILD is noninvariant in group 3 as compared to groups 1 and 2.

APPROXIMATE MEASUREMENT INVARIANCE (NONINVARIANCE) FOR GROUPS

Intercepts
NEIGHB 1 2 3
FRIEND 1 2 (3)
SOCIAL 1 2 (3)
WORK 1 2 3
MARRY 1 2 3
CHILD 1 2 3

Intercept for CHILD
Group Group Value Value Difference SE P-value
2 1 3.378 3.385 -0.006 0.027 0.817
3 1 3.501 3.385 0.117 0.039 0.003
3 2 3.501 3.378 0.123 0.042 0.003
Approximate Measurement Invariance Holds For Groups:
1 2 3
Weighted Average Value Across Invariant Groups: 3.430

Invariant Group Values, Difference to Average and Significance
Group Value Difference SE P-value
1 3.385 -0.046 0.019 0.019
2 3.378 -0.052 0.022 0.017
3 3.501 0.071 0.023 0.002
 Tihomir Asparouhov posted on Tuesday, April 15, 2014 - 10:05 am
The process is explained in Section 4
http://statmodel.com/examples/webnotes/webnote18.pdf
but to summarize the invariance is not determined by pairwise comparison but rather by this: compare group 3 against the average of group 1,2,3. Also due to multiple testing we use smaller p-value 0.001 as the cutoff value.
 Tait Medina posted on Tuesday, April 15, 2014 - 11:58 am
Is the pairwise comparison portion of the output related to the "first step" of the algorithm used to determine a starting set of invariant groups that is described in Section 4? "We conduct a pairwise test for each pair of groups and we "connect" two groups if the p-value obtained by the pairwise comparison test is bigger than 0.01." (pg. 15).

Finally, when dichotomous outcome variables are used, how are scale factors/residual variances handled? Are they fixed to 1 in all groups?

Thank you.
 Tihomir Asparouhov posted on Tuesday, April 15, 2014 - 3:06 pm
Yes on the first question.

The second question also yes - we use the theta parameterization where all residual variances are fixed to 1 during the configural model estimation. After that ... the alignment is done without any consideration for the residual variances, i.e., the alignment is for the intercepts and loadings only and it does not use residual variances in the computations.

I have to also correct my message from Feb 26. To get the
"Item Parameters In The Alignment Optimization Metric"
as your final parameter estimates you have to use a linear scale transformation for each indicator variable Y like this
define:Y=(Y-a)/b;
where a and b are obtained from the configural model estimates as follows

a=average Y intercept across the groups
b=average Y loading across the groups
 Tait Medina posted on Friday, April 18, 2014 - 12:51 pm
Thank you, that makes sense.

I have a follow-up question about Eq. 9 in Webnote 18. I am trying to make sure I understand Eq. 9 by plugging in the estimates taken from the output (using ML, Alignment=Fixed) using my own data. For now, I am using 2 groups. The loading for item 1 is .585 in group 1 and .588 in group 2. Taking the difference of these loadings gives me -.003. Scaling this by the CLF (using the small number .0001) gives me f(x)= .103. The Contribution to the Fit Function for this item, given in the output under Loadings, is -.316. I am not sure how to arrive at that number. The sample size for group 1 is 698 and for group 2 it is 949. The sqrt(N1*N2) is therefore 813.881. Weighting f(x) by 813.881 gives me 83.512. What am I misunderstanding about Eq. 9?
 Tihomir Asparouhov posted on Monday, April 21, 2014 - 8:38 am
The loss function that is reported in the output has a negative sign. See footnote 2 on page 10. You are also using 0.01 not 0.0001. Also the weight is standardized: scaled so the total weight is equal to the total number of cross group comparisons NG*(NG-1)/2 which is 1 in your case. So the actual weight that we use for the tech8 output is
w=((NG-1)*NG/2)*w0/sum(w0)
where
w0=sqrt(N1*N2)
and NG is the number of groups.
The weight standardization of course doesn't affect the optimization since it is a constant multiple. It is done so that all weights are 1 when the groups are of equal sizes. In your case the weight is 1 because there is just one cross group comparision. Thus the loss function for that loading is
-sqrt(sqrt(0.003^2+0.01))=-.316
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