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.
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?
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?
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.
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.
> 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?
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
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?
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?