1. Approaches of Model Specification for Testing Factor Mean Differences in Multi-group Higher-order CFA Model
Jichuan Wang, Ph.D.
Children’s National Medical Center (CNMC)
The George Washington University School of Medicine
Carl Leukefeld, Ph.D
Department of Behavioral Science
University of Kentucky, Lexington, KY

2. Abstract
As the mean structure part of a higher-order CFA model is generally not identified, it is often a challenge in model specification for testing factor mean differences in multi-group higher-order CFA models. Three different parameter constraint approaches (Approaches A, B, and C) are usually proposed to address the identification problems in such modeling. This study is to discuss and demonstrate applications of the three approaches to testing invariance of factor means across groups in a second-order CFA model. Statistical package Mplus is used for modeling.

3. Basics about Multi-group CFA Model
Test if the factorial structure of the constructs of interest are conceptualized similarly across populations/groups.
Test invariance of measurement parameters (e.g., factor loadings, item intercepts, and error variances/covariacnes) across groups.
Test invariance of structural parameters (e.g., factor means and factor variances/covariances) across groups.
Challenge of testing invariance of factor means across groups.

4. Steps to Conduct Invariance Tests in
Multi-group CFA
Determine the baseline model for each group.
Implement the configural model that incorporates the baseline models for all the groups.
Impose equality constrains on parameters of interest across groups.
Compare the unconstrained and constrained models.

5. Example of Multi-group Second-order CFA Model
Data: Two independent samples selected from the natural history studies on rural drug users in Ohio (n =248) and Kentucky (n =225) in the U.S. between 2003 and 2005.
Measures: The BSI-18 scale was used for model demonstration. The three dimensions of psychiatric disorders measured by the BSI-18: somatization (SOM), depression (DEP), and anxiety (ANX).
Baseline CFA models: (see Figures)

6. Ohio SOM GSI DEP ANX Faintness Chest pains Nausea Short of breath
Numb or tingling
Short of breath
Body weakness
Lonely
No interest
Blue
Hopelessness
Worthlessness
Suicidal thought
Nervousness
Tense
Scared
Restlessness
Panic episodes
Fearful
Faintness
SOM
DEP
ANX
GSI
Ohio

7. Kentucky SOM GSI DEP ANX Faintness Chest pains Nausea Short of breath
Numb or tingling
Short of breath
Body weakness
Lonely
No interest
Blue
Hopelessness
Worthlessness
Suicidal thought
Nervousness
Tense
Scared
Restlessness
Panic episodes
Fearful
Faintness
SOM
DEP
ANX
GSI
Kentucky

8. Description of the baseline models:
-- Second-order CFA model was implemented for Ohio and Kentucky samples separately.
-- In both models the BSI-18 items are loaded to three first-order factors (depression, somatization, and anxiety)
-- In both models the three first-order factors are treated as indicators of the second-order factor (global severity index, GSI).
-- Both baseline models fit data very well.
-- Two error covariances were specified in the OH model, while three error covariances were specified in the KY model.
Configural CFA model: 2-group CFA model

9. Testing factorial structure and invariance of factor loadings:
Factorial structure and factor loadings (both first-order and second-order) are invariant across samples.
Our focus here is to test invariance of factor means.
Mean structure of the second-order CFA model:

10. Model parameter specification approach of testing factor mean invariance in the second-order CFA model:
Approach A:
Identification issues:
Total number of observed variable means: 2*18=36
Total number of mean parameters to estimate: = 22
However, we still have identification problem in the second equation above
Additional restrictions is needed in the example:
Set the second-order factor mean to zero in the reference group or
Set two first-order intercepts to be equal.

11. Selected model results:
Model fit: CFI= 0.952; TLI=0.950; RMSEA=0.055; 90% C.I. of RMSEA: (0.046, 0.063); Close test p=0.180; SRMR =0.050
Factor mean difference between OH and KY:
Difference in the 1st-order factor means between Ohio and Kentucky:
SOM: −0.144 (p = 0.019)
DEP: −0.058 (p = 0.535)
ANX: −0.186 (p = 0.049)
In Approach A the second-order factor means were set at zero in both groups. Difference in the second-order factor mean between groups will be tested using different approaches.

12. Approach B:
where
Some of the intercepts of the observed indicators in each group are set to known values (estimated from the baseline models).
The first-order factor intercepts are set equal across groups.
Model fits: CFI = 0.952; TLI = 0.950; RMSEA = 0.055; 90% C.I. = (0.046, 0.063); Close test p= 0.169; and SRMR =
Factor mean difference between OH and KY:
The estimated difference in the mean of the second-order factor GSI between OH and KY: (p = 0.054).

13. Approach C:
where
The first-order factor intercepts are all fixed to zero in both Groups 1 and 2.
The mean of the second-order factor in group 1 is set to 0 by default.
The model fit statistics and indices using Approach C are almost identical to those using Approach B.
The estimated difference in the second-order factor mean is identical to that using the Approach B.

14. Factorial structure of the BSI-18 are valid in both samples examined.
Conclusion
Factorial structure of the BSI-18 are valid in both samples examined.
Factor loadings (both first-order and second-order) are invariant, indicating measurement invariance across the two samples under study.
The parameter constraint approaches are useful for testing factor mean invariance in multi-group second-order CFA.
-- Approach A can be used to test invariance of first-order factor means across groups.
-- Approaches B or C can be used to test invariance of second-order factor means.