David A. Kenny
October 25, 2005
Multiple Group Models
Data Preparation
Start with a covariance matrix and sample size for each group.  Usually it is computationally more efficient to input the correlation matrix with the set of standard deviations.  It is almost always wrong to estimate a multiple group model analyzing the correlation matrices because groups usually differ in their variances.  Alternatively, read the raw data.

Step I: Find a Common Model
Before beginning to estimate invariance models, it must be established that a model without any invariances (i.e., a the same model in all groups, but parameters my vary) is a reasonable model.  The fit of this model equals the sum of the chi squares and degrees of freedom across groups and reveals the extent to which the underlying structure fits the data when no constraints across groups are added.  Before we can decide that parameter estimates are different, we must be sure that the model we are estimating is reasonable.  Once this is done, then that model can be used as a basis for comparison to test for invariance.  In comparing models, one often should use a measure of fit like the Tucker-Lewis index and not the chi square difference. 

Step II: Invariance of Factor Loadings
The first set of values to test for invariance are the factor loadings.  If the factor loadings are not invariant, then it makes no sense to test the equality of the paths because the units of measurement would differ across groups.  So if the loadings vary, proceed to Step III.

Step III: Invariance of Paths
The second set of invariances to test are the paths.  Again this test should only be executed if the loadings are invariant.

Step IV: Invariance of Error Variances
Regardless what has happened above, it is meaningful to test whether the error variances are the same in both groups.  If the paths vary or if both the loadings and paths vary, such variation should be allowed at this step.

Step V: Invariance of Error Covariances
If the error variances are invariant, we can test whether the error covariances are equal.  In essence, this tests the equality of the error correlations.

Step VI: Invariance of Factor and Disturbance Variances
The next test is whether the factor variances are equal.  This test is meaningful only if the loadings are invariant.  Even if the paths or the error variances vary, variation in the variance can still be allowed.

Step VII: Invariance of Factor and Disturbance Covariances
The final test is whether the factor covariances are equal.  This test is only meaningful if the loadings and the factor variances are invariant.   Given equality of the factor variances, this test evaluates equality of the factor correlations.

Interpretation
If a parameter set is deemed to vary across groups, to interpret those differences examine the estimates of a previous model in which that parameter set varies.

Neff Example
This example is taken from

Neff, J. A.  (1985).  Race and vulnerability to stress:  An examination of differential vulnerability.  Journal of Personality and Social Psychology, 49, 481-491.
The same model is estimated for 658 Whites and 171 African-Americans.  The following variables in the model using Neff's notation: There appears to be an error in the standard deviation for education of whites.  It is changed to .75.

The measurement model is as follows: The first two variables are indicators of a life change or stress factor.  The next three are indicators of a mental health factor. The next two are indicators of socio-economic status or SES and the last is a single indicator variable of age.

The structural model is as follows: Age and SES are exogenous and they each cause the endogenous factors.  Stress is assumed to cause mental health. The model is presented in Figure 1 of the paper.

The results from the six models described previously are as follows (Model V cannot be estimated because there are no error covariances):

Model     Chi Square df        c2/df     Tucker Lewis
I           69.81    30         2.33         .940
II          80.52    34         2.37         .938
III         86.06    39         2.21         .946
IV         139.01    46         3.02         .909
VI         186.78    50         3.74         .876
VII        197.33    51         3.87         .871

Parameter                 c2         df      c2/df
Loadings                10.71         4       2.68
Paths                    5.54         5       1.11
Error Variances         52.95         7       7.56
Factor Variances        47.77         4      11.94
Factor Covariances      10.55         1      10.55

The evidence seems to support that the paths and the loadings are invariant, but the variances and covariance are not.

Loadings
Variable   Whites   African-Americans    Summary
   X1        1.000        1.000           Education more important
   X2        .209         .491           for African-Americans
   Y1        1.000        1.000           Total change more important
   Y2          .657         .809            for Whites
   Y3       1.000        1.000            Nervous more important for
   Y4          .988        1.185            Whites
   Y5         1.002        1.229

Note that in comparing loading, we need to compare their relative size.  So if we make Y4 or Y5 the marker, we see more clearly that Y3 is the more variable indicator:

Variable   Whites   African-Americans
   Y3        .998         .814  
   Y4          .986         .964
   Y5         1.000        1.000

Paths
Cause    Effect       Whites  African-Americans    Summary
SES      Stress        .057        .009           SES affects Stress
SES      Mental-Hh    -.081       -.097      ;     more for whites
Age      Stress        .004        .006
Age      Mental-Hh    -.009       -.007
Stress   Mental-Hh     .127        .195           African-Am. more
                                                  affected by stress

Very often the equality of the paths are of central interest.  They can be tested individually by examining the modification indices from Model III.  They evaluate making that path the only one to be unequal across groups.  Below we see that there are no race differences in any of the paths:

Paths
Cause    Effect          Z*
SES      Stress        1.78
SES      Mental-Hh     1.19
Age      Stress       -1.52
Age      Mental-Hh    -1.30
Stress   Mental-Hh    -1.22
*Whites - African-Americans

Error Variance
Variable   Whites   African-Americans     Summary
    X1      5.380        3.620            Whites more variable
    X2       .391         .256
    Y1       .253         .085            Whites more variable
    Y2       .185         .151
    Y3       .158         .227            African-Americans more
    Y4       .109         .191            variable
    Y5       .123         .179

Variance
Variable         Whites   African-Americans   Summary
SES              2.880         1.147          Except for Mental Health
Age            330.876       278.556          Whites more variable
Stress            .784          .402          than African-Americans
Mental-Health     .074          .154

Testing the for the equality of the covariances makes little sense if the variances are not equal, but it is done so for illustrative purposes only.

Covariance/Correlation
Variables    Whites   African-Americans   Summary
SES-Age  -11.832/-.42   -21.184/-.76      r more negative for African-Amer.

Go to the SEM page.