David A. Kenny
September 7, 2011

Multiple Groups

Basic Question and Data Requirement
Does my causal model differ for different groups of persons?  Groups must be categorical and membership must be known.  Ideally there should be 200 persons in each group. If group membership is a latent variable, then we have latent class or mixture analysis.  Note too that group membership should be independent and so if we have heterosexual married couples we cannot treat husbands and wives as independent groups.

Data Preparation
Normally, the raw data are inputted.  If the covariance matrix is to read, usually it is computationally more efficient to input the correlation matrix with the set of standard deviations and means.  It is almost always wrong to estimate a multiple group model analyzing the correlation matrices because groups usually differ in their variances.

Configural Model
Before beginning to estimate invariance models, it must be established that a model without any invariances (i.e., the same model in all groups, but parameters may vary) is a reasonable model.  This model is called the configural model.  The fit of this model equals the sum of the chi squares and the sum degrees of freedom across groups and that fit 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 the same, it must be established that the configural model is reasonable because all other models place constraints on that model.  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 or RMSEA index and not the chi square difference.

To identify latent means and intercepts, each marker variables intercept is fixed to zero.

Ideally, one searches for the common model using both groups.  It is probably inadvisable to use the entire sample because such a strategy uses a mixture of the groups and would be biased toward using the model that favors the larger of the groups.

Invariance of Factor Loadings
Always the first set of values to test for invariance is 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 do not vary, proceed to the next step.

If the loadings are different, the results may be more interpretable if a different marker variable is chosen. Consider for instance a case with four indicators in two groups, the loadings of 2nd through the 4th indicators are invariant. If the 1st indicator were used as the marker, it would appear that the other three loadings are changing across groups when in fact they are invariant. It can be advisable to change the marker variable to determine which loadings are invariant and which are not. One may find that some of the loadings are invariant and others are not; if one has an excess of indictors, one can drop from the model those loadings that differ. To make any claim of invariance, at least one free loading must be fixed to be the same across groups.

Remaining Tests

The next set of tests can be in almost any order, although tests of covariances should be only done if the variances are invariant (and so are tests of equality of correlations). Note that if the parameters are not invariant, then that test might be moved to the bottom list and redone. This is done, because tests of invariance, presume that the parameters tested above are invariant.

Invariance of Paths
The second set of invariances tested is the invariance of the causal paths.  Again this test should only be executed if the loadings are invariant.

Invariance of Intercepts
The next set of invariances tested might be the intercepts of the indicators.   Again this test should only be executed if the loadings are invariant.

Invariance of Error Variances
Regardless what has happened above, it is meaningful to test whether the error variances are the same in both groups, i.e., homogeneity of error variances.  If the paths vary or if both the loadings and paths vary, such variation should be allowed for this model.

Invariance of Error Correlation
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.

Invariance of Disturbance Variances
The next test is whether the disturbance 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 variances can still be allowed. This analysis tests what is essentially the homogeneity of error variance assumption.

Invariance of Disturbance Correlations
The next 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. Note that if this model fits, then all the parameters in the groups are equal and the data could be pooled and group should be ignored.

Invariance of Exogenous Factor 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.

Invariance of Exogenous Factor Correlations
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. Note that if this model fits, then all the parameters in the groups are equal and the data could be pooled and group should be ignored.

Invariance of Factor Intercepts and Means
The final set of invariances tested is the intercepts of the endogenous factors and the means of the exogenous factors.

Model of Complete Invariance
In this model all the above parameters in the groups are set equal. If this is a good fitting model, the grouping variable has no effect and a single model treating persons as if they were from one group can be estimated. There would be complete invariance.

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 Blacks.  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 socioeconomic status or SES and the last is a single indicator variable of age.

The configural 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 and below:

This model has 29 parameters in each group (4 loadings, 5 paths, 4 indicator intercepts, 2 means, 2 latent intercepts, 7 error variances, 2 exogenous variances, 2 disturbance variances, and 1 covariance) and 15 degrees of freedom in each group. The marker variables are Income, Y1, and Y3 and their loadings are set to one and intercepts to zero.

Download the Amos 19 setup and an Excel file with data.

The results from the models described previously are as follows (the model with equal disturbance correlations and equal error correlations are not estimated because they are not contained in the model):

Table 1: Summary of Model Fit with Invariances: A Given Model Has the Constraints of All Models above It

Model

Chi Square

df

p for chi sq.

diff.

TLI

RMSEA

Configural

 69.873

30

----

.940

.040

Equal Loadings

 80.602

34

.030

.938

.041

Equal Paths

 86.162

39

.351

.945

.038

Equal Intercepts

104.963

43

.001

.935

.042

Equal Disturbance Variances

122.923

45

<.001

.922

.046

Equal Exogenous Variances

126.620

47

.157

.924

.045

Equal Exogenous Correlation

132.105

48

.019

.921

.046

Equal Error Variances

218.113

55

<.001

.866

.060

Equal Factor Means and Intercepts

391.181

59

<.001

.746

.083

Configural Model: Although the chi square for this model is statistically significant, the RMSEA is acceptable and the TLI is marginal. Thus, the model is a reasonably good fitting model.

Equal Loadings Model: For the Neff study, it appears that the loadings are invariant. Although we see a slight decline in the CFI and a slight increase in the RMSEA, the fit values remain acceptable.

Variable      Whites         Blacks         Summary
   X1                     1.000           1.000          Education more important for Blacks
   X2            0.209           0.491          
   Y1               1.000           1.000          Total change more important for Whites
   Y2                     0.657           0.809           
   Y3              1.000           1.000          Nervous more important for Whites
   Y4               0.988           1.185           
   Y5                     1.002          1.229

Note that in comparing loadings, their relative size needs to be compared.  So if Y4 or Y5 is made the marker the marker, it would be seen more clearly that Y3 is the more variable indicator:

Variable      Whites         Blacks
   Y3                     0.998          0.814  
   Y4                 0.986           0.964
   Y5                  1.000           1.000

That is, Y3 is considerably lower for Blacks than for Whites.

Equal Paths

Very often the equality of the paths is of central interest.  We see that the fit does not worsen when the paths are set equal.

Cause          Effect                    Whites         Blacks         Summary
SES             Stress                     0.057          -0.009           SES affects Stress more for Whites
SES             Mental-Health        -0.081          -0.097

Age              Stress                    0.004           0.006
Age              Mental-Health       -0.009          -0.007
Stress           Mental-Health        0.127            0.195           Blacks' MH more affected by Stress
                                               

They can be tested individually by two different ways. First, and perhaps the easiest but somewhat rough, is to examine the modification indices from the model with equal paths. The square root of a modification index can be treated as an approximate Z test which evaluates making that one path the only one to be unequal across groups.  This test, denoted as Z1 below, can be too liberal. Better is to use the equal loading model, and fix one path at a time to be equal across groups and use the chi square difference test to evaluate the model: The fit of this model is compared to the model with all paths equal. Again the square root of chi square, denoted as Z2 below, evaluates the equality of loadings. We see below that there are no race differences in any of the paths:

Cause          Effect                     Z1       Z2
SES             Stress                      1.78  0.72
SES             Mental Health          1.19   0.24
Age              Stress                    -1.52 -0.30
Age              Mental Health        -1.30 -0.36
Stress           Mental Health        -1.22 -0.90
*White path minus the Black path

There is a marginally significant difference that higher SES causes greater Stress more strongly for Whites than Blacks, but this uses the liberal modification test.

Equal Intercepts

The non-zero marker variables are allowed to have different intercepts.

Variable   Whites   Blacks      Summary
    X1       0.000         0.000        Whites have relatively more income and less education than Blacks.
    X2       0.570         0.693
    Y1       0.000         0.000        Whites score relatively higher on Y1 and lower on Y2 than Blacks.
    Y2      -0.177        -0.045
    Y3       0.000         0.000            Blacks score relatively higher on Y4 than Whites.
    Y4       0.006         0.185   
    Y5      -0.012         0.054

We note that the fit worsens some imposing these constraints. Likely most of that decline is due to Y4 and if that indicator were dropped the fit decline might disappear.

Equal Disturbance Variances

We force the two disturbance variances to be equal across Whites and Blacks. Variance due to the exogenous variables is removed.

Variable              Whites        Blacks         Summary
Stress                    0.735          0.444           Whites more variable         
Mental-Health        0.077          0.132           Blacks more variable

Note that presuming that the disturbance variances in the two groups are equal results in a worsening of fit.

Equal Variances of Exogenous Variables

We force the two exogenous variances to be equal across Whites and Blacks.

Variable           Whites         Blacks       Summary
SES                     2.842        1.910           Whites more variable
Age                 330.876      278.556          Whites more variable

Note that presuming that the variances in the two groups are equal results in a worsening of fit. Also assuming that the groups have the same means and intercepts leads to falsely finding that Blacks are more variable on SES.

Exogenous Variables Correlation

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

Variables    Whites       Blacks           Summary
SES-Age  -12.727/-.45   -19.257/-.68    r more negative for Blacks

Despite the large difference between the correlations, the difference is not statistically significant.

Equal Error Variances

When we force the error variances to be equal, we find that the fit worsens quite a bit.

Variable  Whites   Blacks      Summary
    X1       5.279        3.079      Whites more variable
    X2       0.419         0.263
    Y1       0.112        -0.102     Whites more variable
    Y2       0.239         0.221
    Y3       0.159         0.308      Blacks more variable
    Y4       0.104         0.221           
    Y5       0.124         0.199

We see that there is more error variance for the SES and Stress indicators for Whites, but there is more error variance for Black for the mental health factor. Note that there is Heywood case for Y1 for Blacks (negative error variance), likely caused by constraining Factor and Disturbance Variances to be equal.

Equal Factor Means and Intercepts

For the Neff data, we can test the equality of the means of the exogenous variables, Age and Stress, as well as the intercepts for the endogenous variables. For the endogenous variables, we have controlled for the two exogenous variables.

Variable      Whites         Blacks         Summary
Age              46.310         43.320          Whites older

SES               5.953         3.369            Whites higher SES.
Stress             0.102         -0.706           Whites have more Stress

Mental Health 1.038         1.015            Blacks have slightly greater Mental Health

We note that imposing these constraints results in much worse fit, and thus there are race difference in the factor means and intercepts.

Summary of the Neff Example

Likely, the best fitting model is Model III, the model with equal loadings and paths.  Perhaps we might wish to additionally allow for equal error variances.


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