David A. Kenny
November 28, 2009

 

Multiple Group Models

 

Basic Question
To what extent is the causal model the same in two or more independent groups.  For instance, is a causal model the same for men and women.  Note that the groups must be independent; if, for instance, both husbands and wives are measured in the same sample, they must be analyzed in one analysis.  The same holds a causal model is estimated for the same people at two different times.  Note also that the model is different for groups of persons, and does not vary as a function of a quantitative variable.  Finally, the “group variable” must be measured.  If it is unmeasured, we have “latent class analysis,” a topic not discussed in this page.

 

Data Preparation
Normally, the raw data are inputted.  If the covariance matrix is to be 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.

 

Basic Strategy
Normally we begin with the same model but the parameters (e.g., paths and loadings are different in the groups.  We then impose equality constraints to test invariance.  We keep imposing more constraints until we obtain a poor fitting model.  If we think it is likely that groups are not difference, i.e., there is complete invariance, we could start the other way.  We begin with a model of invariance and we allow for some differences between groups.

 

There are many parameters that might be invariant (e.g., 29 in the example below), they need to tested sequentially and in groups.  The exact sequence and grouping can vary. 

 

Model I: Find a Common 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 called the configural model) is a reasonable 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, 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 with large sample sizes, one often should use a measure of fit like the CFI or RMSEA index and not the chi square difference.

 

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 two groups.

Model II: Invariance of Factor Loadings
Always 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 do not vary, proceed to Model III.

If the loadings are different, the results very much depend on choice of the marker variable. 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 is used as the marker, it will 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.

Model III: 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.  Normally invariance of the individual paths would be test.  That is, a model would estimated the has all but one set of paths the same in the groups.

Remaining Tests

There is a strong consensus in the literature that the first three models are tested in the order that has been given above.  There is not much consensus about the order of tests of the remaining parameters.

 

One view is that 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.

Model 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 for this model.

Model 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.

Model VI: Invariance of Factor and Endogenous 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.   We may wish to separate the test of variances and disturbances.

Model 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.

Model VIII: Invariance of Relative Intercepts
The next set of invariances tested might be the intercepts of the indicators.  For latent variables with two or more indicators, the indicator intercepts are set equal across groups.  However, the factor means or intercepts are allow to vary across groups.  This is done by fixing the factor mean or intercept to zero in one group, and freeing it in the other groups.  In essence what is being tested here is a group by indicator interaction.

Model IX: Invariance of Factor Intercepts and Means
The next set of invariances tested is the means of the endogenous factors and intercepts of the endogenous factors are the same in the different groups.  In essence what is being tested here is the main effect of group.

Note that if this last model has a good fit, then the groups can be combined into a single sample and group can be ignored.

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 which indicates poor mental health. 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 and below:

 

 

This model has 29 parameters in each group (4 loadings, 5 paths, 7 intercepts, 1 mean, 7 error variances, 2 exogenous variances, 2 disturbance variances, and 1 covariance) and 15 degrees of freedom in each group. The results from the nine models described previously are as follows (Model V is not estimated because there are no error covariances):

 

Model

Tested

Chi Square

df

RMSEA

TLI

I

-----

 69.873

30

.040

.940

II

loadings

 80.602

34

.041

.938

III

paths

 86.162

39

.038

.945

IV

error variances

139.223

46

.050

.909

VI

variance & dist.

187.136

50

.058

.876

VII

exog. covariance

197.723

51

.059

.870

VIII

relative intercepts

218.113

55

.060

.866

IX

factor means & int.

391.181

59

.083

.746

 

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

Model II: For the Neff study, it appears that the loadings are invariant. We see a slight decline in the TLI and a slight increase in the RMSEA.

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.

 

Model III: Equal Paths

 

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

Age      Stress         0.004            0.006
Age      Mental-Hh  -0.009          -0.007
Stress   Mental-Hh  0.127            0.195     Blacks 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. (The square root of a modification index can be treated as an approximate Z test.)  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:

 

Equality of the Individual Paths
Cause          Effect                       Z*
SES             Stress                    1.78
SES             Mental Health         1.19
Age              Stress                    -1.52
Age              Mental Health        -1.30
Stress           Mental Health        -1.22
*White path minus the Black path

 

There is a marginally significant difference that higher SES causes greater Stress more strongly for Whites than Blacks.

 

Model IV: Equal Error Variances

When we force the error variances to be equal, we find that the fit worsens but only slightly.

 

Variable  Whites   Blacks         Summary
    X1       5.190        3.524            Whites more variable
    X2       0.405         0.266
    Y1       0.107         0.002           Whites more variable
    Y2       0.237         0.185
    Y3       0.161         0.237           Blacks more variable
    Y4       0.106         0.196           
    Y5       0.123         0.180

 

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.

 

Model VI: Equal Variances of Exogenous Variables and Endogenous Disturbances

 

Variable           Whites        Blacks   Summary
SES                     2.886         1.147     Whites more variable on everything but

Age                 330.876       278.556    Mental Health         
Stress (U)            0.784          0.402         
Mental-Health (V)  0.074      0.154

 

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

 

Model VII: Covariance/Correlation of Exogenous Variables and Disturbances

 

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.  We also note that the result reverses if we allow for mean and intercept differences between the two groups.

 

Variables    Whites          Blacks                  Summary
SES-Age  -11.822/-.42   -21.173/-.76    r more negative for Blacks

 

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

 

Model VIII: Relative Equal Intercepts

 

Variable   Whites   Blacks        Summary
    X1       5.980         3.570           X1 relatively higher for Whites
    X2       2.190         1.660
    Y1       0.538         0.223                 Y1 relatively higher for Whites
    Y2       0.181         0.101

    Y3       0.616         0.663                 Y4 relatively higher for Blacks
    Y4       0.637         0.864   
    Y5       0.638                0.754

 

What matters here is the difference between indicators, not their absolute size.

Model IX: Equal Exogenous Factor Means and Endogenous Factor Intercepts

Variable                Whites         Blacks         Summary
SES                       0.000           -2.370          Whites higher SES
Age (X3)              46.310           42.176         Whites older

Stress                    0.000          -0.152           Whites more stress

Mental Health        0.000          -0.012           Whites worse mental health

 

Because Age is a single indicator and exogenous, it has a mean for both groups.  Because SES, Stress, and Mental Health are latent variables and we have already constrained their relative intercepts to be equal, we set their factor means and intercepts to zero in one group (Whites) and free them in the other group(s) (Blacks).  

 

We note although the raw means show that Blacks have poorer mental health than Whites, once Age, SES, and Stress are controlled, Blacks have slightly better mental health.  It is noted that the difference between Blacks and Whites on the two endogenous factors (Stress and Mental Health) are not statistically significant.

Summary

Note that because Model IX is not a good fitting model, we cannot pool the data of Whites and Blacks.  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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