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Measurement Model Invariance in Confirmatory Factor Analysis (CFA)

Measurement Model Invariance

Before creating composite variables for a path analysis, configural, metric, and scalar invariance should be tested during the CFA to validate that the factor structure and loadings are sufficiently equivalent across groups, otherwise your composite variables will not be very useful (because they are not actually measuring the same underlying latent construct for both groups).

Configural

Configural invariance tests whether the factor structure represented in your CFA achieves adequate fit when both groups are tested together and freely (i.e., without any cross-group path constraints). To do this, simply build your measurement model as usual, create two groups in AMOS (e.g., male and female), and then split the data along gender. Next, attend to model fit as usual (here’s a reminder: Model Fit). If the resultant model achieves good fit, then you have configural invariance. If you don’t pass the configural invariance test, then you may need to look at the modification indices to improve your model fit or to see how to restructure your CFA.

Metric

If we pass the test of configural invariance, then we need to test for metric invariance. To test for metric invariance, simply perform a chi-square difference test on the two groups just as you would for a structural model. The evaluation is the same as in the structural model invariance test: if you have a significant p-value for the chi-square difference test, then you have evidence of differences between groups, otherwise, they are invariant and you may proceed to make your composites from this measurement model (but make sure you use the whole dataset when you create composites, instead of using the split dataset). If there is a difference between groups, you’ll want to find which factors are different (do this one at a time as demonstrated in the video above). Make sure you place the factor constraint of 1 on the factor variance, rather than on the indicator paths (as shown in the video).

Scalar

If we pass metric invariance, we need to then assess scalar invariance. This can be done as shown in the video above. Essentially you need to assess whether intercepts and structural covariances are equivalent across groups. This is done the same as with metric invariance, but with the test being done on intercepts and structural covariances instead of measurement weights. Keep constraints the same, but for each factor, for one of the groups, make the variance constraint = 1. This can be done in the Manage Models section of AMOS.

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Contingency Plans

If you do not achieve invariant models, here are some appropriate approaches in the order I would attempt them.

  • 1. Modification indices: Fit the model for each group using the unconstrained measurement model. You can toggle between groups when looking at modification indices. So, for example, for males, there might be a high MI for the covariance between e1 and e2, but for females this might not be the case. Go ahead and address those covariances appropriately for both groups. When deleting an item, it does it for both groups. If fitting the model this way does not solve your invariance issues, then you will need to look at differences in regression weights.
  • 2. Regression weights: You need to figure out which item or items are causing the trouble (i.e., which ones do not measure the same across groups). The cause of the lack of invariance is most likely due to one of two things: the strength of the loading for one or more items differs significantly across groups, or, an item or two load better on a factor other than their own for one or more groups. To address the first issue, just look at the standardized regression weights for each group to see if there are any major differences (just eyeball it). If you find a regression weight that is exceptionally different (for example, item2 on Factor 3 has a loading of 0.34 for males and 0.88 for females), then you may need to remove that item if possible. Retest and see if invariance issues are solved. If not, try addressing the second issue (explained next).
  • 3. Standardized Residual Covariances: To address the second issue, you need to analyze the standardized residual covariances (check the residual moments box in the output tab). I talk about this a little bit in my video called “Model fit during a Confirmatory Factor Analysis (CFA) in AMOS” around the 8:35 mark. This matrix can also be toggled between groups. Here is a small example for CSRs and BCRs. We observe that for the BCR group rd3 and q5 have high standardized residual covariances with sw1. So, we could remove sw1 and see if that fixes things, but SW only has three items right now, so another option is to remove rd3 or q5 and see if that fixes things, and if not, then return to this matrix after rerunning things, and see if there are any other issues. Remove items sparingly, and only one at a time, trying your best to leave at least three items with each factor, although two items will also sometimes work if necessary (two just becomes unstable). If you still have issues, then your groups are exceptionally different… This may be due to small sample size for one of the groups. If such is the case, then you may have to list that as a limitation and just move on.
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