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I am conducting a CFA with 5 indicator variables. The theory suggests one or two-factor model, but with 5 indicators, only one-factor model would be viable.

The question is that the one-factor model fit poorly, but when allowing items 1 and 2 to be correlated, the model fit significantly improves and is now acceptable.

I'm trying to understand what does it mean to have two items that have correlated residuals. What are the next steps for the analysis? Should I try a bi-factor model instead? Should I remove one of the two items that have correlated residuals? Any suggestions would much appreciated. Thanks.

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The need for a correlated residual means that these two items are more closely related than they should be, according to the model. It's also called a "bloated specific" or a "local dependency".

For example, if you had a scale that had questions:

  • I like parties.
  • Parties are good.
  • I dislike staying at home and reading.
  • I like hanging out with a lot of friends.
  • I consider myself sociable.

We would expect a residual correlation on the first two items - they are essentially the same item, asked twice. In this case, I'd drop one.

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  • $\begingroup$ Thanks. A follow up question: what if the two items can be differentiated in terms of their relationship with an external variable of interest? So the two items are basically one in the 1-factor model, but it could be of theoretical interest to still keep the two separate items? If this is the case, would it be reasonable to keep the one factor model with 5 items, allowing two items to have correlated residuals? Would there be problems in doing so? Thanks again. $\endgroup$ – Lu Li May 1 '18 at 18:07
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We specify correlated measurement residuals when two or more of the manifest variables share some property that isn't shared by the rest of the variables. This could be, for example, that they were both asked negatively whereas all others were asked positively, that they both address the same facet of the underlying latent dimension, that they were both measured in the same way, etc.

When this is the case, it's reasonable to include correlated measurement residuals. Make sure that you have a good reason for doing this, however, and don't just test the correlated residuals until they improve the fit.

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