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Also, are the standardized regression weights equivalent to the correlation between a factor and a manifest variable?

I write this question with reference to an example on p138-142 of the following document:

Here are illustrative figures and a table: CFA example

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up vote 2 down vote accepted

Yes, if the factors are uncorrelated. This is also true in exploratory factor analysis - if you do an orthogonal rotation (or if you don't rotate) then you have one loading matrix. If you do an oblique rotation, the factors are correlated, and then you have a pattern matrix (which is correlations) and a structure matrix, which is regressions.

If you think about CFA in terms of regression, it becomes clear why.

When you do regression, the regression weights are given by:

$$ \beta = X^{-1}Y $$ Where X is the correlation matrix of the predictors (if we're doing everything standardized). If the predictors are uncorrelated, then $X^{-1}$ is an identity matrix, and so $\beta = Y$. Same with CFA, if your factors are uncorrelated, then regressions are correlations. If not, then they're not.

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