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I have the correlation matrix, sample sizes, and descriptive statistics for a set of variables. I know that it is possible to run principal component analysis (PCA) and exploratory factor analysis (EFA) just using the correlation matrix, but is it possible to do confirmatory factor analysis (CFA) and structural equation modelling (SEM) using just this matrix as well?

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Yes. Pretty much any decent software for CFA/SEM should also accept just the covariance/correlation matrix as input (plus information on the underlying sample size). Lisrel, Mplus, R (the lavaan, sem, and OpenMX packages), and so on, all do. If you are using a correlation matrix as input, you should make sure that the software handles that correctly (i.e., it doesn't just treat it as if it was a covariance matrix that just happens to have all 1s along the diagonal by coincidence). In essence, model estimation then requires constrained estimation (to ensure that the fitted diagonal is also equal to 1s), so that the software gives you correct estimates, standard errors, and test statistics. Check the documentation of your software of choice for details.

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If you use R, I suggest you use the lavaan package. See this tutorial page for using a covariance matrix as input. You need to compute a covariance matrix from your correlation matrix and descriptive statistics (i.e., standard deviations) first. You can also use the sem package.

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