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I read that in order to test whether 2 paths in a structural equation model significantly differ from each other, you have to compare one model in which both paths are allowed to differ with a second model in which they are constrained to be equal. Now I know how to constrain the non-standardized path coefficients to be equal in Mplus. But how can I do this for the standardized path coefficients?

Or do I get the same results concerning a significant difference between the two path coefficients no matter whether I constrain the standardized or the non-standardized coefficients to be equal?

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Significance tests of path coefficients are computed based on unstandardized estimates, and not on standardized estimates (and when they are, they should not be trusted). I don't think there is a way to impose constraints on standardized estimates in Mplus, nor should you need to. Therefore, you do need to constrain the unstandardized estimates to equality, and do a nested-model comparison where you compare this constrained model to a model in which both (unstandardized) paths are freely estimated.

The one exception where you can constrain standardized paths is when you constrain covariances between latent variables and both latent variables have a fixed variance of 1; these covariances are indeed correlations, and correlations are standardized estimates.

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