Quadratic regression among latent variables

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C. M.

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Apr 18, 2019, 10:37:54 AM4/18/19
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Hi,

In lavaan model syntax it is possible to define linear regressions among latent variables. Is there a way to define polynomial (most importantly, quadratic) regression models too? 

Suppose we have want to have the following regression among factors:
 F2 ~ F1 + (F1)^2
My first idea was to introduce a new factor F3 = (F1)^2 and fit a factor model to F1, F2 & F3. 
I am confused how the loadings would behave in this case. Would the loadings of the related factors follow the squared relation of the factors? 

thanks!

Edward Rigdon

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Apr 19, 2019, 10:19:23 AM4/19/19
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If you use a product indicator approach to create an F1 squared, then yes, you will cross multiply the indicators of F1 times each other. The indProd() function in semTools makes this easy. If the indicators are X1, X2 and X3, that would give you 9 new product indicators, but using them all would require a pattern of free residual covariances. A "matching" approach would have you only multiply X1 times X1, X2 times X2, and X3 times X3. Double mean centering will eliminate the need for some more sophisticated constraints. Again, indProd() makes it easy to do this before your sem analysis. After this, then you syntax is straightforward--you just have a new factor with 3 indicators, which represents the quadratic effect. Some literature says that using all 9 cross-products is "better," but I don't think t is worth the hassle. There are also more sophisticated approaches, but their implementations in R are limited.
 

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