# Latent variables in covariance based structural equation modeling

In the partial least squares approach to SEM the latent variables are a weighted sum of their manifest variables: $$LV_{1}=w_{1}X_{1}+w_{2}X_{2}+w_{3}X_{3}$$

I think this refers to the composite factor model. With this I can solve regression equations like: $$LV_{1}=\beta_{21}LV_{2}+\epsilon_{21}$$

Can anybody tell me what the latent Variables in covariance based structural equation modeling are? I think it is called the common factor model and the latent Variables do not have specific values like in PLS but how are the path coefficients $\beta$ computed then? How do the equations look like? It must be some combination of the correlations among the manifest variables x then right? I am not asking for the algorithm but for the plain equations at the end of the algorithm which give the final path coefficients and outer loadings of a CBSEM model.

You might want to check out this JStatSoft paper.

In Covariance based models, the structural equations and latent variable models define a particular covariance struture. This is compared to the actual, observed covariance matrix and parameters are estimated to ensure a good fit. Several approaches are possible, but maximum likelihood and least squares are popular choices. The LV models are of the form:

$$X_1 = \lambda_1 \xi_1 + \epsilon_1$$

say, if $X_1$ only loads on one factor. More complex models are possible, of course:

$$X_1 = \lambda_1 \xi_1 + \lambda_2 \xi_2 + \epsilon_1$$

The $\xi$ are the latent variables and the $\epsilon$'s are the uniquenesses -- We model the observed, manifest variable as a linear combination of unseen latent variables, which affect several manifests, and some noise, which is unique to each manifest. So if $X_1$ is my score on one math test, and $X_2$ is my score on another such test, the idea is that each score is due to my overal math ability (latent variable) and to the vagaries of each test experience.

Regression equations in CBSEM are the same as those used in the partial least squares approach.

In any case, there aren't plain equations'' for the path coefficients, since these do not exist in closed form, but are the result of optimizing an objective function -- typically ML or least squares.

I hope this helps.

• So I think I understood in some way, but let me ask another question to verify: If I would estimate the following model with CBSEM stats.stackexchange.com/questions/156107/… the model would fit the covariance matrix perfect in advance and therefore no optimizing would be needed? Aug 11, 2015 at 14:10
• In that model, the covariance has 21 distinct values, which probably don't correspond to what the model predicts that covariance to be. So you pick the parameters that give the best fit. Aug 11, 2015 at 14:27
• I did not understand this comment, there is a supplied covariance matrix in the first code snippet at the very bottom, which is basically just the paths multiplied so cor(x1,x4)= 0.8*0.8*0.8 = 0.512 and cor(x1,x2) = 0.8*0.8 = 0.64. My question is if I supply CBSEM with that model and the implied covariance matrix, will it estimate all parameters as 0.8 like implied by the model and covariance matrix? Aug 11, 2015 at 14:37
• OK. I get it. I thought you were asking about the original, sample covariance. But yes, the implied covariance would fit the model perfectly. Aug 11, 2015 at 14:40