Solved – How is $R^2$ calculated in the lavaan package for structural equation modeling

lavaanstructural-equation-modeling

I am doing a study for which I tested a model like this:

Model1: Var1 –> var2–> var3 –> var4. All variables are latent variables and have several indicators.

Now, the $R^2$ of var3 is 0.4 in this case. However, something strange happens when I test the model:

Model2: Var1 –> var2 –> var3.

In this case, although the variables BEFORE var3 remain exactly the same, the $R^2$ of var3 becomes much smaller (0.28).

Does anyone have a possible explanation for this? How does lavaan package in R calculate $R^2$? Especially for mediating variables like var3 in this case? The indicators for var3 remain the same. I've already checked whether this is due to missings in var4 but this is not the case.

Best Answer

In Lavaan, $R^2$ is a byproduct of the other parameters, it's not a parameter that is directly calculated. So $R^2$ for Var3 depends on the variance of var3, and the size of the path from var2 to var3. And these all depend on the loading.

When you add something to a model (var4) this can have effects all over the model - people sometimes talk about mis-specification 'flowing through the model'. Let's say that var2 and var4 are not correlated, but var2 and var3 are correlated, and var3 and var4 are correlated. If the estimates stay the same, the predicted correlation between var2 and var3 will be too high, so these the var2-> var3 estimate is reduced, lowering $R^2$, and this would be indicated by a lack of fit.

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Solved – How to compute and interpret the mean of a latent variable in structural equation modeling using lavaan

Means and intercepts of latent variables are kind of weird.

It looks to me like you've constrained the means of the intercepts to zero. If you haven't, then model is not identified.
You estimate the intercepts freely with:
```
A1 ~ 1
```
I don't think you did that. (But post your output if you want me to be sure.)
Means (and intercepts) of latent variables are not directly interpretable, because they are arbitrary. You can set the mean to be any value you want it to be. You can interpret latent variable means relative to each other - e.g. you can constrain one latent mean to be equal to 0 and (as long as there are identifying constraints), a second mean might be 1 - you can then say that the second mean is one unit higher than the first. But you can also constrain the first mean to be 16.5, and then you'll discover that the second mean is 17.5, so it's one unit higher.

Solved – SEM model in lavaan: Can’t compute standard errors

The model is not identified, which means there is no unique solution to the estimation problem. Identification is a challenging topic, one that is often overlooked. First, your graphical model is incorrect. You have manifest variables pointing to the latent variables, when in your model, the manifest variables measure the latent variable. Second, the cause for the lack of identification is the residual covariances among all the indicators and the fact that all the indicators load on both latent variables. With so few indicators and most of them shared between the latent variables, you cannot supply any residual covariances. In general, each latent variable needs two unique indicators (i.e., unique to each latent variable), and residual covariances generally cannot be included without more than four indicators.

I highly recommend you look at the identification chapter in an SEM textbook, like Bollen (1989). There are specific rules to identification and ways to assess whether your model is identified.

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