Calculate R² from regression estimates (beta coefficients, variances of variables etc.)

Cross Validated Asked by phx on December 1, 2020

I need a generalisable way to to calculate $R^2$ from given regression estimates (regression coefficients, variances of variables, covariances between variables, residual variance of the response variable).

Detailed problem and question:

I am estimating a structural equation model with interactions in Mplus. This so called Latent Moderator Analysis is established and works well enough (using the XWITH command in Mplus). However, Mplus is not able to estimate $R^2$ for such models.

Maslowsky, Jager and Hemken (2014, p. 50) argue that one can estimate $R^2$ using a model’s estimated regression coefficients ($beta_{xi}$), variances of the latent predictors ($sigma_{xi}^2$), the covariance(s) between the latent predictors ($sigma_{xij}^2$) as well as the residual variance of the response variable ($sigma_{Yres}^2$).

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Unfortunately, the authors present the equation to estimate $R^2$ only for two independent variables (I tested the given equation for a model with two variables and it works fine). I am, however, not able to generalise this equation and to use it with more than two independent variables. Indeed, I need it for four and eventually seven predictors in a regression model.

How can this equation be generalised to more than two variables?

In addition, the authors present this equation to estimate the $R^2$ for the model with interactions (p. 50):

enter image description here

How can this equation be generalised to more than two variables?

In my case I have four (and eventually with the control variables seven) independent variables and three interaction variables (with a $beta_{xixj}$ each). In other words, one moderator variable moderates three main effects on the response variable.

I would be very grateful for help. Many thanks indeed!


Maslowsky, J., Jager, J., & Hemken, D. (2015). Estimating and interpreting latent variable interactions: A tutorial for applying the latent moderated structural equations method. International Journal of Behavioral Development, 39(1), 87–96.

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