Calculates two multivariate R2 values for multivariate linear regression.
r2_mlm(model, ...)A named vector with the R2 values.
Multivariate linear regression model.
Currently not used.
Joseph Luchman
The two indexes returned summarize model fit for the set of predictors given the system of responses. As compared to the default r2 index for multivariate linear models, the indexes returned by this function provide a single fit value collapsed across all responses.
The two returned indexes were proposed by Van den Burg and Lewis (1988) as an extension of the metrics proposed by Cramer and Nicewander (1979). Of the numerous indexes proposed across these two papers, only two metrics, the \(R_{xy}\) and \(P_{xy}\), are recommended for use by Azen and Budescu (2006).
For a multivariate linear regression with \(p\) predictors and \(q\) responses where \(p > q\), the \(R_{xy}\) index is computed as:
$$R_{xy} = 1 - \prod_{i=1}^p (1 - \rho_i^2)$$
Where \(\rho\) is a canonical variate from a canonical correlation between the predictors and responses. This metric is symmetric and its value does not change when the roles of the variables as predictors or responses are swapped.
The \(P_{xy}\) is computed as:
$$P_{xy} = \frac{q - trace(\bf{S}_{\bf{YY}}^{-1}\bf{S}_{\bf{YY|X}})}{q}$$
Where \(\bf{S}_{\bf{YY}}\) is the matrix of response covariances and \(\bf{S}_{\bf{YY|X}}\) is the matrix of residual covariances given the predictors. This metric is asymmetric and can change depending on which variables are considered predictors versus responses.
Azen, R., & Budescu, D. V. (2006). Comparing predictors in multivariate regression models: An extension of dominance analysis. Journal of Educational and Behavioral Statistics, 31(2), 157-180.
Cramer, E. M., & Nicewander, W. A. (1979). Some symmetric, invariant measures of multivariate association. Psychometrika, 44, 43-54.
Van den Burg, W., & Lewis, C. (1988). Some properties of two measures of multivariate association. Psychometrika, 53, 109-122.
model <- lm(cbind(qsec, drat) ~ wt + mpg + cyl, data = mtcars)
r2_mlm(model)
model_swap <- lm(cbind(wt, mpg, cyl) ~ qsec + drat, data = mtcars)
r2_mlm(model_swap)
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