r2_mlm: Multivariate R2

In performance: Assessment of Regression Models Performance

Multivariate R2

Description

Calculates two multivariate R2 values for multivariate linear regression.

Usage

r2_mlm(model, ...)

Arguments

model	Multivariate linear regression model.
...	Currently not used.

Details

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.

Value

A named vector with the R2 values.

Author(s)

Joseph Luchman

References

• 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.

Examples

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)

performance documentation built on Oct. 19, 2024, 1:07 a.m.