fn_R2RV {Morefi}

14 June 2025

Robust version of the coefficient of determination

Description

The function calculates a robust version of the coefficient of determination $R^{2}_{RV}$ using the Equation Summary values fitted with the robustbase::nlrob() function.

This is the consistency corrected robust coefficient of determination by Renaud and Victoria-Feser (2010) which allows for a possible correction factor a for consistency considerations.

Inside the function:

The vectors of observed (y1), predicted (yc) and weighted (W) values are obtained from the summary model (eq).

The value of the Weighted Estimate Average (ywea), the modified sum of squares for explained (SSEw), total (SSTw), and residual (SSRw) were estimated.

The correction factor value for consistency considerations a for 95 percent was also obtained from summary model (eq).

A vector with the robust version of the coefficient of determination R2wa and its adjusted value R2wa_adj were returned as a vector.

In the function, the following were estimated:

The Weighted Estimate Average $\overline{\widehat{y}}_{w}$ (called as ywea).

$$\overline{\widehat{y}}_{w}=\left(1/\Sigma w_{i}\right)\times \Sigma w_{i} \widehat{y}_{i}$$

The modified sum of squares explained SSEw $$SSEw= \sum_{}^{}w_{i}(\widehat{y}_{i}-\overline{\widehat{y}}_{w}) ^2)$$ The modified sum of squared residuals SSRw $$SSRw= \sum_{}^{}w_{i}(y1_{i}-\widehat{y}_{i}) ^2)$$

The robust version of the coefficient of determination $R^{2}_{w,a}$$$R^{2}_{w,a}=\frac{SSEw}{SSEw+a\times SSRw}$$ The adjusted coefficient of determination $R^{2}_{adj,w,a}$$$R^{2}_{adj,w,a}=1- \left(1- R^{2}_{w,a}\right)\times \left( \frac{n-1 }{n-q} \right)$$

where $y1_{i}$, and $\widehat{y}_{i}$ are the $i_{th}$ observed and predicted values, $a$ is the correction factor value for consistency considerations for 95
percent, $n$ the number of pairs of observations and $q$ the number of variables included in the model.

The function requires defining:

• eq: Summary of the equation fitted using robustbase::nlrob() function.

The function returns the a vector with the robust version of the coefficient of determination $R^{2}_{w,a}$ and its adjusted value $R^{2}_{adj,w,a}$.

fn_R2RV

The function is included in the Morefi package Morphological Relationships Fitted by Robust Regression.

library(Morefi)

The function is detailed below.

fn_R2RV <- function(eq){
      w <- eq$rweights   # w a vector of weights values
      yc <- eq$fitted.values # yc a vector of predicted values
      ywea <-  (1/sum(w)) * sum(w*yc) # a value of the Weighted Estimate Average
      y1 <- eq$model[,1]
      SSEw <- sum(w*(yc-ywea)^2)
      SSRw <- sum(w*(y1-yc)^2)
      a <- environment(eq[["psi"]])[["cc"]]
      R2wa <- SSEw/(SSEw+a*SSRw)
      R2wa_adj <- 1-(1-R2wa)*((length(y1)-1)/summary(eq)$df[2])
      R2was <- c(R2wa,R2wa_adj)
      print(R2was)
    }

Examples

Do not execute, it’s just a syntax example.

R2RV <- fn_R2RV(eq)

References

Renaud, O., & Victoria-Feser, M. P. (2010). A robust coefficient of determination for regression. Journal of Statistical Planning and Inference, 140(7), 1852-1862. doi:10.1016/j.jspi.2010.01.008.