fungible (version 2.2.2)

rarc: Rotate Points on the Surface on an N-Dimensional Ellipsoid

Description

Rotate between two points on the surface on an n-dimensional ellipsoid. The hyper-ellipsoid is composed of all points, B, such that B' Rxx B = Rsq. Vector B contains standardized regression coefficients.

Usage

rarc(Rxx, Rsq, b1, b2, Npoints)

Value

b

N+1 sets of OLS coefficient vectors between b1 and b2.

Arguments

Rxx

Predictor correlation matrix.

Rsq

Model coefficient of determination.

b1

First point on ellipsoid. If b1 and b2 are scalars then choose scaled eigenvectors v[b1] and v[b2] as the start and end vectors.

b2

Second point on ellipsoid. If b1 and b2 are scalars then choose scaled eigenvectors v[b1] and v[b2] as the start and end vectors.

Npoints

Generate ``Npoints'' +1 OLS coefficient vectors between b1 and b2.

Author

Niels Waller and Jeff Jones.

References

Waller, N. G. & Jones, J. A. (2011). Investigating the performance of alternate regression weights by studying all possible criteria in regression models with a fixed set of predictors. Psychometrika, 76, 410-439.

Examples

## Example
## GRE/GPA Data
##-------------------##
R <- Rxx <- matrix(c(1.00, .56, .77,
                      .56, 1.00, .73,
                      .77, .73, 1.00), 3, 3)
                  
## GPA validity correlations                 
rxy <- c(.39, .34, .38)
b <- solve(Rxx) %*% rxy
 
Rsq <- t(b) %*% Rxx %*% b
N <- 200       
                   
b <- rarc(Rxx = R, Rsq, b1 = 1, b2 = 3, Npoints = N) 
 
## compute validity vectors
r <- Rxx %*% b
N <- N + 1
Rsq.r <- Rsq.unit <- rep(0, N)

for(i in 1:N){
## eval performance of unit weights
  Rsq.unit[i] <- (t(sign(r[,i])) %*% r[,i])^2 /
 		           (t(sign(r[,i])) %*% R %*% sign(r[,i]))
 		           
## eval performance of correlation weights               
  Rsq.r[i] <- (t(r[,i]) %*% r[,i])^2 /(t(r[,i]) %*% R %*% r[,i])	
}
 
cat("\nAverage relative performance of unit weights across elliptical arc:",
 	    round(mean(Rsq.unit)/Rsq,3) )     
cat("\n\nAverage relative performance of r weights across elliptical arc:",
 	    round(mean(Rsq.r)/Rsq,3) ) 
 

plot(seq(0, 90, length = N), Rsq.r, typ = "l", 
          ylim = c(0, .20),
          xlim = c(0, 95),
          lwd = 3,
          ylab = expression(R^2),
          xlab = expression(paste("Degrees from ",b[1]," in the direction of ",b[2])),
          cex.lab = 1.25, lab = c(10, 5, 5))
 points(seq(0, 90, length = N), Rsq.unit, 
          type = "l", 
          lty = 2, lwd = 3)
 legend(x = 0,y = .12,
        legend = c("r weights", "unit weights"), 
        lty = c(1, 2),
        lwd = c(4, 3),
        cex = 1.5)