# Rotate Points on the Surface on an N-Dimensional Ellipsoid 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. ```r rarc(Rxx, Rsq, b1, b2, Npoints) ``` ## 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. ## Returns - **b**: N+1 sets of OLS coefficient vectors between b1 and b2. ## Examples ```r ## 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) ``` ## 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. ## Author(s) Niels Waller and Jeff Jones.

rarc function