Estimating Optimal Transformations for Multiple Regression and Correlation

In regression analysis the response variable Y and the predictor variables X 1 …, Xp are often replaced by functions θ(Y) and Ø1(X 1), …, Ø p (Xp ). We discuss a procedure for estimating those functions θ and Ø1, …, Ø p that minimize e 2 = E{[θ(Y) — Σ Ø j (Xj )]2}/var[θ(Y)], given only a sample {(yk , xk1 , …, xkp ), 1 ⩽ k ⩽ N} and making minimal assumptions concerning the data distribution or the form of the solution functions. For the bivariate case, p = 1, θ and Ø satisfy ρ = p(θ, Ø) = maxθ,Øρ[θ(Y), Ø(X)], where ρ is the product moment correlation coefficient and ρ is the maximal correlation between X and Y. Our procedure thus also provides a method for estimating the maximal correlation between two variables.