Estimating Optimal Transformations for Multiple Regression and Correlation

Abstract 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 = Eθ (Y) — Σ Ø j (Xj) 2/varθ (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, Øρθ (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.