Estimating Optimal Transformations for Multiple Regression and Correlation

Abstract

In regression analysis the response variable Y and the predictor variables X 1 …, X p are often replaced by functions θ( Y ) and Ø 1 ( X 1 ), …, Ø p ( X p ). We discuss a procedure for estimating those functions θ and Ø 1 , …, Ø p that minimize e 2 = E {[θ( Y ) — Σ Ø j ( X j )] 2 }/var[θ( Y )], given only a sample {( y k , x k1 , …, x kp ), 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) = max θ,Ø ρ[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.