LINEAR-REGRESSION AFTER SPLINE TRANSFORMATION

In a transformation model h(Y)=X'beta+epsilon for some smooth and usua lly monotone function h, we are often interested in the direction of b eta without knowing the exact form of h. We consider a projection of h onto a linear space of B-spline functions which has the highest corre lation with the design variable X. As with the Box-Cox transformation, the transformed response may then be analysed by standard linear regr ession software. The direction estimate from canonical correlation cal culations agrees with the least squares estimate for the approximating model subject to an identifiability constraint. This approach is also closely related to the slicing regression of Duan & Li (1991). The di mensionality of the space of spline transformations can be determined by a model selection principle. Typically, a very small number of B-sp line knots is needed. A number of real and simulated data examples is presented to demonstrate the usefulness of this approach.