Linear regression after spline transformation

SUMMARY In a transformation model h(Y) = X'/3 + E for some smooth and usually monotone function h, we are often interested in the direction of f, 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 correlation with the design variable X. As with the Box-Cox transformation, the transformed response may then be analysed by standard linear regression software. The direction estimate from canonical correlation calculations 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 dimensionality of the space of spline transformations can be determined by a model selection principle. Typically, a very small number of B-spline knots is needed. A number of real and simulated data examples is presented to demonstrate the usefulness of this approach.