Transformations, regression geometry and R²

In making a least-squares fit to a set of data, it is often advantageous to transform the response variable. This can lead to difficulties in making comparisons between competing transformations. Several definitions of R² statistics have been suggested. These calculations mostly involve the actual and fitted values of the response, after the transformation has been inverted, or undone. Kvålseth (Amer. Statist. 39 (1985) 279) discussed the various R² types and Scott and Wild (Amer. Statist. 45 (1991) 127) pointed out some of the problems that arise. In this paper, we examine such problems in a new way by considering the underlying regression geometry. This leads to a new suggestion for an R² statistic based on the geometry, and to a statistic Q which is closely connected to the quality of the estimation of the transformation parameter.