The additive regression model assumes additivity among explanatory variables and other rigid requirements, which might give poor estimation of regression functions. Transforming response variables is a useful method to diagnose additivity and other requirements. From a practical point of view, parametric transformations such as the Box-Cox power transformation would give more helpful suggestions in interpreting results of analysis than nonparametric transformations.
The power additive smoothing spline (PASS) model is proposed to diagnose the validity of assuming additivity in the additive regression model. The smooth functions (and often regression parameters) are estimated with a penalized likelihood approach, and the power and the smoothing parameters, which govern global nonlinear regression structure, are estimated with the empirical Bayes method, in which a Laplace approximation of the marginal likelihood is developed. The PASS model is applied to some data sets, and also its performance is examined through a simulation experiment. It is shown that the PASS model can extract an appropriate regression structure if true structure is additive after a Box-Cox power transformation of responses.