Increasing the usefulness of additive spline models by knot removal

Modern techniques for fitting generalized additive models mostly rely on basis expansions of covariates using a large number of basis functions and penalized estimation of parameters. For example, a mixed model approach is used to fit a model for children's lung function that allows for non-linear influence of several covariates available in a substantial data set. While the resulting model is expected to have good prediction performance, its handling beyond simple visual presentation is problematic. It is shown how the number basis functions of the underlying B-spline representation can be reduced by knot removal techniques without refitting, while preserving the shape of the fitted functions. The condition for exact knot removal is extended towards approximate knot removal by incorporating the covariance matrix of the initial parameter estimates, resulting in considerable simplification of the model. Covariance matrices for the transformed parameter estimates are provided. It is demonstrated that enforcing the knot removal condition during estimation leads to the difference penalties employed in the P-spline approach for estimation of B-spline coefficients, and therefore provides a further justification for this type of penalty. A final transform to a truncated power basis provides a simple equation for the model. This increases transportability, while retaining properties of the initial fit such as good prediction performance.