Sensible parameters for polynomials and other splines

Splines, including polynomials, are traditionally used to model nonlinear relationships involving continuous predictors. However, when they are included in linear models (or generalized linear models), the estimated parameters for polynomials are not easy for nonmathematicians to understand, and the estimated parameters for other splines are often not easy even for mathematicians to understand. It would be easier if the parameters were differences or ratios between the values of the spline at the reference points and the value of the spline at a base reference point or if the parameters were values of the polynomial or spline at reference points on the x-axis, or The bspline package can be downloaded from Statistical Software Components, and generates spline bases for inclusion in the design matrices of linear models, based on Schoenberg B-splines. The package now has a recently added module flexcurv, which inputs a sequence of reference points on the x-axis and outputs a spline basis, based on equally spaced knots generated automatically, whose parameters are the values of the spline at the reference points. This spline basis can be modified by excluding the spline vector at a base reference point and including the unit vector. If this is done, then the parameter corresponding to the unit vector will be the value of the spline at the base reference point, and the parameters corresponding to the remaining reference spline vectors will be differences between the values of the spline at the corresponding reference points and the value of the spline at the base reference point. The spline bases are therefore extensions, to continuous factors, of the bases of unit vectors and/or indicator functions used to model discrete factors. It is possible to combine these bases for different continuous and/or discrete factors in the same way, using product bases in a design matrix to estimate factor-value combination means and/or factor-value effects and/or factor interactions.