Three types of polynomial mixed model splines have been proposed: smoothing splines, Psplines and penalized splines using a truncated power function basis. The close connections
between these models are demonstrated, showing that the default cubic form of the splines
differs only in the penalty used. A general definition of the mixed model spline is given
that includes general constraints and can be used to produce natural or periodic splines. The
impact of different penalties is demonstrated by evaluation across a set of functions with
specific features, and shows that the best penalty in terms of mean squared error of prediction
depends on both the form of the underlying function and the signal:noise ratio.