More complex functions for time-by-covariate interactions can be used, such as spline functions, which allow for more flexibility in the shape of estimated curves and in particular non-monotonic modeling of changes in relative risks during the follow-up.

A cubic spline function is a smooth piecewise cubic polynomial function. This combination is smooth because the function and its two first derivatives are continuous at the points which are called knots where the polynomial pieces join each other.

Here we don’t use the ordinary spline functions but restricted spline functions, which have been proposed by Stone and Koo in order to avoid the instability of ordinary spline functions before the first knot and after the last knot. In these restricted spline functions linearity is imposed before the first and after the last knot, while still ensuring smoothness between these knots.

If we use v=3 knots , a restricted cubic spline function can be written as follows where t represents the time and t₁, t₂ and t₃ are the time points where the knots are placed. Once the degree of the spline is fixed, the number of parameters depends only on the number of knots.

We proposed to use the restricted cubic spline functions with 3 knots to model the changes in relative risks over time by extending the relative survival model as shown in this slide.

The advantage of the restricted cubic spline functions is that the number of parameters is reduced to the number of knots and so here equals 3.