Deriving an Analytical Solution to Inversion of Royston/Parmar Restricted Cubic Spline Parametric Survival Models for Discrete Event Simulation

OBJECTIVES: Discrete event simulations (DES) simulate times to events rather than use of cumulative survival probabilities for parametric survival models, with inversion of the survival functions required for analytical solutions to derive these event times from given survival estimtes. While numerical methods can be used to approximate event times for more complex survival models, this process may be slow, especially when repeated over large numbers of simulations. We aimed to derive analytical solutions to inverse functions for Royston/Parmar restricted cubic spline (RCS) parametric survival models.

METHODS: 3 initial case types were classified according to the positioning of the given cumulative survival estimate “S” between survival probabilities generated for each knot value. For case 1 (before first knot) and case 3 (after last knot), a linear equation for x=ln(t) is produced and single solutions are derived for time “t” as a function of S. For case 2 (between knots), a cubic equation of the form ax³+bx²+cx+d=0 was derived, with generalized solutions to cubic equations then utilized to derive 3 solutions for x=ln(t). Sub-case types were then defined according to whether an intermediate term C³ is a complex number: for case 2a (C³ real), a single real solution is generated for x=ln(t); while Case 2b (C³ complex) required application of mathematical theorems of complex numbers and trigonometric functions to derive formulas for 3 real solutions for x as a function of S, only one of which produces the correct t solution.

RESULTS: An analytical solution to inverse functions for all case types was derived for Royston/Parmar RCS models to help facilitate direct simulation of event times.

CONCLUSIONS: Analytical solutions to inverse functions of Royston/Parmar RCS models can be derived to allow precise estimation of event times from given survival estimates, and may be useful for faster calculation of event times for DES versus numerical methods.