Spline-based tests in survival analysis

This paper examines a method for testing hypotheses on covariate effects in a proportional hazards model, and also on how effects change over time in regression analysis of survival data. The technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models. The basic idea is to formulate a flexible parametric alternative using fixed knot splines, together with a penalty function that penalizes noisy alternatives more than smooth ones, to focus the power of the tests toward smooth alternatives. The test statistics are the analogs of ordinary likelihood-based statistics, only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log-likelihood. Large-sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases. Numerical results suggest these approximations may be adequate with moderate sized samples.