Estimating Monotonic Hazard Ratio Functions of Time

In non-proportional hazards models, the hazard ratio for a unit increase in covariate value is not constant but varies over time. Existing approaches to estimating time-varying log hazard ratio include various spline approximations and maximum penalised partial likelihood. We consider improvements to these methods under the plausible assumption that the hazard ratio changes from its short-term to long-term value in a monotonic fashion. A monotone B-spline estimate based on equidistant knots with the last few coefficients constrained to be equal works reasonably well. We also propose a constrained maximum penalised partial likelihood approach with the constraints removed through re-parameterisation. A novel feature of the proposed method is that it is based on a log product of spacings penalty rather than the usual roughness penalty, which makes selection of smoothing parameter easier. The utility of the proposed methods is demonstrated using a real data example and simulations.