Abstract
A standard approach for analyses of survival data is the Cox proportional hazards model. It assumes that covariate effects are constant over time, i.e. that the hazards are proportional. With longer follow-up times, though, the effect of a variable often gets weaker and the proportional hazards (PH) assumption is violated. In the last years, several approaches have been proposed to detect and model such time-varying effects. However, comparison and evaluation of the various approaches is difficult. A suitable measure is needed that quantifies the difference between time-varying effects and enables judgement about which method is best, i.e. which estimate is closest to the true effect. In this paper we adapt a measure proposed for the area between smoothed curves of exposure to time-varying effects. This measure is based on the weighted area between curves of time-varying effects relative to the area under a reference function that represents the true effect. We introduce several weighting schemes and demonstrate the application and performance of this new measure in a real-life data set and a simulation study.
Highlights
The Cox proportional hazards model [1] is the standard approach for modelling time to event data
FPT is the part of MFPT which focuses on the selection of a time-varying effect for one covariate using a function selection procedure based on fractional polynomials [11]
In applications with time-varying effects, a measure is required which quantifies, for example, the benefit of a time-varying effect compared to a standard CoxPH effect or a time-varying effect obtained from a different analysis method
Summary
The Cox proportional hazards model [1] is the standard approach for modelling time to event data. When using different approaches for modelling time-varying effects, one may end up with estimates of various types, i.e. interval wise (constant) estimates, piecewise polynomials (often only evaluated at specific time points) or functional forms depending on time. Plotting scaled Schoenfeld residuals [2] from a model is a popular technique to assess whether the effect of a specific variable varies in time. To get an answer to this question, we have to evaluate the similarity of either approach to the truth (e.g. the true effect in simulations or the SSSRs in real data). This stresses the need for a quantitative measure of the difference between the truth and the estimated function(s) under investigation.
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