To overcome this limit of our new model (3), we propose to introduce a time-dependent log HR, b j t), between event-specific hazards.
In our new flexible model (4), the introduction of a time-dependent log HRs b k(t) between event-specific hazards offers the advantage of relaxing the assumption of a common pattern and of estimating jointly the baseline hazards relative to all types of events.
In the flexible model (4), the log of the baseline hazard of event 1, log λ1 t)), and the time-dependent log HRs between event-specific hazard, b j t), are modelled by cubic regression splines, each spline having one knot located at 1 year.
In this simulation the Cox model with the correct specification of the time dependent effect of Z, that is log(t), is used as benchmark estimation.
The Cox proportional hazard assumption was tested by including the interactions of predictors and log survival time as time dependent covariates in the model.
When there are multiple time-dependent coefficients, interpreting the time dependent hazard ratios can be difficult in the log cumulative hazard framework of the Royston-Parmar models [ 20].
Alternatives, including modelling on the log excess hazard scale, may offer more interpretable options when time dependent coefficients are present [ 41].
The proportional hazards assumption was assessed using the log(−log(survival)) plot, and those that failed the assumption or that were deemed to be time dependent were entered as continuous time-dependent covariates (Bradburn et al, 2003).
Trip time dependent on tides.
(a) concentration dependent (b) time dependent.
Plots of time dependent changes in log[s] are now presented in conjunction with the linear regression coefficient and error estimates.
