Modelling time-varying effects in Cox model under order restrictions

The violation of the proportional hazards assumption in Cox model
occurs quite often in studies concerning solid tumours or
leukaemia. Then the time varying coefficients model is its most
popular extension used. The function f(t) that measures the time
variation of a covariate, can be assessed through several smoothing
techniques, such as cubic splines. However, for practical propose,
it is more convenient to assess f(t) by a step function. The main
drawback of this approach is the lack of stability since there is
no standard method of defining the cutpoints of the underlined step
function. The variation in the effect of a predictor can be assumed
to be monotonic during the observational period. In these cases, we
propose a method to estimate f(t) based on the isotonic regression
framework. Applying the idea of Grambsch and Therneau, where
smoothing the Schoenfeld residuals plotted against time reveal the
shape of the underlined f(t) function, we use the Pooled Adjacent
Violators Algorithm as smoother. As a result a set of cutpoints is
returned without any a priori information about their location.
Subsequently, the corresponding step function is introduced in the
model and the standard likelihood-based method is applied to
estimate it while adjusting for other covariates. This approach
presents the advantage that additional decisions that can effect
the result, as the number of knots in cubic splines, do not need to
be taken. The performance of the provided PH test and the stability
of the method are explored in a simulation study.