Tree-based modeling of time-varying coefficients in discrete time-to-event models

Abstract

Hazard models are popular tools for the modeling of discrete time-to-event data. In particular two approaches for modeling time dependent effects are in common use. The more traditional one assumes a linear predictor with effects of explanatory variables being constant over time. The more flexible approach uses the class of semiparametric models that allow the effects of the explanatory variables to vary smoothly over time. The approach considered here is in between these modeling strategies. It assumes that the effects of the explanatory variables are piecewise constant. It allows, in particular, to evaluate at which time points the effect strength changes and is able to approximate quite complex variations of the change of effects in a simple way. A tree-based method is proposed for modeling the piecewise constant time-varying coefficients, which is embedded into the framework of varying-coefficient models. One important feature of the approach is that it automatically selects the relevant explanatory variables and no separate variable selection procedure is needed. The properties of the method are investigated in several simulation studies and its usefulness is demonstrated by considering two real-world applications.

Appendix 1: Augmented data matrices of the TSVC model given in Eq. (14)

For an individual whose event was observed ($\varDelta _i=1$) at time $\tilde{T}_i$ the augmented data matrix after a split in $x_j$ at split point $t^*_{j1}$ is given by

(17)

For an individual that is censored $(\varDelta _i=0)$ at time $\tilde{T}_i$ the augmented data matrix after a split in $x_j$ at split point $t^*_{j1}$ is given by

(18)

The matrices (17) and (18) contain two columns associated with the jth explanatory variable including the values $\varvec{x}_{ij}^\top \,I(t\le t^*_{j1})$ and $\varvec{x}_{ij}^\top \,I(t>t^*_{j1})$.