In tree-based classification, how is including survival time as an ordinal covariate more flexible than including an unrestricted baseline? I am reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016) section 6.3 Recursive Partitioning with Binary Outcome. It presents specification of the data structure for fitting tree-based models. Survival times $T$ are encoded as

*

*$(y_{i,1},\dots,y_{i,t_i-1},y_{i,t_i})=(\underbrace{0,\dots,0,1}_{t_i})$ if $T_i=t_i$ is observed and

*$(y_{i,1},\dots,y_{i,c_i-1},y_{i,c_i})=(\underbrace{0,\dots,0,0}_{c_i})$ if the event of interest does not occur before censoring takes place at $c_i$ (we assume censoring occurs at the end of the interval).

Among the regressors corresponding to observation $i$, one includes a column $(1,\dots,t_i)^\top$ for non-censored data and $(1,\dots,c_i)^\top$ for censored data. At the top of p. 135 (and the last sentence of the picture below) it says

This data structure, which implies that $T$ is treated as an ordinal covariate during tree construction, represents a much more ﬂexible form than the specification of the time trend $\gamma_{0t}$ in a discrete hazard model.

The latter specification corresponds to including a dummy for each time period from 1 to the last time period in the sample, $q$. (See the picture below for a comparison between these two data structures.) Question: Why is the former supposed to be much more flexible?


 A: The extra flexibility is with respect to how time is incorporated into the tree-based survival model. Is time only used to evaluate proposed splits based on (other) covariates? Or do you include time itself as a covariate, to allow the associations of other covariates with outcome to change over time? The latter is the "much more flexible form."
In the tree-based models of Chapter 6, the distinction is whether time can be used explicitly to propose a split at a node.
In the "discrete hazard model" of Section 6.2, time is not used that way; it only appears in the log-likelihood calculations after a proposed split based on (other) covariates. Associations of covariates with survival thus are effectively independent of time.
The data structure explained in Section 6.3 adds time explicitly as a covariate that can be used for splitting. Splitting at a node based on time effectively sets up an interaction between time and other covariates, allowing associations of covariates with outcome to vary with time.
Similarity to Cox models
The above distinction is similar to that between a standard Cox proportional hazards model and a Cox model with time-varying regression coefficients.
In the standard Cox model, time values are used to estimate the baseline hazard function very flexibly, event time by event time, after the model is fit. The associations of covariates with outcome are assumed independent of time.
Time-varying coefficients in a Cox regression are modeled as interactions of covariate values with time. The binary splits based on time at nodes, in the tree models of Section 6.3 of Tutz and Schmid, might be considered an extension of the step-function-in-time coefficient modeling described in Section 4.1 of the R survival vignette on time dependence.
That said, be wary of the danger of survivorship bias when modeling time-varying coefficients or covariates, whether in discrete or continuous time.
