Partition - copyedits

[RaphaelS1]

Did more of a high-level check with this one. It's been reviewed enough times and you're the expert!!

Just two things I wasn't sure about:

1. IID

Can we really say

\Delta_{ij} \stackrel{iid}{\sim} \text{Bernoulli}(\pi_j), \qquad j = 1, \ldots, J_i,
\Delta_{ij} \mid \xx_{i}, t_j \stackrel{iid}{\sim} \text{Bernoulli}(\pi_{ij}),

etc.

Aren't they based on the original data so not truly independent? Got a bit confused.

2. Baseline hazards etc.

In 'representation of time within intervals' you have

• When $t_j$ is treated as a factor variable, the specific choice of $t_j$ (midpoint, endpoint, etc.) is irrelevant. Each interval receives its own parameter (for example, an interval-specific intercept $\beta_{0j}$), as in the examples in @sec-discrete-time-reduction and @sec-piecewise-constant-hazards. This is the classical PEM approach and provides maximal flexibility, but assumes that the hazard in each interval is estimated independently of neighboring intervals. Without penalization, this can lead to implausible hazard estimates, especially when some intervals contain few events. The number of baseline hazard parameters equals $J$, which becomes prohibitive when $J$ is large (for example, when using all unique event times as cut points).
• When $t_j$ enters as a continuous variable, the specific choice of $t_j$ is generally of minor importance, except possibly for the final interval and depending on the interval convention. In this setting, a functional relationship between time and the (log-)hazard is assumed. A linear relationship $\log(h_j) = \beta_0 + \beta_1 t_j$ implies that the log-hazard changes at a constant rate over time, which is often too restrictive in practice. Instead, the relationship could be modeled flexibly, for example as $\log(h_j) = f(t_j)$ using penalized splines [@Bender2018Penalized; @Kopper2022] or tree-based methods [@Bender2021]. In this case, the number of baseline hazard parameters depends on the complexity of $f$, not the number of intervals, making the approach scalable to fine partitions. Penalization encourages hazard estimates in neighboring intervals to be similar unless the data provides strong evidence for rapid changes.

This is quite technical quite quickly and in particular starts talking about regularization, penalized splines, etc. etc. And we haven't really talked about the baseline hazard much in the chapter so it kind of through me off a bit. Happy for you to either add one line to explain or just ignore me

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