there is no recommended method for how the time-grid should be set. Some discussion on the matter has been provided by Sahu et al. [132] and a review is provided by Han et al. [144].
Specific methods for use in a Bayesian framework have been proposed by Demarquie et al. [145, 146], whilst Goodman et al. [48] propose an approach for selecting change points in a piecewise model based on maximum likelihood approaches. Han et al. take a different approach, starting with a saturated model, where each unique event results in a new partition in the time grid and reducing the piecewise model to only include partitions of import. Each of these approaches may not be appropriate for all situations however and generally concentrate on the accurate estimation of a single cumulative hazard function. Interest in this chapter is primarily on the hazard ratio, with the hazard function being considered a set of nuisance parameters. In this context, little discussion is offered on the choice of the correct time-grid in the literature.
Given that the model parameters and the time-grid can not be considered to be independent of one-another, the need is highlighted for further investigation. In this section, a review of some of the techniques used in practice are presented as well as proposing some further methods. In the section that follows this is applied to a simu- lation study with the aim of providing recommendations for future use. Some popular methods from the literature are investigated although the method of Han et al. [144] is not considered as some preliminary exploration shows that initial investigations require the comparison of pairwise disjoint single event intervals. Evaluations on such intervals can sometime be unreliable resulting in time-grids that do not always give an appro- priate fit to the hazard function. Likewise, the method as proposed by Goodman et al. [48] is altered slightly into the ‘split-likelhood’ method presented. This still produces time-grids in a forward step-wise fashion based on maximum likelihood methods but is less stringent than the Wald test approach suggested by Goodman et al [48].
It is assumed that the choice of time-grid is to provide as much flexibility as is required to describe the behaviour of baseline hazard function. Simultaneously, it is desirable to avoid wasting information by allowing too many partitions, this is especially true in a Bayesian framework where including extra parameters may greatly increase the computational burden.
5.4.1 Fixed time grid (Kalb.)
Kalbfleisch [57] proposed that a time-grid is set for the analysis before any of the data are observed. Whilst it requires some underlying knowledge of the data, it may be ad- vantageous as it ensures a level of objectivity is imparted into any analysis. Conversely
there is also some risk of having empty partitions or a time-grid which provides an unsatisfactory fit to the data. In practice, a fixed time-grid may be used by observ- ing an appropriate parametric survival or cumulative hazard functions and choosing reasonable points, or by setting time points based on convenient landmarks.
5.4.2 Fixed number of events (n.event)
Under this strategy, the user sets the number of events that are observed in each partition. The time-grid is then fully dependent on the observed data. This has the advantage that ‘thinner’ partitions will be observed in ‘busier’ areas of the distribution. However, it may cause problems as fixing a small number of events for a large dataset will result in an unnecessary large number of partitions.
5.4.3 Fixed number of intervals (n.part)
An alternative approach to fixing the number of events is to fix the number of inter- vals that are required, partitions are then set to occur at regular intervals. Here the model dimensionality and computation effort required can be directly managed but the possibility remains that partitions may be placed at unsuitable points.
5.4.4 Paired event partitions (paired)
The motivation behind this approach lies in the situation where a trial is set up to determine the difference between two treatment regimens. Here it may be considered beneficial to have partitions which contain events from each treatment group. Defining the observed event times for the control arm and experimental arm of a trial as tc and te respectively, the time-grid is set by
1. Seta0= 0
2. Definea1 =max(min(tc), min(te)) 3. Redefine t≠a1
e and t≠ca1 wheret≠ca1 are the event times for the control arm with the event included in the first partition removed, and likewise fort≠a1
e 4. Repeat steps 2 and 3 to set further partitions
5. Continue until one or both of tc and te has no event times left and set aJ = max(tc, te)
Whilst it may be advantageous to have an event from each arm to influence the estimation of the baseline hazard function, it can result in wide partitions if there are large differences between the treatment arms or a large number of partitions if the two arms are similar to one another.
5.4.5 Random time-grid (Demarqui)
The method as defined by Demarqui [145, 146] is tailored for use in a Bayesian frame- work. Here the-time grid itself is considered to be a random variable to be estimated. Using this approach, each event time is a candidate to define a partition in the time- grid. Demarqui then proposes the use of the posterior predictive distribution to estimate the time-grid. Whilst appealing, a drawback of this approach is it is only set up at present for models without any covariates. Evaluating posterior predictive distributions when covariates are included is more problematic. A two-step approach is taken here whereby the first step estimates the time-grid based on a model with no covariates. This time-gird is then treated as fixed and applied to the model with covariates.
5.4.6 Split likelihood partitions (split.lik)
In a similar fashion to the method as set out for the random time-grid, a time-grid is obtained by searching for a model that provides the best fit. Here, instead of posterior predictive distributions, models are evaluated based on the log likelihood as applied by the following steps
1. Start with a standard exponential model (i.e. a single hazard parameter)
2. Consider each event time as a possible partition in the time-grid and calculate the likelihood for all possible two parameter hazard models
3. Select the partition that gives the best likelihood providing this is an improvement on the single parameter model
4. Repeat this process until no further improvement can be obtained.
This approach, unlike the random time-grid method can easily incorporate covari- ates. Again a two-step approach whereby the time-grid is compiled in the first step and the model is evaluated in the second step.