Here, an exploration of the effects of an informative prior on the baseline hazard func- tion is carried out with specific application to the GemCap trial.
Data to be analysed are taken from the Cunningham trial (GemCap) [186], a trial to investigate the use of Gemcitabine and Capecitabine for the treatment of patients with advanced pancreatic cancer. The trial recruited a total of 534 patients. The final analysis, although showing some survival benefit for the Gemcitabine and Capecitabine arm (P-value = 0.08).
Aside from the GemCap trial, two further trials are available: Herman et al.[187]
and Van Cutsem et al. [188], with total sample sizes of 319 and 301 patients respec- tively, both of which contain a Gemcitabine arm and are used to estimate baseline hazard priors. Table 1 provides summaries of the prior information in terms of sur- vival estimated at given time-points taken from the Herman and Van Cutsem trials along with arithmetic means of the two,{„p} which shall be use to derive prior point estimates.
Time (Months) 1 3 6 9 12 15 18 21
Hermann (2005) 0.98 0.84 0.63 0.40 0.30 0.19 0.10 0.06 Van Cutsem (2009) 0.96 0.78 0.50 0.36 0.21 0.18 0.15 0.15
„p 0.97 0.80 0.55 0.37 0.25 0.18 0.12 0.10 Table 7.1: Derivation of prior survival estimates at given time points.
Using a fixed time grid given by a = (0,3,6,12,18, tú) where tú the maximum observed time, baseline hazard point estimates, È, are obtained using the methods outlined in Section 7.2. Here estimates ofÈ= (≠2.60,≠2.08,≠2.03,≠2.10,≠2.59) are
obtained.
Figure 7.2 shows the Kaplan Meier survival estimates of each treatment arm of the GemCap trial along with the fixed time-grid and the survival function derived from the prior point estimates. Of main interest here is to compare the prior survival function against the estimates obtained from the Gemcitabine arm of the GemCap trial. Figure 7.2 shows a general good level of agreement although the survival estimates in the Gemcitabine arm are slightly below that of the estimates obtained from the prior data.
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Time (Months)
Overall Survival
0 6 12 18 24
Gem.
Gem Cap Prior est.
Figure 7.2: Kaplan Meier estimates from the GemCap data along with an estimate of the survival function obtained from the point estimates of the prior distributions
Some consideration is now given to the level of variability that is attributed to the prior point estimates. Here the effective events approach as defined in Section 7.3.2 is followed. In practice, multiple priors which give varying degrees of belief, reflected in differing total number of effective events may be set to reflect a reference, sceptical and optimistic approach suggested by Speigelhalter et al. [21].
It is noted that an attraction of the MAP approach is to account for uncertainty both within and between studies wheres the pooled baseline estimates effectively ignore any between study variation. With reference to the GemCap study therefore, whilst the
prior information is reasonably consistent, there may still be a desire to use relatively vague priors to ensure that the data remain dominant in the interpretation of the trial.
Despite the prior data being derived from approximately 300 events therefore, effective event sizes of 10, 25, 50 and 100 events are investigated. Figure 7.3 provides a graphical illustration of the survival functions that are obtained from these effective numbers of events.
Figure 7.3: Illustration of the survival functions obtained from informative baseline hazard priors
For the purposes of this analysis, ’ = 50 effective events are chosen to define the prior distributions, being a scenario which will well accentuate the effect of the in- formative priors without being overly influential. The data from the GemCap trial are re-analysed using a PEM. Two models are applied, a ‘reference’ model with vague uninformative priors and a model incorporating an informative baseline hazard prior.
Figure 7.4 shows the prior densities for all parameters along with the posterior densities for the reference model and the model with informative priors.
The effects of the informative priors is shown to reduce the posterior estimate of each
log baseline hazard parameter which is as expected as the prior estimate of the survival function is consistently higher than the Kaplan Meier estimate for the Gemcitabine arm as shown in Figure 7.2. Each baseline hazard parameter is also estimated with a greater degree of precision. The log hazard ratio also has a point estimate which is closer to zero, again as expected as Figure 7.2 illustrates that the prior information will ‘drag’ the survival curve for the control arm closer to the experimental arm. More importantly, the precision of the estimate of the log baseline hazard has increased, without any prior information being included on this parameter.
The full set of parameter estimates from each model are given in Table 7.2. Here there are some results which may be slightly non-intuitive. As an example, the pos- terior distribution for È4 with an informative prior has mean ≠2.25 which may be unintuitive given the reasonably good agreement between the prior distribution and the data respectively (≠2.10 and≠2.12). Upon further inspection, it is clear that there is some correlation between the baseline hazard parameters and that all baseline hazard parameters must be considered collectively. It may also be noted that an informative baseline hazard function has the effect of smoothing the baseline hazard function. This can be shown graphically.
Lastly, the effect on the log hazard rate is again noted with some shrinkage towards zero, in agreement with Figure 7.4. Also associated with the shrinkage is an increase in the parameter precision as the standard deviation decrease from 0.09 for the reference model to 0.06 for the informative model.
Parameter Time-grid È ‘Reference’ Informative (’ = 50) log(⁄1) 3 -2.60 -2.36 (0.17) -2.55 (0.10) log(⁄2) 6 -2.08 -1.91 (0.17) -2.07 (0.11) log(⁄3) 12 -2.03 -1.84 (0.16) -1.99 (0.11) log(⁄4) 18 -2.10 -2.12 (0.20) -2.25 (0.15) log(⁄5) 42 -2.59 -2.33 (0.26) -2.52 (0.21)
— -0.15 (0.09) -0.06 (0.06)
Table 7.2: Results of applying informative baseline hazard priors to the analysis of GemCap
Aside from the results of parameters in Table 7.2, Figure 7.5 shows the resulting survival functions that are obtained from the reference and informative models. In each case, the Kaplan Meier survival estimates for the Gemcitabine arm are included for reference. These show explicitly how the survival function is increased and is estimated with a larger precision compared to the reference model. Note that the Kaplan Meier estimates do not agree entirely with the survival estimates obtained in the reference model, this is as the model estimates are also influenced by the experimental arm as well as the control arm and that some evidence of non-proportionality can distort this estimate.
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Figure 7.4: Illustration of prior and posterior densities for a selection of parameters for the analysis of GemCap data
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Overall Survival
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Figure 7.5: Illustration of fitted survival function for a) vague and b) informative prior distributions