Bayesian Power and Error Estimation in Survival Trials Using BayesPPDSurv | HackerNoon

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Abstract and 1 Introduction: BayesPPDSurv

2 Theoretical Framework

2.1 The Power Prior and the Normalized Power Prior

2.2 The Piecewise Constant Hazard Proportional Hazards (PWCH-PH) Model

2.3 Power Prior for the PWCH-PH Model

2.4 Implementing the Normalized Power Prior for the PWCH-PH Model

2.5 Bayesian Sample Size Determination

2.6 Data Simulation for the PWCH-PH Model

3 Using BayesPPDSurv

3.1 Sampling Priors

4 Case Study: Melanoma Clinical Trial Design

5 Discussion and References

4 Case Study: Melanoma Clinical Trial Design

We consider the high-risk melanoma trial design application in Psioda and Ibrahim (2019), and demonstrate how BayesPPDSurv can be used for coefficient estimation as well as power and type I error rate calculations for time-to-event data in Bayesian clinical trial designs that incorporate historical information.

Interferon Alpha-2b (IFN) is an adjuvant chemotherapy for deep primary or regionally metastatic melanoma. The E1684 trial (Kirkwood et al., 1996) and the E1690 trial (Kirkwood et al., 2000) were randomized controlled trials conducted to evaluate the efficacy of IFN for melanoma following surgery. The studies classified the subjects into four disease stage groups. Following Psioda and Ibrahim (2019), we restrict our attention to patients in disease stage four, i.e., regional lymph node recurrence at any interval after appropriate surgery for primary melanoma of any depth. The primary outcome is relapse free survival. The number of positive lymph nodes at lymphadenectomy is used as a stratification variable (≤ 2 vs. ≥ 3) due to its prognostic value. We compare patients who received the IFN treatment to those who received observation (OBS). Table 1 summarizes the total number of events and the total risk time by treatment group and number of positive lymph nodes for the two studies.

Table 2 displays the posterior mean, standard deviation and 95% credible interval for β and elements of λ. There is weak evidence suggesting a negative association between IFN and time-to-relapse.

Next, our goal is to design a new trial incorporating the E1684 study using the power prior and the normalized power prior. We first specify the characteristics of trial data simulation. Let ν be the number of events at which the trial will stop and let n be the total number of subjects enrolled. For each ν, we take n = 3ν. We assume a subject’s enrollment time follows a uniform distribution over a 4-year period. We allocate 50% of the subjects to the treatment group. For stratum allocation, we sample from the stratum indices of the historical data with replacement. We assume there is only administrative censoring which occurs when ν events have accrued. In the data generation phase, we use the default change points which are determined so that an approximately equal number of events are observed in each time interval for the historical dataset. When analyzing the generated data, we use the default change points which are determined so that an approximately equal number of events are observed in each time interval for the pooled current and historical datasets. The baseline hazard parameters are not shared between the current and historical data. The same set of default priors are used for β, λ and λ0 as before.

We compute the Bayesian power and type I error rate for a few sample sizes for tests of the hypotheses

H0 : β ≥ 0

and

H1 : β < 0.

Now we are ready compute the power and type I error rate using the power prior via the function power.phm.fixed.a0(). The following code computes the power for ν = 350 and a0 = 0.6 using the default alternative sampling prior.

Table 3 displays the power and type I error rates for ν = 350 and ν = 710 for a0 values of 0, 0.2 and 0.6 using the default sampling priors and the point-mass sampling priors. We obtain 50,000 posterior samples using the package’s custom slice sampler after 200 burn-ins for each of the 10,000 simulated datasets. The results in Table 3 are comparable to the results in Figure 2 in Psioda and Ibrahim (2019) (i.e., the same up to Monte Carlo error). We observe that the power increases with a0 and sample size as expected. We can also see that the default sampling priors yield average rates that are often lower than rates based on point-mass priors.

5 Discussion

BayesPPDSurv facilitates Bayesian power and type I error rate calculations using the power and normalized power prior for time-to-event outcomes using a PWCH-PH model. We implement a flexible stratified version of the model, where the historical data can be used to inform the treatment effect, the effect of other covariates in the regression model, as well as the baseline hazard parameters. We develop a novel algorithm for approximating the normalized power prior that eliminates the need to compute the normalizing constant. The package also has features that semi-automatically generate the sampling priors from the historical data.

Future versions of the package will accommodate cure rate models. Another possible feature is the computation of optimal hyperparameters for the beta prior on a0 to ensure that the normalized power prior adapts in a desirable way to prior-data conflict or prior-data agreement, based on the work of Shen et al. (2024).

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