Objectives

Fractional polynomials (FPs) are traditionally analysed in a Bayesian framework with WinBugs using a Gibbs sampler. The 2nd-order FPs include the power parameters p1 and p2, usually run for the recommended testing values [−2,−1,−0.5,0,0.5,1,2,3], which result in 36 combinations of p1 and p2. FPs are fit on aggregated events over predefined periods instead of individual patient-level-data (IPD). RSTAN, a novel R-package for Bayesian analyses, was created to overcome converging problems WinBugs has around complex models. We compare traditional Winbugs FPs and novel RSTAN FPs.

Methodology

We developed a novel FP framework in RSTAN, fitting directly on IPD using the Hamiltonian MCMC-sampler, with p1 and p2 optimized by RSTAN instead of using the testing values, requiring 4 FPs covering the 36 combinations. The comparison was based on a PFS dataset with fluctuating hazards over time from a multiple-myeloma trial. Additionally, in RSTAN four new FP-formulations are implemented where combinations of time and log(time) are explored. The fit was compared in terms of ΔMean Absolute Deviation (MAD) of observed vs predicted.

Results

The p1=p2=0.5 Winbugs-model had the lowest ΔMAD with 3.08 and converged within 100,000 iterations (13 minutes). Running all 36 combinations took 7.8 hours. RSTAN-model with the lowest ΔMAD of 1.26 gave (p1=0.2, p2=-0.66) over the four models and required 10,000 iterations (6 minutes and 24 minutes in total). Within RSTAN, the new FP formulations had a better ΔMAD (1.08 vs 1.26) than the standard FP formulations. This formulation is defined as β₀+β₁t^p1+β₂ t^p2ln(t) with p1=1 and p2=1.5.

Conclusion

The novel RSTAN fitted better, converged faster, and required only four formulations instead of 36. The difference in fit might be explained using IPD within RSTAN directly or by more precise powers in RSTAN. Although most papers use WinBugs, we expect our novel FP framework in RSTAN to be the future.