Assessing the Uncertainty in QUANTEC's Dose–Response Relation of Lung and Spinal Cord With a Bootstrap Analysis
Purpose: To apply a statistical bootstrap analysis to assess the uncertainty in the dose–response relation for the endpoints pneumonitis and myelopathy reported in the QUANTEC review. Methods and Materials: The bootstrap method assesses the uncertainty of the estimated population-based dose-response relation due to sample variability, which reflects the uncertainty due to limited numbers of patients in the studies. A large number of bootstrap replicates of the original incidence data were produced by random sampling with replacement. The analysis requires only the dose, the number of patients, and the number of occurrences of the studied endpoint, for each study. Two dose–response models, a Poisson-based model and the Lyman model, were fitted to each bootstrap replicate using maximum likelihood. Results: The bootstrap analysis generates a family of curves representing the range of plausible dose–response relations, and the 95% bootstrap confidence intervals give an estimated upper and lower toxicity risk. The curve families for the 2 dose–response models overlap for doses included in the studies at hand but diverge beyond that, with the Lyman model suggesting a steeper slope. The resulting distributions of the model parameters indicate correlation and non-Gaussian distribution. For both data sets, the likelihood of the observed data was higher for the Lyman model in >90% of the bootstrap replicates. Conclusions: The bootstrap method provides a statistical analysis of the uncertainty in the estimated dose–response relation for myelopathy and pneumonitis. It suggests likely values of model parameter values, their confidence intervals, and how they interrelate for each model. Finally, it can be used to evaluate to what extent data supports one model over another. For both data sets considered here, the Lyman model was preferred over the Poisson-based model.