Purpose: The beta distribution was recently proposed as an appropriate statistical distribution for modelling gamma index passing rates (GIPRs) computed during patient-specific quality assurance (PSQA). As such, it has been used to infer control limits, which improve the robustness of PSQA by defining thresholds beyond which abnormal GIPRs are automatically identified. This study presents a more appropriate statistical distribution for GIPRs which produces control limits with better theoretical justification.
Methods: Our presented distribution is a modified beta distribution, derived from the beta distribution with an additional nonzero probability at 100% GIPR. To represent this probability, the distribution includes a separate parameter equal to the fraction of fitting data points at 100% GIPR. This parameter accounts for the cluster of plans that fall well within the gamma index’s pass thresholds and achieve exactly 100% GIPR. The beta distribution assumes that these plans are infinitesimally unlikely to exist and models them incorrectly. The data points between 0% and 100% GIPR are then used to fit a separate beta distribution. The modified beta distribution was compared to the beta distribution according to the control limits produced by each distribution. These control limits were generated assuming a false positive rate of 1/370 from two years of individual plan field GIPR data from PSQA of volumetric modulated arc therapy.
Results: The control limits derived from the patient GIPR data were 92.8% and 93.6% for the beta distribution and modified beta distribution, respectively. This difference is comparable to the differences between outliers in the data and it can impact abnormal plan detection.
Conclusion: The modified beta distribution produces control limits which are meaningfully different from the control limits produced by the original beta distribution. Given its stronger theoretical justification, the modified beta distribution should be used over the beta distribution to improve the robustness of PSQA.