Simple Approach to Calculate Random Effects Model Tolerance Intervals to Set Release and Shelf-life Specification Limits of Pharmaceutical Products

Abstract

Tolerance intervals are used to statistically derive acceptance limits that drugs must conform to upon manufacture (release) and throughout shelf-life. The single measurement per lot in release data and repeated measurements per lot longitudinally for stability data have to be considered in the calculation. Methods for the one-way random effects model by Hoffman and Kringle (2005) [HK] for two-sided intervals and Hoffman (2010) [H] for one-sided limits are extended to a random intercepts, fixed slope model in this paper. The performance of HK and H were evaluated via simulation by varying the following factors: i) magnitude of stability trend over time, ii) sample size, iii) percentage of lot-to-lot contribution to total variation, iv) targeted proportion, and v) data inclusion. The performance metrics are average width (for two-sided) or average limit (for one-sided) and attained confidence level. HK and H maintained nominal confidence levels as originally developed, but H is too conservative (i.e., achieved confidence level exceeds the nominal level) in some situations. The HK method adapted for an attribute that changes over time performed comparably to the more computationally intensive generalized pivotal quantity (GPQ) and Bayesian posterior predictive (BayesPP) methods. Mathematical formulas and example calculations as implemented using R statistical software functions are provided to assist practitioners in implementing the methods. The calculations for the proposed approach can also be easily performed in a spreadsheet given basic regression output from a statistical software package. Microsoft Excel spreadsheets are available from the authors upon request.