On Issues in the Estimation of Quantiles in the Presence of Random Effects with Applications to Shelf Life Estimation
A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Statistics, Under the Supervision of Professor Walter W. Stroup. Lincoln, Nebraska: June, 2010
Copyright 2010 Michelle Quinlan
Shelf life is defined as the length of time a pharmaceutical product is expected to remain within approved specifications, provided it is stored under specified conditions (ICH Q1A (R2)). The goal of shelf life estimation is to determine the storage time during which the entire product meets specification with an acceptably high probability. The estimated shelf life should also be “applicable to all future batches” (ICH Q1E, p. 1).
There is compelling evidence of serious issues with the ICH guidelines for shelf life estimation. These issues include: treating batch effects as fixed; addressing batch-to-batch variability via tests for poolability; and, using a confidence interval for the mean to estimate shelf life. A fixed effects model does not allow inference to be made to future batches. Sampling more batches increases the likelihood batches cannot be pooled, resulting in a shorter shelf life and providing a disincentive to sample more batches.
Two main conclusions from evaluating the performance of ICH are: (1) batch effects should be random; and, (2) focus should be on a quantile of the distribution. The linear mixed model treats batch effects as random, but does not focus on a quantile. A mixed model tolerance interval treats batch effects as random and focuses on a quantile, but models the mean instead of a quantile directly. Mixed model quantile regression (MMQR) is an emerging area of research that combines the mixed model with quantile regression to model a quantile directly.
The focus of shelf life estimation should be on a quantile of the distribution of batch shelf lives, where a batch shelf life is the time the batch specific regression line intersects the acceptance limit. Exploiting the relationship between quantiles of the distribution of batch shelf lives on the x-axis and batch means over time, MMQR can be used to estimate a suitably small quantile of the distribution of batch shelf lives. The MMQR estimated shelf life addresses the ICH objective of estimating the minimum batch shelf life and is applicable to future batches. The theory and methodology for estimating shelf life using MMQR are developed in this dissertation.