Statistical considerations for using tolerance interval to set product specification for normally distributed...

Conventionally, the product quality specification and control chart limits are determined as the mean plus and minus 3 sample standard deviations with the assumption that the quality data is normally distributed. These limits correspond to an interval centered at the mean, covering approximately 97.3% of the population. The estimate of such an interval is called the $\beta$-content tolerance interval. It has been proposed to use a two one-sided $\beta$-content tolerance interval approach for determining drug product quality specifications. For a given confidence level, $1-\alpha ,$ and a coverage percentage p, the $\beta$-content tolerance interval is not precise when the sample size is small. For the derivation of a precise $\beta$-content tolerance interval, Faulkenberry and Daly proposed a "goodness" criterion for sample size determination. In order to avoid overestimating the $\beta$-content tolerance interval when p is large, we propose to define the precision requirement as the probability of the tolerance interval covering more than $\frac{\left(1+p\right)}{2}$ is restricted to a pre-specified significance level ${\alpha }^{\text{'}}$. Quality specification studies are often not planned with proper sample sizes. To obtain precise $\beta$-content tolerance intervals for quality specification studies, the proper coverage p satisfying the "goodness" criterion and the minimum sample sizes were also determined with the pre-specified significance level ${\alpha }^{\text{'}}$. With this approach, one may properly set the product specificationwhile avoiding over-specifying the quality limits.