Constructing acceptance limits
The LBOUND calculation derived in the previous section can be used to develop acceptance limits. This is done by first constructing
a simultaneous confidence region for μ and σ from the data. If a 90% confidence region is constructed for μ and σ, and the
entire region is below the 95% LBOUND, then at least 95% of the samples tested would pass the USP test with 90% confidence.
Construction of the confidence region depends on the sampling plan used to collect the samples. There are two sampling plans
that are generally used when testing blends or final product. In the first plan (Sampling Plan 1), a single test result is
obtained from each location sampled. For example, in a blending step, a single test result would be obtained from each of
a number of different locations within the blender. In a drum, a single test result might be obtained from the different locations
within the drum or from each of a number of different drums. For final tablets, a single tablet may be tested from various
time points throughout the tableting run. In the second plan (Sampling Plan 2), more than one test result is obtained from
each of the sampled locations. For example, during the tableting operation, if a cup is placed under the tablet press at specific
time points during the tableting run, several of the tablets from each cup sample would be tested for content uniformity.
Sampling Plan 2 allows for the estimation of between-location and within-location variability.
For Sampling Plan 1, the sample mean and sample standard deviation estimate the population parameters μ and σ. Lindgren gives
a simultaneous confidence region for μ and σ (10). The region and the 95% LBOUND are visible in Figure 2, where ULS is the
upper confidence limit for σ, and Z is a standard normal critical value.
Figure 2. 95% lower bound with 95% simultaneous confidence region for μ and σ.
Once the confidence region is constructed, it must fall completely below the specified LBOUND. One can generate an acceptance-limit
table by finding the largest sample standard deviation for a fixed sample mean, such that the resulting confidence region
remains below the prespecified LBOUND. Note that the only two points to evaluate on the triangle are the two points with
the maximum value of σ.
Table II provides an example of an acceptance-limit table. SAS program version 8.2 was written to generate the acceptance
limits. The acceptance limit table corresponds to a target value of 100% label claim, a sample size of 30, a 95% confidence
region, and a 95% lower bound.
Table II: Acceptance limits for content uniformity.
Suppose that a random sample of 30 tablets is taken from a batch and tested for content uniformity. Suppose the sample mean
is 98.4% label claim with a coefficient of variation (CV) of 3.01%. Since the acceptance limit for the CV is 3.85%, the sample
passes. This means that with 95% confidence, any set of tablets taken from the batch has at least a 95% probability of passing
the USP test.
For Sampling Plan 2, the variance of a single observation is the sum of the between-location and within-location variances.
The standard deviation of a single observation, σ, is estimated by calculating the square root of the sum of the between-
and within-location variance components. Graybill and Wang give a confidence region for σ (11).
= mean square between locations from one-way analysis of variance (ANOVA)
= mean square within locations from one-way ANOVA
L = number of locations