Large sample size UDU test proposals
The hUSP test is designed to accommodate a small sample size (i.e., as many as 30 units), but PAT-associated on-line methods
permit a much larger sample size from each commercial batch. An alternative "large n" UDU test is thus needed. Several proposals have appeared in recent years, and a brief review is provided below.
Sandell et al. proposed a nonparametric counting test for sample sizes between 100 and 10,000 units (2). The maximum number
of defect units, whose content exceeds the range of 85–115% LC, is equal to the median of the binomial distribution defined
by a sample size (n), and a defect probability of 1 – 0.952 = 0.048. The authors demonstrated that the test has lower acceptance probability
than that of the hUSP test as long as the probability is at or below 50%. However, the choice of the binomial median and defect
probability of 0.048 is somewhat arbitrary and remains a point of controversy.
Bergum and Vukovinsky proposal.
Bergum and Vukovinsky suggested a nonparametric test with a defect-count limit equal to the largest integer less than or
equal to 3% of total sample size (3). OC curves are used to make various comparisons with the Sandell and hUSP tests. However,
as with Sandell's proposal, the choice of 3% seems somewhat arbitrary.
Diener et al. proposed several parametric alternatives for sample sizes in the range of 31 to 99 (4). As they did with Sandell's
approach, the authors demonstrated with OC curves that their tests were more stringent for batches with at most 95.2% coverage.
But, again, it is not clear that the inherent quality levels that the tests represent are widely accepted.
The key to any large sample UDU test is to define the underlying quality threshold that distinguishes an acceptable batch.
Each of the above proposals makes assumptions about the acceptable coverage or defect rate. For example, both Sandell's and
Diener's approaches chose 95.2% coverage as the quality requirement, while the Bergum and Vukovinsky proposal led (in the
large sample limit) to a coverage of 97% as the quality requirement. The quality requirement of a large-sample UDU test should
ensure good product quality and should be at least as stringent as the quality requirement of the hUSP test. OC curves are
often used to evaluate large-sample UDU tests, but they depend on test characteristics (e.g., sample size) and parameters
of the population distribution, thus requiring the generation and unwieldy comparison of a large number of OC curves. It would
seem desirable to summarize the performance of any proposed large-sample UDU test using a single quality criterion.
Limiting discriminatory threshold
Formally, a limiting discriminatory threshold (LDT) is defined as the large sample limit of the inverse of the probability
of acceptance function in Eq. 1. The concept of LDT for a large-sample UDU test is motivated by the following ideas and assumptions:
1. The key critical quality parameter of content uniformity is coverage.
2. A coverage threshold of less than 100% that unambiguously distinguishes between acceptable and unacceptable batches should
be agreed upon.
3. UDU tests exhibit both Type I (i.e., rejection of acceptable batches) and Type II (i.e., acceptance of unacceptable batches)
errors. As sample sizes increase, the test becomes more discriminatory. In the ideal limit of infinite sample size (i.e.,
the true coverage of each lot is known with absolute certainty), the error probabilities are zero, and the OC curve is a step
function. In this article, the coverage at which this transition occurs is called the LDT, defined in the following equation
as the converging limit with increasing sample size:
4. The LDT of a given UDU test represents its inherent quality-discriminating threshold in the ideal limit of complete knowledge
(i.e., infinite sample size) and can be determined algebraically or by computer simulation.
5. To be considered equally stringent, UDU tests, at a minimum, should have the same LDT. Tests with the same LDT may approach
the LDT at different rates as sample size increases. When comparing two large-sample UDU tests that have the same LDT, the
test that approaches the LDT more rapidly can be considered more efficient.
An LDT of 100%, while certainly desirable, is not realistic because of inherent analytical uncertainty. The LDT and the rate
of approach to the LDT will depend not only on the test itself, but, in some cases (e.g., parametric tests), also on the distribution
of the population from which samples are drawn.
The LDT concept may not be limited to single coverage, but is applied to the 85–115% LC coverage here for illustration. In
principle, the concept could be extended to multivariate quality metrics (e.g., joint coverage of 75–125% and 85–115% LC ranges),
although that approach is not considered here.
It is legitimate and useful to determine the LDT for compendial tests (e.g., hUSP test) that employ fixed sample sizes. Such
tests rarely define a quality requirement (i.e., the required coverage to pass the test) explicitly. When a definitive quality
metric, such as coverage, can be identified, LDT provides a reasonable way to reverse engineer the intended standard of quality.
Multistage tests are often designed such that the acceptability range is widened as the fixed sample size (i.e., stage) increases
because the estimation of the population information has improved. This design reduces the risk of the Type I error and maintains
a nominal Type II error rate. Ultimately, the criteria applied at the final stage set the standard for expected quality and
decision error rates. Consequently, when evaluating the LDT of fixed-sample-size tests, the acceptance criteria of the final
stage should be kept constant with changing sample size. If the content of all individual units were known, the coverage would
be known exactly and could be compared with the LDT that is inherent to the final stage of the test. This procedure effectively
assumes that batch acceptance would be based on the perfect knowledge of the batch uniformity, had it been available.