LDT estimation for the hUSP zero-tolerance criterion
Eq. 2 is used to estimate the LDT of the nonparametric zero-tolerance (ZT) criterion in Stage 2 of the hUSP test. Assuming
that a batch true mean is at the target potency of 100% LC, the ZT criterion requires all tested units to be within the range
of 75–125% LC. If Q represents the true 75–125% LC coverage of the batch, the probability of acceptance (Eq. 1) is expressed in the following
The above equation may be inverted to solve for Q, thus yielding the following result:
Substituting the above equation into Eq. 2 and solving for the limit yields the following result:
Thus, the hUSP ZT criterion inherently would require a 75–125% LC coverage of 100%, which, as argued above, cannot be attained
with existing technologies. The ZT criterion is useful as a failsafe in the current hUSP test to protect against an accidental
serious failure that is not otherwise detected because of the small sample size. However, the ZT criterion in large-sample-size
UDU tests may be unnecessarily stringent and dissuade personnel from performing testing.
LDT estimation for the hUSP test without the ZT criterion
The hUSP test does not explicitly define the inherent quality requirement. Because the test only applies to a small sample
size, it will exhibit nonzero Type I and Type II errors. The test effectively establishes an acceptable standard for the Type
I error probability that is determined by the Stage 2 criteria. Therefore, the LDT for the hUSP test without the ZT requirement,
referred to as hUSP (–ZT) can be determined solely based on the Stage 2 acceptance-value requirement. This LDT can serve as
a benchmark of the quality level that is expected in production batches.
Assuming a target potency (T) of 100% LC, the Stage 2 AV requirement can be expressed by the following equation:
Figure 1 contains definitions of L, M, S, and X.
The batch means above 100% LC are the mirror images of the ones below 100%, and thus are not considered in Eq. 7. The limiting
coverage at large sample size is expressed in the following equation:
in which N(x|μ,σLDT) is the normal probability density function. The LDT for a normal population distribution (Eq. 8) is shown in Figure 3 as
a function of population mean.
Figure 3: Limiting discrimination threshold (LDT) coverage of the hUSP(-ZT) test as a function of batch means, assuming a
normal population distribution and target assay of 100% label claim (LC).
The LDTs are 95.45% and 95.96% for batch means of 100% LC and 96% LC, respectively. The dip in LDT curve at batch mean of
98.5% LC results from the indifference zone of the hUSP test. Clearly, the most stringent coverage requirement of about 96%
occurs at a batch mean near 96% LC. This requirement is more conservative than that of Sandell (i.e., 95.2%), but less conservative
than that of Bergum and Vukovinsky (i.e., 97%).