Determining the efficiency of the hUSP (–ZT) test
Monte–Carlo simulation was used to determine the approach to LDT for the hUSP (–ZT) test as sample size increases. In these
simulations, half of the total units (n) were tested at Stage 1 and the other half at stage 2, if necessary. Therefore, a test with sample size n may only use half of its total units. The coverage required to achieve a specified probability of acceptance, given a normal
batch mean, was determined iteratively. Figure 4 illustrates the coverage required to achieve 10% and 90% probability of acceptance
(P10 and P90, respectively) for two cases of batch means (96% LC and 100% LC). P10 and P90 coverage are plotted against the inverse of the square root of sample size.
P10 and P90 of the hUSP test (n = 30, including the ZT requirement) is also given in Figure 4. A batch with 89% coverage will have 10% probability to pass
the hUSP test, while coverage of 98% is needed to pass the hUSP test with 90% probability. These coverage values are essentially
the same for the two batch means, as implied by the overlapping OC curves for batch means between 94% and 100% LC.
Other data points in Figure 4 are simulated without the ZT criterion. With increasing sample sizes, P10 and P90 coverage converge to the LDTs identified in Figure 3, thus indicating the increasing discrimination power with increasing
sample sizes. With a batch mean of 100% LC, P10 and P90 converge to an LDT of 95.4%. This LDT is the inherent quality-level requirement of the hUSP (–ZT) in the ideal state where
the content of all units in a batch are known. Figure 4 also shows that the LDT is 96% for a batch mean of 96% LC, thus matching
the data in Figure 3. The differences in the LDTs for various batch means indicate the hUSP test is not totally independent
of the batch mean.
Figure 4: Batch coverage to achieve 10% or 90% probability of acceptance for the hUSP(-ZT) test. Coverage from hUSP test provided
as references. Normal distribution is assumed.
The choice of acceptance probabilities of 10 and 90% (i.e., P10 and P90) to represent the rate of convergence is arbitrary. It is desirable to choose probabilities that are extreme enough to
illustrate convergence yet are not so extreme as to require excessive computer simulation time. Although all coverage lines
should converge to the same LDTs, it is possible that other probability pairs (e.g., 5% and 95% or 20% and 80%) could lead
to different conclusions about test efficiency. The chosen probability pairs should be consistent across the tests being compared.
Together, Figures 3 and 4 establish the inherent quality requirements of the hUSP test for the content range of 85–115% LC,
as well as the convergent rates of the hUSP test toward the inherent quality requirements (LDTs). Jointly, these figures serve
as useful tools for assessing large-sample UDU tests. A satisfactory large-sample UDU test should have LDTs no less than those
of the hUSP test and should converge relatively quickly toward the LDTs. These two assessment criteria can be demonstrated
using Sandell's proposal as an example.