Evaluation of the LDT of Sandell's proposal
As stated above, the Sandell limit is equal to the median of the binomial distribution defined by a sample size n and a defect probability p of 1 – 0.952 = 0.048. For a large n, this binomial distribution approaches (by the central limit theorem) the normal distribution with mean (median) of np. Thus, at infinite sample size, an acceptable batch will have no more than 4.8% of all units outside 85–115% LC. Thus, the
LDT coverage of Sandell's proposal is 95.2%.
Figure 5: Sandell’s proposal assessment: (a) Sandell’s limited discrimination threshold (LDT) is fixed at 95.2% coverage;
(b) Sandell’s proposal coverage for 10% and 90% probability of acceptance versus the coverage from the hUSP(-ZT) test (batch
mean = 100% label claim [LC]) (2).
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As a nonparametric test, the Sandell LDT is independent of assumed batch mean. Figure 5a superimposes the LDT of Sandell's
proposal as a horizontal line on the LDTs from the hUSP (–ZT) test for a normal population distribution. Clearly, Sandell's
proposal is not uniformly as stringent as the hUSP test.
The P10 and P90 coverage lines for Sandell's proposal are shown in Figure 5b. For comparison, the coverage lines from hUSP (–ZT) test assuming
a batch mean of 100% LC are also plotted. The angle of convergence of Sandell's proposal is only slightly greater than that
of the hUSP (–ZT) test. For tests that include as many as 1000 samples, the P
90 coverage line from Sandell's test is at or above the corresponding line from the hUSP (–ZT) test, thus indicating that Sandell's
approach has a higher risk of Type I errors than that of the hUSP (–ZT) test. Furthermore, the P10 coverage line from Sandell's proposal is lower than that of the hUSP (–ZT) test, thus indicating that Sandell's approach
has a higher risk of Type II error as well. Sandell's coverage lines can be compared with those of the hUSP test for other
batch means as well, although this article will not discuss this comparison.
In summary, a nonparametric test that does not rely on the batch mean must have at least a 96% LDT coverage for the test to
be considered as stringent as the hUSP test for all batch means. Furthermore, the P10 and P90 coverage lines of a proposed test should be contained within those of the hUSP (–ZT), as shown in Figure 5b. A P
90 coverage line above that of the hUSP (–ZT) test indicates a higher risk of Type I error, with potential negative effect on
the business operation. Similarly, a P10 coverage line below that of the hUSP (–ZT) test indicates a higher risk of Type II error, potentially harming the marketed
product's quality.
Comparison of hUSP (–ZT) test and Sandell proposal with a non-normal population distribution
Given the concern over greatly deviating units, it is of interest to determine the LDT assuming population distributions with
"fatter tails" than the normal distribution. Such distributions might be present in batches for which the unit-dose potency
variance is not constant during manufacturing. The nonstandard t-distribution, indexed by μ, σ, and df (i.e., degrees of freedom), is such a distribution. In this case, the limiting coverage at large sample size is expressed
in the following equation:
in which t(x|μ, σLDT, df) is the nonstandard t probability density function.
Figure 6: Limited discrimination threshold (LDT) coverage for the hUSP(-ZT) and Sandell tests when used to test units from
nonstandard t-distributions. A target mean of 100% label claim (LC) is assumed (2).
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Figure 6 shows how LDT coverage varies as a function of the assumed batch mean. The LDT for the hUSP (–ZT) was calculated
for various nonstandard t distributions (indexed by the t-distribution parameter df) using Eq. 9. The LDT for the nonparametric Sandell test does not depend on assumptions about the population distribution
and thus remains constant. The hUSP (–ZT) LDT for the normal distribution is also shown for comparison.
With increasing degrees of freedom, the LDT coverage profile of a nonstandard t distribution approaches that of the normal population distribution, as expected. With lower degrees of freedom, the LDTs
are less stringent for batches with means at or above 96% LC, but more stringent for batches with lower means. Between 96%
and 100% LC batch means, the effect of df on the LDTs is surprisingly minimal, thus indicating the robustness of the hUSP test with respect to the changes in the greatly
deviating units. Again, Sandell's proposal does not provide as stringent a requirement as the hUSP test does under the assumption
of a nonstandard t distribution.
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