Using the inequality that, for two events A and B:
P(A and B) = P(A) + P(B) – P(A or B) ≥ P(A) + P(B) – 1
It follows that:
P(S
2) = P(C
21 and C
22) ≥ max {P(C
21) + P(C
22) – 1, 0}
Since criterion C
21 is similar to S
1 except for n = 30 and k = 2.0 in the former while n = 10 and k = 2.4 in the latter, the calculation of P(C
21) is carried out similarly as in P(S
1) with n = 30 and k = 2.0. Therefore:
Therefore, a lower bound on the probability of passing USP requirements is max {P(S
1), P(S
2)}.
Figure 1. 95% lower bound on passing the USP test for dose uniformity (target 5 100%).
|
For a given value of μ and a given value of σ, a lower bound (LBOUND) can be determined using the above calculations. Figure
1 shows a contour for the combinations of μ and σ that have an LBOUND of 95% assuming a target, T, of 100. Any combination of μ and σ at or below the contour results represents at least a 95% chance of samples passing the
USP content uniformity test.
The "true" probability of passing the USP test can be found by simulation. Table I gives a comparison of the simulated probabilities
and the LBOUND calculation.
Table I: Simulated (SIM) versus lower bound (LB) probabilities of passing content-uniformity test.
|
As can be seen in Table I, the LBOUND calculations are fairly close to the simulated results across various population means
and standard deviations.
|
