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May 2, 2010

*A method for determining sample size is finally getting some respect.*

Dr. James Bergum is now getting the recognition he deserves. His technique fundamentally changes the way we should use *US Pharmacopeia* (USP) standards and in-house specifications. Bergum, an associate director of nonclinical biostatistics at Bristol Myers Squibb, published a landmark paper in 1990 that, with his follow-on work, is now the basis for a new American Society for Testing and Materials (ASTM) Standard, E11 2709-09, adopted in 2009.

Lynn D. Torbeck

People often ask statisticians about sample sizes, particularly for *USP* standards such as <905> Uniformity of Dosage Units and <711> Dissolution. For example, "Is a sample of size 10 or 30 tablets adequate for a lot of 2.1 million tablets?" Or, "How many capsules—6, 12, or 24—are enough to measure dissolution profiles?"

But, how do we decide the sample size? How do we relate the sample size to the chances of passing a standard? Suppose we collect and test a representative sample of 30 tablets from a lot. Suppose further the specifications are 90% to 110%, and the 30 values are near 100% with a low standard deviation. What is the probability that the lot, if tested again or even several times, would meet the *USP* standard? Intuitively, we would say the probability is quite high. In this situation, we may not have control over the sample size, so rather than ask what should the sample size be, perhaps the question should be "What is the probability?" Bergum's method can be used to answer this question.

Bergum's method is a statistical procedure for calculating the probability of passing multistage *USP* standards such as dissolution or uniformity of dosage units. Passing a *USP* standard once is no assurance that the lot would pass again if tested later, either on stability, by another company, or the US Food and Drug Administration. But, by using Bergum's method, a lot can be tested for dissolution, and the probability of meeting the standard in the future can be determined. His approach converts an absolute standard into a workable practical "in-house" specification.

Thus, the test data for multistage dissolution criteria are converted into a probability of passing the standard with a specified confidence level. If the probability is less than a specified criterion, say, 95% with 95% confidence, then a larger sample size may be needed to better assure passage of the standard for a future sample. Or, perhaps the lot is not acceptable and should be rejected. Passing a *USP* standard once does not ensure that subsequent samples would also pass the standard. The probability of future tests from the lot can be calculated and reported.

It is necessary to stress that *USP* chapters such as the one for dissolution are standards and not working specifications. The standard must be met at any time up to the expiry date. It is a bright line that can't be crossed. It is absolute; there is no probability associated with a standard. By contrast, specifications are developed in-house by companies to provide a high degree of confidence of meeting the standard at any time. Specifications have a probability associated with them.

Unfortunately, many companies don't develop in-house specifications, but rather use the *USP* standard as if it is a specification. Because there is no probability associated with the standard, passing the standard once doesn't provide any assurance of passing again in the future.

Bergum's method evaluates the variability in the dissolution data and calculates a lower probability of passing the standard with a specified confidence. So instead of a pass/fail response, using the *USP* standard as a specification, Bergum's method gives a lower bound on the probability.

Dr. Bergum has provided a valuable service to our industry. He has developed a sound statistical basis for creating in-house specifications based on *USP* standards. This paradigm shift fundamentally changes how our industry should develop in-house specifications.

**Sources**

1. J. S. Bergum, Drug Development and Industry Pharmacy, 16 (14), 2153–2166 (1990).

2. J.S. Bergum and M. L. Utter, “Process Validation,” Encyclopedia of Bio-pharmaceutical Statistics (Marcel Dekker, New York, 2000).

3. J. S. Bergum and M. L. Utter, “Statistical Methods for Uniformity and Dissolution Testing,” in Pharmaceutical Process Validation, 3rd ed., Eds. R. Nash, R. and A. Wachter (Marcel Dekker, New York, 2003).

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4. P. Cholayudth, Pharm. Technol. 28 (9), 86–98 (2004).

5. P. Cholayudth, Pharm. Technol. 30 (1),( 88-94 (2006).

6. P. Cholayudth, Jrnl. of Validat. Technol. 13 (4), (2007).

7. J.S. Bergum and L. Hua, Pharm. Technol. online, http://pharmtech.findpharma.com/pharmtech/article/articleDetail.jsp?id=463577), accessed Apr. 14, 2010, October 2007.

8. P. Cholayudth, Jrnl. of Validat. Technol. 14 (2), 62–72, (2008).

9. P. Cholayudth, Jrnl. of Validat. Technol. 14 (3) 67–80 (2008).

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10. P. Cholayudth, Jrnl. of Validat. Technol. 14 (4), 72–78 (2008).

11. P. Cholayudth, Jrnl. of Validat. Technol. 15 (4), 77–90, (2009).

12. P.D. Lunney and C. A. Anderson, Jrnl. of Pharm. Innov. 4 (2009). PT

**Lynn D. Torbeck** is a statistician at Torbeck and Assoc., 2000 Dempster Plaza, Evanston, IL 60202, tel. 847.424.1314, Lynn@Torbeck.org, www.torbeck.org.