Journal of Pharmacokinetics and Pharmacodynamics, Vol. 29, No. 1, February 2002 ( 2002)

LETTER TO THE EDITOR

Sample Size Calculation in Bioequivalence Trials

Peter Blood

Received August 30, 2001—Final January 30, 2002

I needed to determine some sample sizes and turned to the recent article byChow and Wang (1). However, I had some difficulties with the article andwould like to mention them.

First, there are a number of typographical errors. Second, the log trans-formed sample size values in Tables II and I are high, particularly for σ eG

10% and 15%. I have recalculated the table values, using a SC programwith the formulae used by Chow and Wang. I list an extract from Tables IIand IV in Tables A and B below. Also listed in the same two tables, are theoutput from two pieces of software, PASS 200 and nQuery 4, which arecapable of determining sample sizes for bioequivalence trials. Only outputfrom Tables II and IV are shown below, as the sample s sizes in Tables Iand III are double the values of Tables II and IV respectively. Table IV andby implication Table III show similar values across all methods (C&W table,PASS2000, nQuery, SC) for all σ values (0, 5, 10, 15%). However for TableII, and by implication Table I, there is increasing discrepancy on movingfrom θG0% to 15% for both 80% and 90% power. Of the four methodsused, the values of Chow and Wang stand out as different from the otherthree solutions. The formulae, as pointed out by Chow and Wang, go backto earlier papers by Schuirmann and Westlake.

Table IV is reproduced in a reduced format in Liu and Chow (2), Phil-lips (3) and Dilietti (4) with good agreement. Hauschke (5) has a table forthe multiplicative model similar to the Table II in Chow and Wang but thevalues agree only for 0% and 5%. For the 10% and 15% values, they followthe PASS2000, nQuery 4 and SC results. Table II and by implication TableI, do not seem to be correct unless there has been a further correction, notmentioned in the paper. In the article (p. 160), Chow and Wang mentionthat for the formula ‘‘When 1n�θ �Z0, this is a good approximation.’’ Butit appears that the formula is only a good approximation for low values of

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Blood96

Table A. Total Sample Size—Cross-over with Log Transformed Data [Corresponds to Table IIin (1)]

�θ′� (80% power) �θ′� (90% power)

σ e(%) 0% 5% 10% 15% 0% 5% 10% 15%

10 C&W 6 6 10 24 8 8 14 34PASS2000 6 8 10 20 8 8 14 28nQuery4 6 8 10 20 8 8 14 28S+ 5 6 9 19 6 8 14 26

20 C&W 16 18 34 94 20 24 46 130PASS2000 16 20 32 74 20 26 44 102nQuery4 16 20 32 74 20 26 44 100S+ 16 18 32 72 19 24 44 100

30 C&W 32 38 74 210 42 54 104 290PASS2000 34 40 70 162 42 54 96 224nQuery4 34 40 70 162 42 54 96 224S+ 33 38 70 161 41 52 96 223

40 C&W 56 68 132 372 72 94 182 514PASS2000 58 70 122 286 72 94 170 396nQuery4 58 70 124 290 72 94 170 400S+ 57 67 122 286 71 92 169 395

�θ � in Tables I and II. Using the mistaken value of log (1.262)G0.2331mentioned on page 165 does not correct the problem at the 10 and 15%levels.

The last equation in Appendix B (B2 not B1) involves ‘‘2CV2’’ and thearticle states that B2 is consistent with Eqs. (1) and (2) in the earlier paper

Table B. Total Sample Size—Cross-over with Raw Data [Corresponds to Table IV in (1)]

�θ′� (80% power) �θ′� (90% power)

σ e(%) 0% 5% 10% 15% 0% 5% 10% 15%

10 C&W 6 8 14 52 8 10 20 70PASS2000 8 8 14 52 8 10 20 70nQuery4 6 7 14 51 7 9 18 70S+ 7 7 14 51 8 9 19 70

20 C&W 20 24 52 200 24 32 70 276PASS2000 20 24 52 200 24 32 70 276nQuery4 18 24 51 199 23 32 70 275S+ 19 24 51 199 24 32 70 276

30 C&W 40 52 112 446 50 70 156 618PASS2000 40 52 114 446 52 72 156 618nQuery4 40 51 112 446 50 70 155 618S+ 40 51 113 447 51 70 156 618

40 C&W 70 90 200 792 88 124 276 1098PASS2000 70 92 200 792 88 124 276 1098nQuery4 70 92 200 794 88 124 276 1098S+ 70 89 199 793 88 123 276 1098

Letter to the Editor 97

by Liu and Chow (2). However, this earlier paper has equations for samplesize involving ‘‘CV2’’. The last sentence in Chow and Wang states the equa-tions in the two papers are consistent despite the missing ‘‘2’’ in theequations of the earlier paper. An explanation is not given. The reasonis, the CV (coefficient of variation) is defined differently in the two papers.In Liu and Chow, CVG[(1MSE)�µR ]B100, i.e., using the notation ofChow and Wang (σ e�µR )B100. However in Chow and Wang CV is usedas (σd�µR)B100, where σ2

eG2σ2d .

It should also be mentioned that Tables I and II need σd whereasTables II and IV expect the input of σ e .

REFERENCES

1. S.-C. Chow and H. Wang. On sample size calculation in bioequivalence trials. J. Pharmaco-kinet. Pharmacodyn. 28:155–169 (2001).

2. J.-P. Liu and S.-C. Chow. Sample size determination for the two one-sided tests procedurein bioequivalence. J. Pharmacokinet. Biopharm. 20:101–104 (1992).

3. K. F. Phillips. Power of the two one-sided tests procedure in bioequivalence. J. Pharmacoki-net. Biopharm. 18:137–144 (1990).

4. E. Diletti, D. Hauschke, and V.W. Steinijans. Sample size determination for bioequivalenceassessment by means of confidence intervals. Int. J. Clin. Pharmacol. Ther. Toxicol. 29:1–8(1991).

5. D. Hauschke, V. W. Steinijans, E. Diletti, and M. Burke. Sample size determination forbioequivalence assessment using a multiplicative model. J. Pharmacokinet. Biopharm.20:557–561 (1992).