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Re: st: Sample size for equivalence trials in Stata

From   Philip Ryan <>
Subject   Re: st: Sample size for equivalence trials in Stata
Date   Tue, 04 Oct 2005 09:42:05 +0930

In a recent contribution to this thread Joseph Coveney wrote:


You'd need to check with Philip Ryan to be sure, but -equivsize- looks like
it's based upon Blackwelder's* formula.  It's not necessarily intended to be
an endorsement of the practice, but a Web search turns up examples of sample
size estimation based upon the Blackwelder formula with proportions smaller
than 0.2 (larger than 0.8).

Joseph Coveney

*W. C. Blackwelder, Proving the null hypothesis in clinical trials.
_Controlled Clinical Trials_ 3:345-53, 1982.

I based -equivsize- on the formulae in Machin & Campbells' Statistical tables for the design of clinical trials; Blackwell 1987. In turn these authors quote Makuch & Simon's Sample size requirements for evaluating a conservative therapy in Cancer Treatment Reports Vol 62 pp1037-40. The formula given in Machin & Campbell is also identical to that in Table 1 of Blackwelder, as Joseph surmised.

As I recall, equivsize- was written as a hurried reply to another Lister's request for help. I didn't even put it on SSC as it has no proper help file, nor was it exhaustively debugged or tidied. (Nonetheless it appears to work. As nQuery Advisor also quotes Machin & Campbell, and -equivsize- yields results identical to nQA, I seem to have coded things OK, algebraically if not aesthetically).

I appended a caution regarding the normal approximation since Blackwelder states:

"Assuming sufficiently large samples to justify the normal approximation
to the binomial, the statistic z' indicated in Table 1 can be used to test the
null hypothesis H0'; the same statistic, without delta subtracted in the numerator,
can be used to test H0. (Note that the denominator of z' is slightly different
from what might be used in testing the usual hypothesis H0, but for com-
parison the same estimate of variance is used for both cases. Also, a correction
for continuity can easily be made in either case.) Dunnett and Gent [11]
compared several asymptotically appropriate test statistics for H0', including
z' corrected for continuity, and suggested use of a chi-square statistic. How-
ever, the statistic z' is especially useful since it is easily calculated and leads
to explicit sample size and confidence interval expressions. It will be appro-
priate in most practical situations, since the specified difference delta will usually
be relatively small and a sample large enough to justify the normal approx-
imation will therefore be required in order to provide acceptable statistical
power. If, however, the study is too small for use of the normal approximation,
the problem of an appropriate test statistic is more difficult. One approach,
discussed by Dunnett and Gent [11] and useful particularly when the prob-
ability _pi_ of success with standard therapy is known fairly accurately, involves
expressing the hypothesis in terms of an odds ratio and constructing confi-
dence intervals."

Having read that, I (probably over-conservatively) threw in the comment on proportions less than 0.2, but I freely confess being uncertain as to the performance of the asymptotic normal-based tests in this circumstance. Wellek, in his book: Testing statistical hypotheses of equivalence,Chapman & Hall 2003, provides a detailed discussion of exact tests in his chapter 2, and also provides a Fortran program to calculate sample sizes for exact Fisher type tests of non-inferiority. see and look for the bi2ste2 program. This is source code, so you will need a compiler. I am not into Fortran so have not used it.


Philip Ryan
Associate Professor,
Department of Public Health
Associate Dean (Information Technology)
Head, Data Management & Analysis Centre
Faculty of Health Sciences
University of Adelaide 5005
South Australia
tel 61 8 8303 3570
fax 61 8 8223 4075
CRICOS Provider Number 00123M
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