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# st: Sample size for a test of equivalence of proportions

 From Ángel Rodríguez Laso To statalist@hsphsun2.harvard.edu Subject st: Sample size for a test of equivalence of proportions Date Wed, 26 May 2010 13:24:11 +0200

```Dear Statalisters,

I want to calculate the sample size for an experiment to compare the
refusal rate of two strategies to contact participants in a survey. In
the usual procedure, I expect a refusal rate of 17%. I would like to
detect a 10% increase in refusal with the alternative procedure, with
alpha 0.05 (one sided) and power 80%. I consider this a test of
equivalence between procedures.

After searching Statalist, I’ve come across two possibilities to
calculate sample sizes that yield different results: I would
values for the programs and the reasons for the differences in the
results obtained.

1) The user-written program equivsize by Phil Ryan v0.02 2002-02-06
(http://www.stata.com/statalist/archive/2003-02/msg00204.html)

. equivsize .17 .17 .1 .05 .8

n (per group) = 175

2) The user-written artmenu program version 1.0.4 SB/PR 13jan2005:

. artbin, pr(.17 .27) ngroups(2) aratios(1 1) distant(0) alpha(0.05)
power(0.8) onesided(1) ni(1)

ART - ANALYSIS OF RESOURCES FOR TRIALS (version 1.0.0, 3 March 2004)
------------------------------------------------------------------------------
A sample size program by Abdel Babiker, Patrick Royston & Friederike Barthel,
MRC Clinical Trials Unit, London NW1 2DA, UK.
------------------------------------------------------------------------------
Type of trial Noninferiority - binary outcome
Statistical test assumed Unconditional comparison of 2
binomial proportions
Number of groups 2
Allocation ratio Equal group sizes

Anticipated event probabilities 0.170, 0.270

Alpha 0.050 (one-sided)
Power (designed) 0.800

Total sample size (calculated) 451
Expected total number of events 100

That is, 225 individuals per group.

3) In addition, in a research methods text book (Argimón, Jiménez.
Métodos de investigación clínica y epidemiológica. Elsevier, Madrid,
2004) I’ve found the following formula to calculate sample sizes for
equivalence tests (no further reference for the formula is provided,
but maybe someone will identify its origin):

N(per group)= 2*P*(1-P)*(Zalpha+Zbeta)squared) / (difference of interest)squared

Where P would be refusal rate in the usual procedure group (0.17)
Zalpha=1.645 (one-sided)
Zbeta=0.842 (corresponding to a 0.8 power)
Difference of interest=0.1

N(per group)= 2*0.17*0.83*(1.645+0.842)squared / 0.1*0.1 = 175

This matches perfectly the result of 1). Nevertheless, this option
does not allow unequal group sizes that could be of interest in this
experiment.

To complicate things further, the experiment will deal with clustered
samples. To correct sample size for clustering, would it be enough to
multiply the sample size obtained from these methods by the expected
DEFF?

Thank you very much for your time and interest.

Angel Rodriguez-Laso

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