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Calculate Sample Size Needed to Test Odds Ratio: Equality

This calculator is useful for tests concerning whether the odds ratio, OROR, between two groups is different from the null value of 1. Suppose the two groups are 'A' and 'B', and we collect a sample from both groups -- i.e. we have two samples. We perform a two-sample test to determine whether the odds of the outcome in group A, pA(1−pA)p_A(1-p_A), is different from the odds of the outcome in group B, pB(1−pB)p_B(1-p_B), where pAp_A and pBp_B are the probabilities of the outcome in the two groups. The hypotheses are
H0:OR=1H_0:OR=1
H1:OR≠1H_1:OR\neq1
. where the ratio between the sample sizes of the two groups is
κ=nAnB\kappa=\frac{n_A}{n_B}
and OR=pA(1−pB)pB(1−pA)OR=\frac{p_A(1-p_B)}{p_B(1-p_A)}.

Formulas

This calculator uses the following formulas to compute sample size and power, respectively:
nA=κnB   and   nB=(1κpA(1−pA)+1pB(1−pB))(z1−α/2+z1−βln⁡(OR))2 n_A=\kappa n_B \;\text{ and }\; n_B=\left(\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-p_B)}\right) \left(\frac{z_{1-\alpha/2}+z_{1-\beta}}{\ln(OR)}\right)^2
1−β=Φ(z−z1−α/2)+Φ(−z−z1−α/2),z=ln⁡(OR)nB1κpA(1−pA)+1pB(1−pB)1-\beta= \Phi\left(z-z_{1-\alpha/2}\right)+\Phi\left(-z-z_{1-\alpha/2}\right) \quad ,\quad z=\frac{\ln(OR)\sqrt{n_B}}{\sqrt{\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-pB)}}}
where
OR=pA(1−pB)pB(1−pA)OR=\frac{p_A(1-p_B)}{p_B(1-p_A)}
and where κ=nA/nB\kappa=n_A/n_B is the matching ratio Φ\Phi is the standard Normal distribution function Φ−1\Phi^{-1} is the standard Normal quantile function α\alpha is Type I error β\beta is Type II error, meaning 1−β1-\beta is power

R Code

1pA=0.40
2pB=0.25
3kappa=1
4alpha=0.05
5beta=0.20
6(OR=pA*(1-pB)/pB/(1-pA)) # 2
7(nB=(1/(kappa*pA*(1-pA))+1/(pB*(1-pB)))*((qnorm(1-alpha/2)+qnorm(1-beta))/log(OR))^2)
8ceiling(nB) # 156
9z=log(OR)*sqrt(nB)/sqrt(1/(kappa*pA*(1-pA))+1/(pB*(1-pB)))
10(Power=pnorm(z-qnorm(1-alpha/2))+pnorm(-z-qnorm(1-alpha/2)))

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 106.