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Calculate Sample Size Needed to Test Odds Ratio: Equivalence

This calculator is useful when we wish to test whether the odds of an outcome in two groups are equivalent, without concern of which group's odds is larger. Suppose we collect a sample from a group 'A' and a group 'B'; that is we collect two samples, and will conduct a two-sample test. For example, we may wish to test whether a new product is equivalent to an existing, industry standard product. Here, the 'burden of proof', so to speak, falls on the new product; that is, equivalence is actually represented by the alternative, rather than the null hypothesis.
H0:∣ln⁡(OR)∣≥δH_0:|\ln(OR)|\ge\delta
H1:∣ln⁡(OR)∣<δH_1:|\ln(OR)|<\delta
where δ\delta is the superiority or non-inferiority margin and the ratio between the sample sizes of the two groups is
κ=nAnB\kappa=\frac{n_A}{n_B}

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+z1−β/2δ−∣ln⁡(OR)∣)2n_A=\kappa n_B \;\text{ and }\; n_B=\left(\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-pB)}\right) \left(\frac{z_{1-\alpha}+z_{1-\beta/2}}{\delta-|\ln(OR)|}\right)^2
1−β=2(Φ(z−z1−α)+Φ(−z−z1−α))−1,z=(∣ln⁡(OR)∣−δ)nB1κpA(1−pA)+1pB(1−pB)1-\beta= 2(\Phi(z-z_{1-\alpha})+\Phi(-z-z_{1-\alpha}))-1\quad ,\quad z=\frac{(|\ln(OR)|-\delta)\sqrt{n_B}}{\sqrt{\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-p_B)}}}
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 δ\delta is the testing margin

R Code

1pA=0.25
2pB=0.25
3delta=0.50
4kappa=1
5alpha=0.05
6beta=0.20
7(OR=pA*(1-pB)/pB/(1-pA)) # 1
8(nB=(1/(kappa*pA*(1-pA))+1/(pB*(1-pB)))*((qnorm(1-alpha)+qnorm(1-beta/2))/(delta-abs(log(OR))))^2)
9ceiling(nB) # 366
10z=(abs(log(OR))-delta)*sqrt(nB)/sqrt(1/(kappa*pA*(1-pA))+1/(pB*(1-pB)))
11(Power=2*(pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))-1)

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 107.