Sample Size, $n_B$

Power, $1-\beta$

Type I error rate, $\alpha$

X-axis

min

max

Documentation
Test 1 Mean
Compare 2 Means
Compare k Means
Test 1 Proportion
Compare 2 Proportions
Compare Paired Proportions
Compare k Proportions
Test Time-To-Event Data
Test Odds Ratio
Test Relative Incidence in Self Controlled Case Series Studies
Other

## 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.

$H_0:|\ln(OR)|\ge\delta$
$H_1:|\ln(OR)|<\delta$

where $\delta$ is the superiority or non-inferiority margin and the ratio between the sample sizes of the two groups is
$$\kappa=\frac{n_A}{n_B}$$

### Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$n_A=\kappa n_B \;\text{ and }\; n_B=\left(\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-p_B)}\right) \left(\frac{z_{1-\alpha}+z_{1-\beta/2}}{|\ln(OR)|-\delta}\right)^2$$
$$1-\beta= \Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right) \quad ,\quad z=\frac{(|\ln(OR)|-\delta)\sqrt{n_B}}{\sqrt{\frac{1}{\kappa p_A(1-p_A)}+\frac{1}{p_B(1-p_B)}}}$$
where $$OR=\frac{p_A(1-p_B)}{p_B(1-p_A)}$$ and
where

### R Code

R code to implement these functions:

pA=0.25
pB=0.25
delta=0.50
kappa=1
alpha=0.05
beta=0.20
(OR=pA*(1-pB)/pB/(1-pA)) # 1
(nB=(1/(kappa*pA*(1-pA))+1/(pB*(1-pB)))*((qnorm(1-alpha)+qnorm(1-beta/2))/(abs(log(OR))-delta))^2)
ceiling(nB) # 366
z=(abs(log(OR))-delta)*sqrt(nB)/sqrt(1/(kappa*pA*(1-pA))+1/(pB*(1-pB)))
(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.