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I am planning a study where the endpoint of the current practice is 40% successful. The null hypothesis will be rejected if the intervention produces a 60% success rate (treatment effect of interest). We are happy with a significance criterion of 0.05 and a $\beta$ 0.8. The study will be to intervention or to current treatment, randomised and blinded. What is the sample size needed?

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I'm assuming you're taking $H_0: p_1 = p_2 = 0.4$, $H_a: p_1 < p_2$, the specific alternative $p_2 - p_1 > 0.6 - 0.4 = 0.2$, and $n_1 = n_2$. Under these assumptions, the test statistic is $$z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}}.$$

With $\alpha = 0.05$, you're saying you want $P(z \leq z^*) = 0.05$ when $p_1 = p_2 = 0.4$, which gives $z^* = -1.645$. With $\beta = 0.8$, you're saying you want $P(z \leq z') = 0.8$ when $p_2 - p_1 = 0.2$, which gives $z' = 0.84$. Based on the assumptions on $z^*$ and $z'$ we have $$z^* = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\frac{0.4(0.6)}{n} + \frac{0.4(0.6)}{n}}} = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\frac{0.48}{n}}}$$ and $$z' = \frac{(\hat{p}_1 - \hat{p}_2) - (-0.2)}{\sqrt{\frac{0.4(0.6)}{n} + \frac{0.6(0.4)}{n}}} = \frac{\hat{p}_1 - \hat{p}_2 + 0.2}{\sqrt{\frac{0.48}{n}}}.$$

Thus to find $n$ we solve $$-1.645 + \frac{0.2}{\sqrt{\frac{0.48}{n}}} = 0.84,$$ which (rounding up) yields $n = n_1 = n_2 = 75$.

Incidentally, there are online tools that will do this calculation for you, such as this one. They give the sample size as $n_1 = n_2 = 77$. I'm not sure the reason for the discrepancy; maybe they are using a continuity correction or using slightly different values from the normal distribution or taking the conservative $p_1 = p_2 = 0.5$ in the $z$ score calculation.

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