Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
We define the power 1 − β to be the probability of rejecting the null hypothesis at the two-sided α level of significance. Τhe null hypothesis is that the outcome probabilities in the two groups are equal and the alternative hypothesis is that they take the unequal anticipated probabilities p1 and p2. If the trial has equal sample sizes n in each group, then a popular formula for the total sample size required is (Julious and Campbell, 2012; Marley-Zagar et al., 2023).
n =(z_{1-α/2}*SQRT(2*pa*(1-pa))+z_{1-b}*SQRT(p1*(1-p1)+p2*(1-p2)))^2 / ((p2-p1)^2)
where z_{1-α/2}=1.96 for α=5%, z_{1-β}=0.84 for β=80% and pa= (p1+p2)/2
Woerner et al. (2024) find a 62.9% vote for carbon pricing support across all conditions and we take this value as p1 = 0.629. Given a minimum detectable difference of 15% between the two groups (p2 = 0.779), we need at least 145 subjects per treatment.
Julious, Steven A., and Michael J. Campbell. "Tutorial in Biostatistics: Sample Sizes for Parallel Group Clinical Trials with Binary Data." Tutorial in Biostatistics 31, no. 24 (2012): 2904–2936. Special Issue in Honor of Jerome Cornfield on the Centennial of His Birth.
Marley-Zagar, Ella, Ian R. White, Patrick Royston, Friederike M.-S. Barthel, Mahesh K. B. Parmar, and Abdel G. Babiker. "Artbin: Extended Sample Size for Randomized Trials with Binary Outcomes." The Stata Journal 23, no. 1 (March 2023): 24–52.
Woerner, Andrej, Taisuke Imai, Davide D. Pace, and Klaus M. Schmidt. "How to Increase Public Support for Carbon Pricing with Revenue Recycling." Nature Sustainability 7 (2024): 1633–1641.