Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
The primary outcome is willingness to pay (WTP) for climate mitigation, elicited as a monthly euro amount. With approximately 4,000 respondents divided equally across 10 randomisation cells (5 treatment arms × 2 WTP question versions), each treatment arm contains approximately 800 respondents when pooling across WTP versions.
We assume a standard deviation approximately equal to the mean monthly WTP, consistent with the right-skewed distribution with a large mass at zero typical of stated WTP variables. Under this assumption, at 5% significance (two-sided) and 80% power, the minimum detectable effect for the main treatment vs. control comparison pooling across WTP versions (n ≈ 800 per arm) is approximately 14% of the control group mean, and for analyses within each WTP version (n ≈ 400 per arm) approximately 20% of the control group mean. Assuming a control group mean of €8–15 per month, this corresponds to an MDE of approximately €1.40–2.80 per month for the pooled comparison and €2.00–3.96 per month for the within-version comparison.