Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
All calculations assume 80 percent power, a 5 percent two-sided test, and equal allocation, using MDE = 2.80 × σ × √(2/n) per arm. Item-level models cluster standard errors by respondent, so the effective unit is the participant, with twenty proposals each. Assumed standard deviations are stated so they can be checked against realized data. Retention is assumed at 50 percent for Wave 2 and 40 percent for Wave 3. The range in each row spans the expected sample of 2,500 to 5,000 completions.
Primary outcome, attribution effect on alignment. Estimated on the full Wave 1 sample with depolarization fixed effects, giving 1,250 to 2,500 participants per attribution arm.
Alignment rate, per candidate, 0–100 scale, assumed σ = 18 points: minimum detectable effect of 2.0 to 1.4 points, or 0.11 to 0.08 standard deviations.
Difference between own-candidate and opposing-candidate alignment rate, assumed σ = 25 points: 2.8 to 2.0 points, or 0.11 to 0.08 standard deviations.
Item-level agreement, attribution by authorship interaction, five-point scale. The relevant quantity is the within-participant difference between mean agreement with own-candidate and opposing-candidate proposals, assumed σ = 1.1 scale points given an item-level σ of 1.3 and moderate within-person correlation: 0.12 to 0.09 scale points, or 0.09 to 0.07 item-level standard deviations.
Secondary outcomes.
Feedback effect on the reported vote, Wave 2, binary indicator with p ≈ 0.5 and σ = 0.5, 625 to 1,250 per arm: 7.9 to 5.6 percentage points.
Feedback effect on Wave 3 policy preferences, item level, adjusted for the Wave 1 rating of the same proposal, assumed residual σ = 0.54 at the participant level, 500 to 1,000 per arm: 0.10 to 0.07 scale points, or 0.07 to 0.05 item-level standard deviations.
Feedback effect on long-run alignment beliefs, Wave 1 to Wave 3, 0–100 scale, assumed σ = 20 points: 3.5 to 2.5 points.
Depolarization effect on affective polarization, measured as the difference between the Wave 2 left and right thermometers, assumed σ = 40 points: 6.3 to 4.5 points.
Exploratory. The interaction between feedback and the misperception gap requires roughly four times the sample of a main effect of equal standardized size. On the Wave 2 vote this gives a minimum detectable differential of approximately 16 to 11 percentage points per standard deviation of the gap. This is larger than any effect the information-provision literature reports, so the interaction is prespecified as exploratory rather than confirmatory. Power for it depends on the dispersion of the misperception gap rather than on sample size alone: if most participants prove approximately correct about their own alignment, the interaction is weakly identified at any sample size. The distribution of the gap is the first quantity we report.