Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Assumptions: two-sided test at the 5% level, 80% power, individual-level randomization with no clustering, and outcomes standardized against the control group. Minimum detectable effects (MDEs) are stated in control-group standard deviations (SD), with and without covariate adjustment. The adjusted figures assume that the stratum indicator and pre-treatment covariates explain 50% of outcome variance (R-squared of 0.5), which is realistic for the partisan-affect outcomes given that party identification and ideology are covariates; the unadjusted figures are conservative. All figures were verified by Monte Carlo simulation of the primary specification with HC2 standard errors (2,000 replications per scenario; rejection rate at the computed MDE between 79% and 81%; size 5% under the null).
Any gun video versus control (1,200 versus 800 respondents): 0.128 SD unadjusted, 0.090 SD adjusted. One treated arm versus control (600 versus 800): 0.151 SD unadjusted, 0.107 SD adjusted. Gun video plus nudge versus gun video alone (600 versus 600): 0.162 SD unadjusted, 0.114 SD adjusted. Any gun video versus control within one ownership stratum (600 versus 400): 0.181 SD unadjusted, 0.128 SD adjusted. Difference in treatment effects between gun owners and non-owners, or between men and women (two groups of about 1,000 each): 0.256 SD unadjusted, 0.181 SD adjusted. Difference between Republicans (about 950) and Democrats (about 850): 0.270 SD unadjusted, 0.191 SD adjusted.
In native units, assuming control-group standard deviations typical of national surveys: for a single party feeling thermometer (SD about 33 points), the pooled MDE is 4.2 points unadjusted and 3.0 points adjusted; for net partisan affect, the Republican minus Democratic thermometer (SD about 58 points), 7.4 points unadjusted and 5.2 points adjusted, and 10.5 points (7.4 adjusted) within one ownership stratum; for the gun-policy attitudes index (SD of 1 by construction), 0.13 SD unadjusted and 0.09 SD adjusted; for the five-point purchase-intention item (SD about 1.2), 0.15 scale points unadjusted and 0.11 adjusted. For the binary purchase-intention indicator, assuming a control-group rate of 30%, the pooled MDE is 5.9 percentage points; within the non-owner stratum at an assumed control rate of 15%, 6.5 percentage points; within the owner stratum at 45%, 9.0 percentage points. For the Republican-preference indicator at a control rate near 45%, the pooled MDE is 6.4 percentage points and the MDE for the change in the gender gap is 12.7 percentage points.