Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Assumptions: two-sided test, α = 0.05, power = 0.80, no clustering (individual-level randomization). With equal-sized arms (n ≈ 4,000 vs 4,000), the minimum detectable effect (difference in means) for a standardized continuous outcome is 0.063 SD. For a binary main outcome with baseline rate p = 0.50, the MDE is approximately 0.031 (3.1 percentage points).
Comparisons against the pure-control sub-arm only (e.g., n ≈ 2,800 vs 4,000) yield an MDE of about 0.069 SD for continuous outcomes and about 3.5 percentage points for binary outcomes (conservative, ignoring covariate adjustment and blocking gains).