Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Power analysis was conducted in StataNow 19.5 for a cluster-randomized two-sample z test of means (sd₁ = sd₂ = sd). We tested H₀: m₂ = m₁ versus Hₐ: m₂ < m₁ with α = 0.050 (one-sided). Based on pilot data (Δ = -0.14, SD = 0.35, ICC = 0.09), the target power was 0.80. The required minimum cluster size was estimated to be 5, with 20 clusters in the treatment group and 20 clusters in the control group.