Math for Causal Inference
All simulators

Cluster Designs & Power

Minimum detectable effect against the number of schools for any ICC, cluster size and baseline R².

Day 40: Cluster Randomisation & Design Effects
00.250.50.75150100150200250300number of schools Jminimum detectable effect (sd)
design effect
3.40
1 + (m − 1)ρ
effective pupils per school
7.4
limit 1/ρ = 10.0
schools needed
69
1,725 pupils
schools needed as m → ∞
51
MDE with clusteringif pupils were independenttarget effect

Schools are randomised, pupils are measured. With intraclass correlation ρ, each school of m pupils counts like 1 + (m − 1)ρ fewer pupils. Adding pupils per school hits a ceiling; adding schools does not. Power 80%, two-sided 5%, half the schools treated, outcome in standard deviations.