All simulators
Non-Compliance: ITT & CACE
An insurance lottery with compliers, always-takers and never-takers: only the Wald ratio recovers the CACE.
Day 41: Non-Compliance & Intention-to-TreatCompliers 40%
Always-takers 15%
Never-takers 45%
first stage
0.40
P(D=1|Z=1) − P(D=1|Z=0)
ITT
0.60
CACE
1.50
A health-insurance lottery (Z = won) and actual enrolment (D). Outcome: doctor visits per year. Population values, no sampling noise. Only the Wald ratio ITT ÷ first stage recovers the complier effect (CACE); as-treated and per-protocol comparisons mix in who chooses to enrol. Assumes no defiers.