Regaining power lost by non-compliance via full probability modelling

Non-compliance, or non-receipt of randomized intervention, is a common problem in randomized controlled trials. An intention-to-treat (ITT) analysis, which compares individuals as randomized, underestimates the efficacy of the intervention and leads to a loss of power. We explore the possibility of regaining some of this power in a setting with all-or-nothing compliance, without making any assumptions about the comparability of compliers and non-compliers. Efficacy may be specified as the complier average causal effect (CACE), which is the difference in mean outcome among compliers. Compliance is only partially observed, but under an exclusion restriction assumption, the CACE may be estimated using maximum likelihood. In order to quantify the possible gain in power, we derive an expression for the asymptotic relative efficiency (ARE) of the CACE relative to the ITT effect with a Normally distributed outcome. Under the assumption of a common CACE across covariate strata, the CACE estimate is at least as powerful as ITT analysis. The inclusion of covariates that predict compliance enables an additional gain in power, which is investigated algebraically. Using data from three clinical trials, we obtain values of the ARE ranging up to 1.05 due to covariates alone, and 1.13 due to CACE modelling alone, corresponding to gains in power of up to 5 per cent. This implies that a large gain in power obtained using as-treated or per-protocol analyses is likely to be due to the strong and often implausible assumptions such analyses require to be valid.