Intention to treat, per protocol, as treated and instrumental variable estimators given non-compliance and effect...

We consider the behaviour of three approaches to efficacy estimation--using so-called 'as treated' (AT), 'per protocol' (PP) and 'instrumental variable' (IV) analyses--and of the Intention to Treat estimator, in a two-arm randomized treatment trial with a Normally distributed outcome when there is treatment effect heterogeneity and non-random compliance with assigned treatment. Formulae are derived for the bias of estimators when used either to estimate average treatment effect (ACE) or to estimate complier average treatment effect (CACE) under several models for the relationship between compliance and potential outcomes. These enable the expected values of AT, PP and IV estimators to be ranked in relation to ACE, and show that AT and PP estimators are generally biased for both ACE and CACE even under homogeneity. However, we show that the difference between any pair of (AT, PP, IV) estimates can be used to estimate the correlation between the latent variable determining compliance behaviour and one potential outcome. In the absence of measures that predict compliance, bounds for ACE can only be set given strong assumptions. Regarding the Intention to Treat estimator, while this is 'biased towards the null' if viewed as a measure of CACE, we show that it is not always so in relation to ACE. Finally we discuss the behaviour of the estimators under weak and strong null hypotheses.