  In Randomised Controlled Trials (RCT) with treatment non-compliance,
instrumental variable approaches are used to estimate complier average causal
effects. We extend these approaches to cost-effectiveness analyses, where
methods need to recognise the correlation between cost and health outcomes.
  We propose a Bayesian full likelihood (BFL) approach, which jointly models
the effects of random assignment on treatment received and the outcomes, and a
three-stage least squares (3sls) method, which acknowledges the correlation
between the endpoints, and the endogeneity of the treatment received.
  This investigation is motivated by the REFLUX study, which exemplifies the
setting where compliance differs between the RCT and routine practice. A
simulation is used to compare the methods performance.
  We find that failure to model the correlation between the outcomes and
treatment received correctly can result in poor CI coverage and biased
estimates. By contrast, BFL and 3sls methods provide unbiased estimates with
good coverage.
