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In clinical trials where patients are randomized between two treatment arms, not all patients comply with the treatment they were randomly assigned to. The reasons for (non)compliance may be associated with the outcome variable and thereby act as confounders. The standard way of analysing such trials is by the 'intention-to-treat' principle, which allows the use of permutation tests. Conclusions drawn from such tests do not depend on untested assumptions such as absence of confounding. However, this approach may yield biased estimators for the causal effects of treatments. We consider the estimation of such effects for clinical trials where non-compliers can be considered to have switched to the other trial arm. The most important example of this is the placebo-controlled clinical trial where no substantial placebo effects are anticipated. We consider the situation where the relationship between compliance, and thus treatment received, and outcome is influenced by unobserved confounders. The residual of the regression of the actual treatment indicator variable on the randomization arm indicator variable is shown to 'intercept' the effect of such confounders. Inclusion of this residual in a multivariate analysis, in conjunction with the treatment indicator variable, should thus adjust for confounding. Examples are given. In those examples, the results are similar to those obtained by more complex methods.
Nagelkerke et al. (Sat,) studied this question.