The Multiple Risk Factor Intervention Trial (MRFIT) was a major trial that examined the efficacy of a multifactor intervention on reducing coronary heart disease mortality. After a mean follow-up of 7 years, treatment was not significantly different from control, and extensive investigation was undertaken to explain the result. This article examines how the effect of treatment varies with compliance with treatment. The basic idea is that the binary characteristic "would comply with treatment" is balanced between groups by randomization, even though it is unobservable before randomization and only observed after randomization in the treatment group. Using baseline covariates, a prediction model for this observed characteristic is developed in the treatment group, and estimated probabilities of treatment compliance are calculated for all patients in both groups. The assumption that patients who would nut comply with treatment must have the same mean response in both groups is avoided. Two methods for examining how the effect of treatment depends on treatment compliance are described. Under the first approach, the predicted probabilities of treatment compliance are used to specify a treatment by covariate interaction. Under the second approach, a regression model is specified where the covariate "would comply with treatment" has an interaction with treatment. This produces a standard regression model in the treatment group and a mixture model with mixing over the unobserved binary covariate in the control group. The two methods are applied to MRFIT and the results compared.
