In randomized clinical trials, we often encounter the situations in which there are some patients who do not comply with their assigned treatments and also some confounders that are needed to control in assessing a treatment effect. To account for both noncompliance and confounders, we developed four asymptotic test procedures: (1) the test procedures based on the risk difference (RD) using the intention-to-treat approach with the optimal weight derived by the weighted least squares method; (2) the test procedure based on the instrumental variable (IV) estimator for the RD with the corresponding weighted least squares optimal weight; (3) the test procedure based on the IV estimator for the RD with tanh(-1) (x) transformation; and (4) the test procedure based on the Mantel-Haenszel estimator for the RD using the intention-to-treat approach. We apply Monte Carlo simulation to evaluate the finite sample performance of these test procedures in a variety of situations. Except for the test procedure using tanh(-1) (x) transformation, all the other procedures can perform well with respect to type I error even when the mean stratum size is moderate. We further find that when the mean stratum size is moderate, the test procedure based on IV estimator directly is generally preferable to the others with respect to power subject to maintaining type I error less than or approximately equal to the nominal a-level. When both the probability of compliance and the mean stratum size are large, however, we find that the test procedure based on IV estimator with tanh(-1) (x) transformation is more powerful than the others without losing the accuracy of type I error. Finally, we use the data taken from afield trial of studying the effect of a multifactor intervention program on the mortality incidence from coronary heart disease in middle-aged men to illustrate the practical use of these test procedures.
