Testing homogeneity of risk difference in stratified randomized trials with noncompliance

When assessing a treatment effect in the presence of confounders, we often employ stratified analysis and obtain a summary estimate of the risk difference (RD) under the assumption that the underlying RD is homogeneous across strata. In a randomized clinical trial (RCT), we may commonly come across the data in which there are patients who do not comply with their assigned treatments. Thus, to avoid reaching a misleading conclusion due to overlooking an interaction between treatments and strata, it is important that we can incorporate noncompliance into examining the homogeneity of the RD. In this paper, we develop four statistics for testing the homogeneity of the RD in a stratified RCT with noncompliance. These include the test statistic derived from the weighted-least-squares (WLS) method, the test statistic using the WLS method and tanh-1(x) transformation, the test statistic using the weight similar to the Mantel-Haenszel (MH) estimator, and the test statistic using an optimal weight and the MH point estimator. We apply Monte Carlo simulation to evaluate the performance of these test statistics with respect to Type I error and power in a variety of situations. We use the data taken from a multiple risk factors intervention trial and a numerical example of simulated data to illustrate the practical use of these test statistics. Finally, we do a sensitivity analysis and discuss why applying test statistics for the ITT analysis to test the homogeneity of RD as focused in this paper can lead us to make an incorrect inference.