Test homogeneity of odds ratio in a randomized clinical trial with noncompliance

The odds ratio (OR) has been recommended to measure the relative treatment effect in therapeutic equivalence or meta-analysis. When controlling the confounding effect due to strata formed by centers (or trials) on patients' response in a multicenter study (or a meta-analysis), we commonly employ stratified analysis and obtain a summary estimate of the treatment effect. In practice, it is not uncommon to come across data in which there are patients not complying with their assigned treatment. To avoid obtaining a misleading summary estimate due to overlooking the interaction between the stratum and treatment effects as well as the selection bias from noncompliance, it is important to develop test statistics accounting for noncompliance for testing the homogeneity of the OR across strata. In this article, we develop five asymptotic test statistics and employ Monte Carlo simulation to evaluate the performance of these statistics in a variety of situations. We note that the weighted-least-squares (WLS) test statistic can be liberal when the number of strata is moderate or large (>/=5). We find that the logarithmic transformation of the WLS (LWLS) test statistic, the squared-root transformation of the LWLS (SLWLS) test statistic, and Fisher's logarithmic transformation of LWLS (LLWLS) test statistic can perform well with respect to Type I error in all the situations considered here. We further find that the Z-transformation of LWLS (ZLWLS) test statistic can be liberal when the number of strata is small or moderate. We note that the LWLS test statistic is likely preferable to the others for a small number of strata, while the ZLWLS test statistic can be the best for a moderate or large number of strata. Finally, we use the data taken from a multiple-risk-factor intervention trial to illustrate the use of these test statistics.