Notes on estimation of proportion ratio under a non-compliance randomized trial with missing outcomes

When measuring the efficacy of an experimental treatment in dichotomous data, we often employ the proportion ratio () of a positive response (or an adverse event) between the experimental and standard treatments. In this paper, we consider a non-compliance randomized clinical trial with outcomes missing at random (MAR). We derive the maximum likelihood estimator (MLE) for the under the model in which missingness depends only on assigned treatments, and the model in which missingness depends on both assigned and received treatments. We develop three asymptotic interval estimators for the under the latter situation. We apply Monte Carlo simulation to evaluate and compare the performance of these estimators in a variety of situations. We note that the point estimator assuming outcomes missing completely at random (MCAR) can be subject to a serious bias under various models with MAR, and the point estimators developed here may consistently perform reasonably well when the number of patients per assigned treatment is large. We find that the interval estimator using Wald's statistic tends to lose accuracy with respect to the coverage probability, while the interval estimator using the logarithmic transformation tends to lose precision with respect to the average length. We also find that an ad hoc combination of the previous two interval estimators can consistently perform well in many situations. Finally, we use data taken from a multiple risk factor intervention trial for reducing mortality of coronary heart disease to illustrate the bias of the point estimator for MCAR and the use of these point and interval estimators for the under the assumed MAR.