Interval estimation of risk ratio in the simple compliance randomized trial

Consider the simple compliance randomized trial (SCRT), in which patients assigned to an experimental group may switch to receive a control treatment, but patients assigned to a control group are assumed to all receive their assigned treatment. We develop five asymptotic interval estimators for the relative risk (RR) of probabilities of response among patients who would comply with the experimental treatment under the SCRT. We employ Monte Carlo simulation to evaluate the performance of these interval estimators in a variety of situations. We note that the interval estimator using Wald's statistic and the interval estimator derived from a quadratic equation based on asymptotic properties of the maximum likelihood estimator (MLE) can lose accuracy, while the most commonly-used interval estimator using a logarithmic transformation of the MLE for the RR suggested elsewhere can lose efficiency. We further note that the probability of failure to apply the interval estimator derived from an idea used in Fieller's Theorem to produce a confidence interval can be non-negligible even when the number of patients in both comparison groups is not small. Finally, we find that an interval estimator using a simple ad hoc procedure of combining two interval estimators with and without a logarithmic transformation of the MLE can consistently perform well with respect to the coverage probability even when the number of patients per treatment is not large. In fact, this estimator uniformly outperforms all the other estimators considered here and thereby is recommended for general use. We include an example regarding the study of vitamin A supplementation to reduce the mortality among preschool children to illustrate the use of interval estimators discussed in this paper.