Likelihood inference on the relative risk in split-cluster designs

Abstract

Split-cluster experiments are being used by investigators in health sciences when naturally occurring aggregate of individuals with nested sub-groups may be assigned to different treatments. Cited examples include the split-mouth trials, in which a subject's mouth is divided into two segments that are randomly assigned to different treatment groups. In other situation, randomization to treatment conditions may be possible at the person level within the cluster. In this case, when the treatment conditions are available within each cluster, the design is referred to as a multisite or split-cluster design (SCD). The major attractiveness of this design is that it removes a large portion of the inter-subject variation from the estimates of the treatment effect; hence, it has the potential to require lesser number of measurements than a parallel arm design with the same power. When the response variable of interest is binary, statistical methods developed to evaluate the effect of interventions depended on nonparametric methods. Though these methods are simple to apply, they are known to be less efficient. Taking the relative risk (RR) as an effect measure, we construct a bivariate-correlated model under which a score test is applied to test H(0): RR = 1.0. Moreover, we construct Wald- and Fieller-based confidence intervals on RR. Since the efficiency of SCD increases when the interclass correlation coefficient (ρ₁₂) is high, we present a goodness-of-fit procedure for testing H(0):ρ₁₂ = 0, which may be helpful in choosing a design for a future study. For illustrating the proposed methodology, we consider two application data from the published literature; the first from a split-mouth trial on 23 patients evaluating the effect of chlorhexidine in the treatment of gingivitis, and second from study of mental health (depression and anxiety) as outcome measure obtained on 173 patients evaluated by two screening instruments. Moreover, we discussed the efficiency gained using our approach in these design settings. The likelihood approach makes more assumptions as compared to previous approaches that have been described. We have developed a bivariate beta-binomial model, from which we can conduct a full likelihood statistical inference. Based on this model, we may construct Wald's confidence intervals and score tests, which are known to possess optimal statistical properties. For the purpose of comparison with nonparametric methods, we constructed the Fieller's confidence interval.