Inference of equivalence for the ratio of two normal means with unspecified variances

Equivalence trials aim to demonstrate that new and standard treatments are equivalent within predefined clinically relevant limits. We focus on when inference of equivalence is made in terms of the ratio of two normal means. In the presence of unspecified variances, methods such as the likelihood-ratio test use sample estimates for those variances; Bayesian models integrate them out in the posterior distribution. These methods limit the knowledge on the extent to which equivalence is affected by variability of the parameter of interest. In this article, we propose a likelihood approach that retains the unspecified variances in the model and partitions the likelihood function into two components: F-statistic function for variances, and t-statistic function for the ratio of two means. By incorporating unspecified variances, the proposed method can help identify a numeric range of variances where equivalence is more likely to be achieved, which cannot be accomplished by current analysis methods. By partitioning the likelihood function into two components, the proposed method provides more inference information than a method that relies solely on one component. Using a published set of real example data, we show that the proposed method produces the same results as the likelihood-ratio test and is comparable to Bayesian analysis in the general case. In a special case where the ratio of two variances is directly proportional to the ratio of two means, the proposed method yields better results in inference about equivalence than either the likelihood-ratio test or the Bayesian method. Using a published set of real example data, the proposed likelihood method is shown to be a better alternative than current analysis methods for equivalence inference.