Wald tests for variance-adjusted equivalence assessment with normal endpoints

Equivalence tests may be tested with mean difference against a margin adjusted for variance. The justification of using variance adjusted non-inferiority or equivalence margin is for the consideration that a larger margin should be used with large measurement variability. However, under the null hypothesis, the test statistic does not follow a t-distribution or any well-known distribution even when the measurement is normally distributed. In this study, we investigate asymptotic tests for testing the equivalence hypothesis. We apply the Wald test statistic and construct three Wald tests that differ in their estimates of variances. These estimates of variances include the maximum likelihood estimate (MLE), the uniformly minimum variance unbiased estimate (UMVUE), and the constrained maximum likelihood estimate (CMLE). We evaluate the performance of these three tests in terms of type I error rate control and power using simulations under a variety of settings. Our empirical results show that the asymptotic normalized tests are conservative in most settings, while the Wald tests based on ML- and UMVU-method could produce inflated significance levels when group sizes are unequal. However, the Wald test based on CML-method provides an improvement in power over the other two Wald tests for medium and small sample size studies.