245 – Planning and Analysis of Noninferiority Studies
Inferiority Index and the Behrens-Fisher Problem for Noninferiority Trials
George Y H Chi
Janssen Pharmaceuticals R&D
The classical Behrens-Fisher problem poses the question regarding the distribution of the test statistics for the null hypothesis defined by the mean difference when the variances are not equal under normality. The Behrens-Fisher distribution for the test statistic that is defined as the observed mean difference divided by the square root of the sum of the sample variances divided by their respective sample size has been derived [see Kim and Cohen (1998)]. Welch (1938) proposed an approximate t-test and derived its distribution. Dannenberg et al (1994) derived the corresponding extended Behrens-Fisher distribution for the test statistic under heterogeneity of variances for the equivalence hypothesis with a pre-specified equivalence margin for bioequivalence trials. In this paper, we will apply the theory of inferiority index under normal distributions to derive an extended Behrens-Fisher distribution assuming heterogeneity of variances under the inferiority null of a non-inferiority trial for the relative difference measure where the non-inferiority margin is actually a function of the standard deviation of the control distribution defined at a specified level of the inferiority index. Based on this extended Behrens-Fisher distribution, we can then derive the corresponding Welch approximate t-test and its associated distribution. Further improvement of this extended Welch approximate t-test can be made analogous to the improvement made for the classical Welch distribution by Bhoj (1993).
