S14.2: Simultaneous confidence sets and confidence intervals for multiple ratios

Abstract

For the two-sample scale problem we propose an adaptive test which is based on four rank tests, the tests of Klotz and Mood and two new ones which have high power for short-tailed and right-skewed distributions, respectively. The proposed adaptive test uses the concept of Hogg (1974), i.e.,first,to classify the unknown distribution function with respect to two measures, one for skeweness and one for tailweight and second,to apply an appropriate test for this classified type of distribution. Two parametric tests, the F-Test and the test of Levene (1960), are included in our study. If we assume normality of the data the F-test is the uniformly most powerful unbiased test for one-sided alternatives. Normality, however, appears to be the exception rather than the rule and it is well known that the F-test is extremely sensitive with respect to level under nonnormality, see Tiku et al. (1986) and Büning (1991). There are a lot of robustified versions of the F-test, e.g. the tests of Levene (1960) and Brown and Forsythe (1974). But these tests are generally not robust, too, as it will be shown. We investigate the actual levels of the tests considered and compare the power of the two-staged adaptive test with the four rank tests in the adaptive scheme and with two further rank tests,the Ansari-Bradley test and a new test for long tails. The power comparison is carried out via Monte Carlo simulation assuming four symmetric distributions with increasing tailweight, the uniform, normal, logistic and t-distribtuion with 3 degrees of freedom, as well as four distributions skewed to the right with increasing strength of skewness, the chi-square distribution with 3 degrees of freedom, the exponential and two lognormal distributions. It turns out that, on the whole, the adaptive test is the best one for the broad class of distributions considered. References:Brown, M.B. and Forsythe, A.B. (1974).The small sample behavior of some statistics which test the equality of several variances.Technometrics 16, 129-132.Inferences concerning ratios of means or ratios of regression coefficients frequently appear in clinical trials. Applications include (i) relative potency estimation in drugs compared with standard (ii) multiple assays and (iii) ratio-based inference for noninferiority and superiority trials. In the simplest case of only one ratio parameter, Fieller's theorem can be used to construct a confidence interval for the single ratio. For multiple ratios, there is no method available to construct simultaneous confidence intervals that guarantee the pre-specified familywise confidence level. Some of the methods in use, namely, Bonferroni and Scheffe are conservative. In this talk, first we consider exact simultaneous confidence sets based on the multivariate t-distribution. Two equivalent approaches of determining the exact simultaneous confidence sets are outlined. Second, approximate simultaneous confidence intervals based on the multivariate t-distribution with estimated correlation matrix and a resampling approach are discusse...