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This paper addresses the problem of deriving one-sided tolerance limits and two-sided tolerance intervals for a ratio of two random variables that follow a bivariate normal distribution, or a lognormal/normal distribution. The methodology that is developed uses nonparametric tolerance limits based on a parametric bootstrap sample, coupled with a bootstrap calibration in order to improve accuracy. The methodology is also adopted for computing confidence limits for the median of the ratio random variable. Numerical results are reported to demonstrate the accuracy of the proposed approach. The methodology is illustrated using examples where ratio random variables are of interest: an example on the radioactivity count in reverse transcriptase assays and an example from the area of cost-effectiveness analysis in health economics., (© 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

The construction of adequate confidence intervals for adaptive two-stage designs remains an area of ongoing research. We propose a conditional likelihood-based approach to construct a Wald confidence interval and two confidence intervals based on inverting the likelihood ratio test, one of them using first-order inference methods and the second one using higher order inference methods. The coverage probabilities of these confidence intervals, and also the average bias and mean square error of the corresponding point estimates, compare favorably with other available techniques. A small simulation study is used to evaluate the performance of the new methods. We investigate other extensions of practical interest for normal endpoints and illustrate them using real data, including the selection of more than one treatment for the second stage, selection rules based on both efficacy and safety endpoints, and the inclusion of a control/placebo arm. The new method also allows adjustment for covariates, and has been extended to deal with binomial data and other distributions from the exponential family. Although conceptually simple, the new methods have a much wider scope than the methods currently available., (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

A conditional likelihood-based approach is proposed to construct confidence intervals for the parameters of interest in a two-stage design with treatment selection after the first stage. Both a Wald confidence interval and a confidence interval based on inverting the likelihood ratio test are proposed. The operating characteristics of these confidence intervals: the coverage probabilities and average confidence interval lengths, as well as the average bias and mean-square error of the corresponding point estimates, compare favorably with other available techniques. Possible extensions and an alternative unconditional approach based on the likelihood with missing at random mechanism are also briefly described.

The problem of constructing a confidence interval for the ratio of two regression coefficients is addressed in the context of multiple regression. The concept of a Generalized Confidence Interval is used, and the resulting confidence interval is shown to perform well in terms of coverage probability. The proposed methodology always results in an interval, unlike the confidence region generated from Fieller's theorem. The procedure can easily be implemented for parallel-line assays, slope-ratio assays, and quantal assays under a probit model. Furthermore, this approach can also be extended to compute confidence intervals based on data from multiple bioassays. The results are illustrated using several examples.