A note on confidence bounds after fixed-sequence multiple tests

We are concerned with the problem of estimating the treatment effects at the effective doses in a dose-finding study. Under monotone dose–response, the effective doses can be identified through the estimation of the minimum effective dose, for which there is an extensive set of statistical tools. In particular, when a fixed-sequence multiple testing procedure is used to estimate the minimum effective dose, Hsu and Berger (1999) show that the confidence lower bounds for the treatment effects can be constructed without the need to adjust for multiplicity. Their method, called the dose–response method, is simple to use, but does not account for the magnitude of the observed treatment effects. As a result, the dose–response method will estimate the treatment effects at effective doses with confidence bounds invariably identical to the hypothesized value. In this paper, we propose an error-splitting method as a variant of the dose–response method to construct confidence bounds at the identified effective doses after a fixed-sequence multiple testing procedure. Our proposed method has the virtue of simplicity as in the dose–response method, preserves the nominal coverage probability, and provides sharper bounds than the dose–response method in most cases.