Stepwise Con dence Intervals without MultiplicityAdjustment for Dose Response and Toxicity

Not all simultaneous inferences need multiplicity adjustment. If the sequence of individual inferences is predeened, and failure to achieve the desired inference at any step renders subsequent inferences unnecessary, then multiplicity adjustment is not needed. This can be justiied using the closed testing principle to test appropriate hypotheses which are nested in sequence, starting with the most restrictive one, but what hypotheses are appropriate may not be obvious in some problems. We give a fundamentally diierent, conndence set-based, justiication by partitioning the parameter space naturally and using the principle that exactly one member of the partition contains the true parameter. In dose response studies designed to show superiority of treatments over a placebo (negative control) or a drug known to be eecacious (active control), the conndence set approach generates methods with meaningful guarantee against incorrect decision while previous applications of the closed testing approach have not always. Application of the conndence set approach to toxicity studies designed to show equivalence of treated groups with a placebo is also given. Suppose data Y has a distribution determined by the parameter 2 ; the parameter space, vector of mean eeects of k treatments, where 1 is the mean of the control. Then in one-sided comparisons with the control where signiicant diierence inference is of interest (as in dose response studies, for example), the desired inferences are i > 1 + where deenes practical signiicant diierence, so c i = f i ? 1 > g, i = 2; : : : ; k: In multiple comparisons with the control where practical equivalence inference is of interest (as in toxicity studies, for example), if i and 1 can be considered practically multiple comparisons do not need multiplicity adjustment in some situations. One such situation occurs when it is desirable to give the inferences in a speciied order, and failure to achieve the desired inference at any step renders subsequent comparisons unnecessary.