An argument, based on expanded confidence intervals, is given for always performing one-sided tests. However, if goals are not one-sided and "follow-up" inferences are required, then the usual two-sided confidence intervals (corresponding to two-sided tests) are generally appropriate. When equivalence testing is required (the goal being to show that treatments are "not too different"), expanded confidence intervals are more efficient than Westlake's symmetrical confidence intervals.

Summary A goodness‐of‐fit test is constructed using sample quantiles for a p.d.f. F(x, θ) (where θ is to be estimated) which is continuous and strictly increasing at the quantiles of interest. It is shown to be asymptotically equivalent, under alternatives “close” to the null, to the usual goodness‐of‐fit test based on grouped observations, the advantage being that we may avoid the difficulties of choosing cell boundaries for grouping.

Step down" or "sequentially rejective" procedures for comparisons with a control are considered for both one sided and two sided comparisons. Confidence bounds (in terms of the control) are derived for those (location) parameters not in a selected set. Special results are derived for the normal distribution with unknown variance where the sample numbers are (possibly) unequal.

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SUMMARY The problem of selecting those treatments which are equivalent (or bioequivalent) to a control treatment is investigated by using single‐step, step‐down and step‐up multiple‐testing procedures. For each of the tests considered the null hypothesis is that of non‐equivalence and the hypotheses are rejected in either a single step or a stepwise fashion while controlling the family wise error rate. The single‐step procedure is based on ‘expanded’ confidence intervals, as discussed by Bofinger, the step‐down procedure is based on the method of Naik and the step‐up procedure on the method of Dunnett and Tamhane. Almost all the critical constants used in the procedures are based on tables already published.