AbstractIf the variables in MANOVA problem can be arranged according to the order of their importance, then J. ROY'S(1958) step‐down procedure may be more appropriate than the conventional invariant inference techniques. However, it may often be possible only to identify subsets such that variables within subsets are equally important and subsets are of unequal importance. In experimental situations, it is common to have a set of variables of primary interest and another of “addon” variables. The step‐down reasoning is extended to such cases and a set of simultaneous confidence bounds based upon the procedure which uses the largest root criterion at each stage are derived. The confidence bounds are on all linear functions of means only that do not involve nuisance parameters, and are therefore suitable for studying the configuration of means. This method yields shorter intervals for contrasts among the means of the variables of primary interest compared with the conventional intervals based upon the largest root. The method is illustrated using BARNARD'Sdata (1935) on skull cha