Generalizations of Steel's Treatments-Versus-Control Multivariate Sign Test
将Steel的多元符号检验推广到允许对照组在每区组中出现多次、处理组出现次数不等的情形，提出符号-处理-对照组平衡设计，并给出渐近最优设计及同时置信区间。
Abstract Steel (1959) introduced a nonparametric multivariate sign test to allow one to compare p ≥ 2 test treatments simultaneously with a control or standard treatment in a randomized-block setting. Steel proposed using blocks of size p + 1 composed of one observation from each treatment. Intuitively, it seems that the control should be observed more often that a test treatment, because the control is involved in all treatment-control comparisons and an individual test treatment is involved in only one. Studies in the corresponding parametric analyses bear out this intuition. In this article Steel's multivariate sign test is generalized to allow the control to appear more than once in each block and to allow the test treatments to appear an unequal number of times in each block. A class of block designs known as sign-treatment-control balanced designs is proposed. This class is a counterpart of the classes of supplemented balance designs and balanced-treatment incomplete block designs that have been popular for parametric treatment-control comparisons. Using a modification of Pitman asymptotic relative efficiency appropriate for simultaneous inference, an asymptotically optimal design in this new class is found. Asymptotically optimal designs for the parametric and nonparametric analyses are shown to differ. Nonparametric simultaneous confidence statements for all treatment-control differences are presented.