Page's statistic in balanced incomplete block designs

Abstract

A block is a collection of experimental units that are as nearly alike as possible relative to the extraneous variable. Each treatment is then randomly assigned to one experimental unit within each block. If the experimental units within blocks are relatively alike and units in different blocks are relatively different, then the randomized complete block design is usually more sensitive to differences in treatment means than the one-way classification design, a design which assumes all experimental units are relatively homogeneous.

Often, one may not be able to run all of the treatments in each block. Possible reasons may be due to shortages of experimental units, the physical size of the blocks, or that the cost is too great to use a complete block design. Assuming a complete block design cannot be used, the designer must turn to alternative methods. One popular choice is the randomized incomplete block design, a design which allows for analyzing treatment effects without running every treatment within each block. By assigning treatments in a balanced manner among the experimental units in a block, accurate analysis of treatment effects can be accomplished while reducing the number of treatment runs needed in each block.

When performing a normal-theory F test to analyze the treatment effects, the experimenter must assume that the error variables are normally distributed. However, a design could occur in which the normality assumption is invalid, and the designer may wish to use a distribution-free procedure. Nonparametric methods for determinig differences in treatment effects have been proposed for complete randomized block designs by Friedman [3] and for balanced incomplete block designs by Durbin [2]. An analyst may be interested in detecting some specific relationship among the treatment effects. In particular, one may be interested in the simple order alternative, which is useful for testing treatment effects versus a control. For the complete block design, a statistic for the simple order alternative was proposed by Page [5]. We intend to use Page's statistic with incomplete block designs while creating exact distributions for small designs and simulated distributions for larger designs. In the absence of exact or simulated tables, the normal approximation can be used to make a decision with respect to the rejection of the null hypothesis.