Incomplete Block Designs for Comparing Treatments with a Control: General Theory

Abstract

In this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control. This class of designs is appropriate for comparing simultaneously p > 2 test treatments with a control treatment (the so-called multiple comparisons with a control (MCC) problem) when the observations are taken in incomplete blocks of common size k < p + 1. For this problem we propose a new general class of incomplete block designs that are balanced with respect to (wrt) test treatments. We shall use the abbreviation BTIB to refer to such designs. We study their structure and give some methods of construction. A procedure for making exact joint confidence statements for this multiple comparisons problem is described. By using a new concept of admissibility of designs, it is shown how "inferior" designs can be eliminated from consideration, and attention limited to a small class of BTIB designs that can be constructed from so-called generator designs in the minimal complete class of such designs. Some open problems concerning construction of BTIB designs are posed.

Robustness to the unavailability of data in block designs

“…Reinforced block designs were first discussed by Das (1958), and the idea can be extended to BBDs. Another type is the balanced treatment incomplete block design (BTIBD), first discussed by Bechhofer and Tamhane (1981). In these designs the test treatments are compared to the control in blocks of constant size k < v + 1, and the plots for the test treatments make a pairwise balanced block design (call it Dp, possibly with blocks of size 1).…”

Some optimal designs for comparing a set of test treatments with a set of controls

“…In fact, a good many results have been obtained in recent years for comparing a set of test treatments to a single control (see e.g., Bechhofer and Tamhane (1981), Majumdar and Notz (1983), Majumdar (1984, 1985), Notz (1985), Jacroux (1986Jacroux ( , 1987aJacroux ( , 1987bJacroux ( and 1989, Stufken (1987) and Cheng et al (1988)). For a more detailed summary of the known results on this topic, the reader is referred to Hedayat et al (1988).…”

On bounds for the efficiency of block designs for comparing test treatments with a control

“…If we call this solution t 1 we know that g(t,O) is decreasing for 0 < t < t 1 and increasing for t 1 < t < k. From g(toO" < g(t 0 -l,0) it follows that t > t O -1, while g(to,O) < g(t0+l,0) implies that t I < t + 1.…”