Exact Tests That Recover Interblock Information in Balanced Incomplete Blocks Designs

Consider a balanced incomplete blocks design in which treatment effects are fixed and blocks are random. Assume that the number of blocks is greater than I, the number of treatments. The objective is to test the null hypothesis that there are no differences in treatment effects. Other null hypotheses are that v treatment contrasts are 0, where v may be any number from 1 to (I − 1). The usual exact test for no treatment effects is the F test, based entirely on intrablock estimates. A long-standing problem has been to find good exact tests that use both intrablock and interblock estimates (e.g., see Houtman and Speed 1983, p. 1078; Scheffé 1959, p. 175). In this study such exact tests are found. Furthermore, they are easy to implement and perform much better than the usual F test.