Extension of Multiple Range Tests to Group Correlated Adjusted Means

Abstract

In recent years several writers have developed multiple range tests of differences among group (e.g. treatment) means that have equal variances and zero covariances. Duncan [1] describes various multiple range tests and points out their superiority over multiple t-tests when no relation among treatments is specified. Kramer [3] extended the multiple range test to group means with unequal numbers of replications. The extension to adjusted means that are correlated will be explained here in reference to Duncan's New Multiple Range Test, although it is applicable also to the tests of Keuls, Newman, and Tukey. In Duncan's test, the difference between two means is significant if it exceeds a shortest significant range. The shortest significant range, R. , is obtained by multiplying the standard error of a mean, s1i by a value z,n,, of the studentized range, which Duncan has tabled for both the 5% and 1% levels. In Duncan's notation, n2 is the number of degrees of freedom of the error mean square and p = 1, 2, * , k is the number of treatments. Consider an experiment with five treatments, A, B, C, D, and E, each replicated r times. Suppose the ranked means are