Multiple Range Tests for Correlated and Heteroscedastic Means

Miultiple range tests have been developed by several writers, for example, D. Newmiian 8, M. Keuls 5, J. W. Tukey 10 anid D. B. Duincani 3, for testinig differenices between several treatmenlt means in cases inl which all such differences are of equal a priori interest. These tests, which are also described in recent textbooks, for example, W. T. Federer 4, chapter 2, have been worked out for data in which the treatment means are homoscedastic (have equal variances) and are uncorrelated. Recently, C. Y. Kramer 6 has presented a simple miethod for extending these procedures to give useful tests for differences between means with unequal replications, the method being applicable to aniy set of heteroscedastic unicorrelated means. In a subsequent paper 7, the same authorl has given further extensions to tests of mneanis which are also correlated, such as the adjusted means from analyses of covariance or from incomplete block designs. Similar work has also been done by E. Bleicher 1 and P. G. Sanders 9 in extendinig a multiple F test to making tests in lattice and rectangular lattice designs. One purpose of this paper is to present a more complete method for these extensions which lnecessarily sacrifices a little in simplicity but is more powerful, especially in cases in which the differences between the mneans have appreciably different varianices. Another purpose is to incdicate briefly the closeniess of the properties of these complete tests of heteroscedastic and correlated meanis to those of the corresponding tests of homoscedastic and unicorrelated mieans. Incidcental to these main ptirposes, a shor t-cut skippinig priniciple, useful in applyilng multiple ranige tests to a large lntum-ber of treatmenit means (or totals), is also presented.