Confidence Bounds Connected with Anova and Manova for Balanced and Partially Balanced Incomplete Block Designs

Abstract

It is well known [3, 4] how, in the case of any general strongly testable [5] linear hypothesis for either ANOVA or MANOVA one can put simultaneous confidence bounds on a particular set of parametric functions, which might be regarded as measures of deviation from the "total" hypothesis and its various components. The parametric functions are such that, in each problem, one of these can be appropriately called the "total" and the rest "partials" of various orders. For each problem the "total" function, (i) in the univariate case, is related to, but not quite the same as, the noncentrality parameter of the usual $F$-test of the "total" hypothesis in ANOVA, and (ii) in the multivariate case, is the largest characteristic root of a certain parametric matrix which is related to, but not quite the same as, another parametric matrix whose nonzero characteristic roots occur as a set of noncentrality parameters in the power function for the test (no matter which of the standard tests we use) of the "total" hypothesis in MANOVA. The same remark applies to "partials" of various orders considered in the proper sense. In this note, for both ANOVA and MANOVA, the hypothesis considered is that of equality of the treatment effects--vector equality in the case of MANOVA. Starting from such a hypothesis, explicit algebraic expressions are obtained for the total and partial parametric functions that go with the simultaneous confidence statements in the case of both ANOVA and MANOVA and for balanced and partially balanced designs. It is also indicated how to obtain, in a convenient form, the algebraic expression for the confidence bounds on each such parametric function, without a derivation of these expressions in an explicit form.