Exact Confidence Bounds for all Contrasts of Three or More Regression Lines

Abstract

Abstract It is desired to compare κ ≥ 3 treatments. Under the assumption of iid normal errors, it is well known that the Scheffé method produces exact simultaneous confidence bounds for all contrasts of the treatment means. Furthermore, it is known that the Scheffé method is conservative when one desires confidence bounds for a specific subset of contrasts of means. Exact methods, such as those due to Tukey and Dunnett, yield tighter bounds than the Scheffé method for specific subsets of contrasts of means. In this article, multiple comparisons of the κ treatments are done not in terms of their means, but rather in terms of a parametric function. The parametric function of interest is the simple linear regression model, E(Y|x). It is desired to find simultaneous confidence bounds for all contrasts of the κ simple linear regression models. Although the Scheffé method can be used to find such bounds, this is extremely conservative. The union-intersection method is used to develop simultaneous confidence bounds for these contrasts under the assumption of equal design matrices for each treatment. The method is based on a pivotal quantity whose distribution function is a linear combination of F distribution functions. Thus probability points can be computed using standard computing packages. The Scheffé bounds are about 5% wider than the exact bounds for κ = 3 and about 13% wider for κ = 6.