An extension of the Williams trend test to general unbalanced linear models

The trend test of Williams is one of the most common approaches to test for monotone dose-response relationships. It has originally been introduced for one-way layouts. Applications of the Williams test beyond balanced block designs have not been described yet, what severely restricts its use in practice. An extension of Williams’ test to general unbalanced linear models is presented which allows the inclusion of covariates and/or factorial treatment structures. The test statistic of Williams is rewritten as the maximum of a finite number of standardized linear combinations of the means. The associated distribution function is seen to be available in closed form. Accordingly, pp-values/critical values and power/sample sizes can be computed thoroughly. The methods are also applied to a modified version of Williams’ test. Data examples and numerical comparisons suggest a good performance of both extended approaches when compared to current competing methods.