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

Abstract

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, $p$-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.

Introduction

Dose-response studies play an important role in many areas of applications. Particularly in the context of pharmaceutical drug development the assessment of a dose related effect and the subsequent estimation of a target dose is of central importance. Clinical dose response studies are designed to investigate the effect of a certain drug at several doses, possibly in comparison with a placebo group. Current clinical practice typically aims at two major goals when performing a dose-response study. First, one is interested in assessing an overall dose related trend. The goal is thus to ensure that with increasing dose the effect to increase (or decrease) as well. Once such an overall trend has been established, the second goal is to estimate a target dose of interest. One common target dose is the minimum effective dose, i.e., the smallest dose revealing a statistically significant and clinically relevant effect (Ruberg, 1995).

Both the problem of testing on an overall dose related effect as well as the estimation of a target dose may require the application of trend tests. The likelihood ratio test of Bartholomew (1961) is a common and powerful approach, although the numerical tractability of its distribution function reduces its application to balanced one-way layouts and other simple designs (Robertson et al., 1988). Single contrast tests are a popular alternative (Ruberg, 1989, Tamhane et al., 1996). However, these tests are potentially less powerful than competing methods if the true dose-response shape deviates substantially from the vector of contrast coefficients (Stewart and Ruberg, 2000, Hothorn et al., 1997). Another widely used test on trend in the one-way layout is due to Williams (1971). The Williams test compares the highest dose group with the placebo group in a two-sample $t$-test fashion by replacing the usual arithmetic mean of the highest dose group with the corresponding maximum likelihood estimate (MLE) under the given order restriction. This approach has been generalized to a variety of situations, including non-parametric rank-based methods (Shirley, 1977, Williams, 1986), applications to binomial data (Williams, 1988) and robustifying approaches (Hothorn, 1989, Tsai and Chen, 2000). Williams (1971) also considered a modification by additionally replacing the arithmetic mean of the placebo group with the corresponding MLE. Marcus (1976) investigated this test in detail and concluded from a simulation study that the modified Williams test is more powerful for many parameter constellations. Both the Williams and the modified Williams tests are restricted in their applications, however, since the distribution functions of their test statistics are difficult to handle. The PROBMC function in SAS, for example, provides the Williams test only for balanced one-way layouts. Kuriki et al. (2002) described an approach to compute the distribution function of the modified Williams test, again restricted to one-way layouts. Thus, both tests remain unavailable for unbalanced designs taking covariates and/or factorial treatment structures into account.

This paper aims at bridging this gap. It presents an extension of both tests to general unbalanced linear models allowing the inclusion of covariates and factorial treatment structures. To simplify the representation, the paper first focuses on the Williams test. To this end, Williams’ test is rewritten as a maximum of several, individually standardized test statistics. This representation allows a deeper insight into the behavior of the test. In particular, we will show that the regular $t$-tests used by Williams can be improved by applying a complete studentization. We use recent numerical results to compute accurately the joint distribution of the resulting individual test statistics for generally correlated means, thus allowing the application of Williams’ test in general linear models. Moreover, sample size determination can be performed prior to a designed experiment at pre-specified type I and type II error rates. We also apply the previous methods to the modified Williams test, thus extending it in a similar way to unbalanced designs in general linear models. The methods are illustrated with data examples and the resulting approaches are compared with each other in a power study.