A ratio‐to‐control Williams‐type test for trend

Abstract

Under certain conditions, many multiple contrast tests based on the difference of treatment means can also be conveniently expressed in terms of ratios. In this paper, a Williams test for trend is defined as ratios-to-control for ease of interpretation and to obtain directly comparable confidence intervals. Simultaneous confidence intervals for percentages are particularly helpful for interpretations in the case of multiple endpoints. Methods for constructing simultaneous confidence intervals are discussed under both homogeneous and heterogeneous error variances. This approach is available in the R extension package mratios. The proposed method is used to test for trend in an immunotoxicity study with several endpoints as an example.

A statistical method for estimating the no-observed-adverse-event-level

“…In the original paper of example 2 ratio-to-control tests under variance heterogeneity were used [12]. Therefore a closure test is used by simultaneous ratio-to-control tests [15] (R-Code see Appendix 2). The severity score endpoint in example 3 is analysed by a nonparametric multiple contrast test for the relative effect size [16] (p-value in brackets) and alternatively by contrast test using the most likely transformation model for count data [18] (R-Code see Appendix 3).…”

Claiming trend in toxicological and pharmacological dose-response studies: an overview on statistical methods and related R-Software

“…This makes the Williams test the recommended trend test in pharmacology/toxicology: sensitive to some monotonic and partially non-monotonic forms, a comparison to the control, easily interpretable confidence intervals for the required effect sizes (difference of proportions [20], risk ratio or odds ratio of proportions [21], ratio-to-control estimates [18], relative effect sizes [31], hazard rates [15], multiple endpoints [14], heteroscedastic error terms [16], poly-k-adjusted tumor rates [42]). Sometimes the question arises whether to use the Dunnett-test (unrestricted H 1 ) or the Williams-test (restricted H 1 )?…”

Similarity of multiple dose-response curves in interlaboratory studies in regulatory toxicology

“…In regulatory toxicology not only the difference of means is relevant as effect size [23], such as proportions or counts. Therefore, the Williams test is available for difference [10], risk ratio or odds ratio of proportions [9], ratio-to-control estimates [11], the nonparametric relative effect sizes [14], hazard rates [7], multiple endpoints [6], heteroscedastic error terms [8] and poly-k-adjusted tumor rates [20]).…”