Ludwig A. Hothorn and Frank Bretz
The analysis of dose–response relationships is an important objective in toxicology, and one in which both modelling and testing approaches are used. One particular question is whether a threshold exists at low doses. The concept of a pragmatic threshold is used, i.e. low doses with biologically unimportant effects are assumed to be threshold doses. “Biologically unimportant” means, in statistical terms, a lower effect than the effect of the negative control, or at least a just-tolerable margin δ higher than the effect of the negative control. Therefore, threshold doses can be tested in terms of a one-sided hypothesis of equivalence. A new approach is proposed, assuming, at the least, that the low dose is a threshold dose, and the highest dose is superior to the negative control. By analogy to the k-fold rule commonly used in mutagenicity studies, tests on ratio-to-control are used. The a priori definition of the threshold margin is inherently needed. A further approach proposes the analysis of dose–response relationships by means of order-restricted inference (the so-called trend test). A modification of a multiple-contrast test is used, in which only those contrasts are included that are sensitive for no effects at low doses. A further modification treats the complicated, but real, problem of simultaneous existence of a threshold, a monotonic increase, and a downturn effect at high dose(s). A parametric procedure is considered, together with an extension for proportions. The important problem of a priori sample size definition is discussed. The approaches are demonstrated by means of examples based on real data.
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