Fig. 2.

Comparison of cumulative and independent models. With a logistic dose–response, the toxicity of repeated challenges may be higher (a) or lower (b) for the cumulative model. Probabilities of infection are for one challenge (green, cumulative model) and two (blue), four (red), and six (brown) challenges under the independent model. In a, the dilution required to reduce toxicity from 50% to 20% is 2 (logistic parameters: α = 0, β = 2); in b the dilution factor is 10 (α = 0, β = 0.6). (c) Total sample size in each group (bars) to distinguish the cumulative and independence hypotheses (80% power, 5% significance level) and the probability of infection with the independence model (lines) for three scenarios. Dilution factors are 4 (blue), 2 (red, independence model less toxic), and 10 (gray, independence model more toxic). The specific dose for each value of the required dilution factor was chosen to minimize the total sample size required (see text; e.g., the dose for the red scenario is the single-challenge ID₅₀).