Power and sample size calculations for linear hypotheses associated with mixtures of many components using fixed-ratio ray designs

Response surface methodology, often supported by factorial designs, is the classical experimental approach that is widely accepted for detecting and characterizing interactions among chemicals in a mixture. In an effort to reduce the experimental effort as the number of compounds under study is increased, ray designs have been proposed to study combinations of chemicals. When interest is restricted to relevant mixing ratios, we are only interested in making inference along the specific rays of interest, as opposed to methods which use designs that require more experimental effort to support the estimation of a response surface over a broader experimental region. Methods have been developed for the test of additivity along multiple fixed-ratio rays. Of primary importance is the detection of interactions with reasonable power. The objective of this paper is to address power and sample size issues related to the hypothesis of no interaction.