Biological models for the independent action of two drugs imply that if the drugs are administered in various combinations of doses, then the corresponding probabilities of response must satisfy certain inequalities if the drugs are acting independently. The hypothesis that the probabilities do satisfy these inequalities can be tested using the likelihood ratio test or using the bootstrap test proposed by Wahrendorf and Brown in 1980. In the simplest dose design, only one dosage of each drug is used. The three combination doses required to test the hypothesis are each drug singly and the combination of the two drugs. The asymptotic distribution of the bootstrap test is derived. The asymptotic distribution of the likelihood ratio test is obtained by applying Feder's results. The calculation of the asymptotic critical values and powers is presented.
"Large Sample Properties of Two Tests for Independent Joint Action of Two Drugs." Ann. Statist. 18 (4) 1634 - 1650, December, 1990. https://doi.org/10.1214/aos/1176347870