23. Rank-sum test. Let YI , . . . , YN be independently distributed according to the binomial distributions b(p;, n;), i = 1, . . . , N, where 1 P;= 1 + e-ca+!JXj) . This is the model frequently assumed in bioassay, where X; denotes the dose, or some function of the dose such as its logarithm, of a drug given to n, experimental subjects, and where Y; is the number among these subjects which respond to the drug at level x.. Here the x, are known, and a and fJ are unknown parameters . (i) The joint distribution of the Y's constitutes an exponential family, and UMP unbiased tests exist for the four hypotheses of Theorem 3, concerning both a and fJ .