Rank-sum test. Let Y1,..., YN be independently distributed according to the binomial distributions b(pi, ni),i = 1,..., N where pi = 1 1 + e−(α+βxi) .

This is the model frequently assumed in bioassay, where xi denotes the dose, or some function of the dose such as its logarithm, of a drug given to ni experimental subjects, and where Yi is the number among these subjects which respond to the drug at level xi . Here the xi are known, and α and β 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 4.4.1, concern both α

and β.

(ii) Suppose in particular that xi = i, where  is known, and that ni = 1 for all i. Let n be the number of successes in the N trials, and let these successes occur in the s1st, s2nd,...,snth trial, where s1 < s2 < ··· < sn. Then the UMP unbiased test for testing H : β = 0 against the alternatives β > 0 is carried out conditionally, given n, and rejects when the rank sum n i=1 si is too large.

(iii) Let Y1,..., YM and Z1,..., ZN be two independent sets of experiments of the type described at the beginning of the problem, corresponding, say, to two different drugs. If Yi is distributed as b(pi, mi) and Z j as b(πj, n j), with pi = 1 1 + e−(α+βui) , πj = 1 a + e−(γ+βvj) , then UMP unbiased tests exist for the four hypotheses concerning γ − α and δ − β.