Sigmoidally Constrained Maximum Likelihood Estimation in Quantal Bioassay

Abstract

Abstract In quantal bioassay, doses x 1, …, xk of a test substance are administered to n 1, …, nk animal subjects. Let II(x) denote the probability that an arbitrary subject responds in a specified way to a dose x. In this article we consider maximum likelihood estimation of II(x 1), …, II(xk ) subject to the constraint that II is sigmoid. Of particular interest is the nature of II at very small x. Upper confidence bounds for II(x), x ≤ x 1, are discussed. These confidence bounds depend on an inequality that holds when II is convex on [0, x 1]. A test of concavity against the alternative under sigmoidicity, and related tests are discussed. Rejecting concavity requires evidence that II is convex on [0, x 1].