On the bias of various estimators of the logit and its variance with application to quantal bioassay

Abstract

SUMMARY The bias of several logit estimators and their corresponding variance estimators is investigated in small samples. Their use in quantal bioassay is similarly explored. The logit transformation has been suggested in the analysis of higher dimensional contingency tables, by Woolf (1954) and many others more recently, and also in estimating the parameters of the logistic function in the quantal bioassay problem (Berkson, 1944, 1953). Various modifications of the logit have been suggested by Berkson (1953), Haldane (1955), Anscombe (1956), Tukey, mentioned by Anscombe (1956), and Hitchcock (1962). Modifications of its usual variance estimator have been proposed by Haldane, Goodman (1964) and Gart (1966). More recently Goodman, in an unpublished paper, has derived several further modifications of his estimator. In this paper we present a numerical comparison of the bias of these estimators and give conditions under which one or the other may be preferred. The use of logits in the quantal bioassay problem is briefly explored with particular reference to the asymptotic results of Hitchcock regarding the bias of the estimators.