Minimum X2and Maximum Likelihood Solution in Terms of a Linear Transform, with Particular Reference to Bio-Assay

Abstract

An old and commonly used device for fitting a curve y = F(x, α, β) is to find a function of y which is linearly related to x, or a function of x which is linearly related to y, or functions of y and x which are linearly related to each other. The linearly related functions are plotted against one another, and a line is fitted to the points, usually by eye, sometimes by “least squares” in terms of the transforms. Texts dealing with empirical curve-fitting characteristically use this scheme [6]. Before Bliss and Fisher [2], no attempt was made to adjust the transforms systematically in order to achieve a fit that fulfilled defined criteria in terms of the original measures y and x, nor was it known that such adjustments were possible. These authors presented for the bio-assay experiment a method in terms of probits, which as shown by Garwood, [5] accomplished a maximum likelihood estimate of the integrated normal curve, when the observations were distributed binomially. Following the procedures used by Bliss and Fisher for the integrated normal curve, I [1] formulated the similar adjustments applicable to “logits” for a maximum likelihood solution of the logistic function, but did not advocate using this solution. Finney [4] has recently presented the adjustments for a maximum likelihood solution of several functions of interest in bio-assay. So far as I know the similar method for a solution fulfilling the criteria of minimum X² rather than maximum likelihood has not been presented. In the following paper this is given for several functions in common statistical use, and at the same time a résumé is given of the maximum likelihood adjustments for the same functions.