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This paper is concerned with the estimation of the toxicity of individual poisons which may be administered singly or jointly to a population of living organisms under conditions of simple similar action, with particular reference to the analysis of uncontrolled data of the kind frequently encountered in studies of human populations. A brief description is given of the methods applied in the analysis of quantal responses to a single poison and the extension of these techniques to mixtures of poisons is examined. A model to represent the action of mixtures of poisons is derived and the application of maximum likelihood and minimum logit x2 estimation is discussed. It is shown that both procedures involve the use of an iterative process to calculate the estimates of the various parameters. With the Newton—Raphson process of successive approximation, the calculation of the minimum logit x2 estimates is, in general, less complicated than the evaluation of the maximum likelihood estimates. If, however, the individual tolerances are assumed to follow a logistic distribution the difference is marginal. In view of the close agreement between the logistic function and other functions (such as the normal) which have been proposed to describe the tolerance distribution it is concluded that, as the maximum logit x2 procedure offers no appreciable advantages (either theoretical or practical) over the maximum likelihood procedure under the assumption of a logistic distribution of tolerances, the latter method is to be preferred. The calculations involved are, however, likely to be too complex for the conventional methods of statistical computing, although the use of an automatic digital computer offers a practicable method of analysis. The application of the proposed procedures is illustrated by data relating to the prevalence of pneumoconiosis amongst coal miners at one of the collieries operated by the National Coal Board.
J. R. Ashford (Wed,) studied this question.