A Unified Theory for Quantal Responses to Mixtures of Drugs: The Fitting to Data of Certain Models for Two Non-Interactive Drugs with Complete Positive Correlation of Tolerances

Abstract

SUMMARY Three mathematical models for the joint action of two drugs are considered, each model being a special case of a more general model for the joint action of drugs showing complete positive correlation of tolerances. The models are likely to be of most use in the analysis of responses to drugs acting similarly. The maximum likelihood procedures for fitting the models to data, and for calculating errors, are discussed. In general these procedures consist of iterative operations whereby corrections to initial estimates of the parameters are calculated; the computations can be done with a good desk calculating machine. Numerical examples are given. In the preceding paper of this series, Hewlett and Plackett [1959] constructed very general models for the quantal responses to mixtures of drugs operating non-interactively 'on biological material, giving fully the biological basis for the models. The models take into account the degree of similarity between the biological effects of the two drugs, the relations between the doses of the drugs and the amounts acting, and the bivariate distribution of tolerances of the individual organisms to the two drugs. The fitting of these models in general would be complex, requiring an electronic computer, but there are certain useful special cases for which a good desk calculating machine can suffice. It is these special cases that we are concerned with here. They are likely to be useful mainly in the analysis of data for responses to drugs acting similarly. All the models measure quantal response in terms of the normal equivalent deviation (N.E.D.), defined as follows. If p is the proportion of organisms responding to a drug and 4(x) is standard normal cumulative distribution function, then the N.E.D. corresponding to p is