An Approach to the Analysis of Data for Semi-Quantal Responses in Biological Assay

In a typical biological assay a population of living organisms is exposed to varying quantities of the material under investigation and the reaction of the various individual members of the population is subsequently correlated with the applied dose. The methods of statistical analysis used to handle data of this kind may be divided into two broad classes, depending on whether the reaction of the individual organism is expressed in quantitative or quantal terms. In certain circumstances it is apparent that the underlying reaction is essentially continuous, although owing to difficulties of observation it can only be measured in terms of a quantal response. When these conditions apply, the observed quantal response may be defined by one or more subdivisions of the underlying reaction scale. A single subdivision corresponds to a binomial response, the subject being classified as or responding according to whether or not the underlying reaction is in excess of some specified value. Two or more subdivisions correspond to a multinomial or semi-quantal response. The problem of relating an observed quantal response to an underlying quantitative reaction has been considered briefly by Finney 1952 and, in more detail, by Hewlett and Plackett 1956 who show that the quantal dosage-response relationship may be derived from the corresponding quantitative dosage-response relationship. The ideas put forward by Hewlett and Plackett, though plausible, have not, however, been verified experimentally in any particular bioassay situation. Aitchison and Silvey 1957 describe a method of analysis for multiple response data, but these authors do not consider the problem from the point of view of an underlying quantitative reaction. The purpose of this paper is to examine the application in reverse of some of the results given by Hewlett and Plackett, first to determine