Matrices in Quantal Analysis

Fisher (1954) has recently discussed the various transformations of probability used in the analysis of binomial data, and in that paper a full account of the statistical theory is given. While the assumption is often made that a distribution of thresholds must be postulated before efficient analysis may be made of binomial data (Finney, 1952a), Fisher (1954) has clearly demonstrated that this is unnecessary. Transformations of the expected proportion responding may thus be simply regarded as a different scale for the measurement of response. In this paper practical methods of relating the binomial variable to the coordinates of experimental designs are given, and matrix methods are employed so that the results may immediately be applied to any experiment with known design matrix. A considerable time has been devoted in the past to methods supposed to give estimation of parameters in quantal analysis. These graphical or semigraphical methods are usually employed in order to avoid efficient but tedious probit analysis in routine work. In this paper it will be shown that efficient solution is afforded by use of the angular transformation. Recently Berkson (1953) has advanced a simplified and quick method for the estimation of parameters of binomial data by means of a modified logit technique. The method is still tedious when compared with the method exemplified here, which gives estimates equivalent to those derived by Berkson's method.