Abstract

Quantal biological assays are traditionally performed on grouped data. However, grouping disguises the fundamental all-or-none nature of the responses, is not necessary and may not be possible. The statistical method relevant to quantal assays is reconsidered from the viewpoint of the individual responses and with a view to getting the apparatus for their explicit analysis. Specifically, the logistic function is used as a model for the distribution of the probability of responding as a function of dose or concentration. The parameters of the logistic function are estimated by the method of maximum likelihood. Since the function is nonlinear in both the scale and location parameters, solution of the normal equations is achieved by an iterative technique base on a Taylor series expansion. Observations are not grouped so the method is applicable to cases in which several observations are not available at the same dose level as well as to grouped data. Three variants on the basic model are considered: the classical model in which the parameters estimated are the intercept and slope of the associated linear regression and two models in which the parameter of prime pharmacological interest—the ED50—is estimated directly. In one of the latter the ED5O is considered to be symmetrically distributed on a logarithmic scale, in the other on a linear scale. Similarly, when two curves are compared, potency ratios or their logarithms are estimated directly rather than indirectly as the difference or ratio of two variables. One advantage of such direct estimation is that the error valiance can then be obtained directly from the covariance matric obtained during the solution of the normal equations. Numerical examples are given and illustrative computer programs are given in an appendix.