A New Method of Testing Hypotheses and Estimating Parameters for the Logistic Model

Abstract

Data collected in many types of research often consist of the proportion of experimental units having a specified attribute. In this paper we shall be concerned with the analysis of this type of data when it can be arranged in a multiway classification as for example, a factorial arrangement. An example of this type of data is given in Table 1. Several methods may be used in analyzing these data. Regardless of the method, a model relating the proportion having the attribute to the treatments must be assumed for a meaningful interpretation of the experimental results. The problem of the most appropriate model becomes particularly acute when some of the treatments are applied at several levels and it is desired to investigate the nature of treatment effects with regard to both main effects and interactions. If the sample sizes in the cells are equal, the observed proportions are often analyzed by the analysis of variance. A more desirable procedure is to analyze the arc sine transformation of the proportion. It is well known that this transformation stabilizes the variance if the sample sizes are equal and not too small, and, for a large class of data, it provides a unit of measurement on which treatment effects are approximately linear except at values of the proportion near zero or one. In bioassay both the logit and the probit transformations have a long history of use. With appropriate extensions these models can be used in analyzing data of the type being discussed. Since the proportion responding has been found to increase sigmoidally with increasing stimulus for many phenomena, these transformations are particularly effective in providing a scale on which treatment effects are linear. Dyke and Patterson [1952] gave an example of how the logit trans