Estimating the weights of an integration model from binary data: A discrete geometrical approach

ABSTRACT: A particular kind of data from psychophysical experiments is considered, which may be depicted as a two-colour set of points in the plane, each point recording one answer of a subject in a 2AFC task (the pair of colours corresponds to the pair of possible answers in the task). The ultimate target of scientific inference is the estimation of three parameters – two weights and one threshold – of a linear integration and dichotomization model associated with an experiment of this kind. An intermediate target of inference may be identification of a line in the plane, optimally separating the points of one colour from those of the other, since the parameters of the model can easily be computed from the coefficients of the line. The paper presents a method for finding such an optimally separating line, with special reference to the situation in which a straight line completely separating points of one colour from points of the other does not exist. The method is divided into two stages, one for finding an optimal linear bipartition (using topological information) and the other for determining a representative linear separator (using geometrical information). Concepts required to formulate and justify the method are gradually introduced and its use is illustrated on a simple example.