Effects of a limited class of nonlinearities on estimates of relative weights

Perturbation analyses have been applied in recent years to determine the relative contribution of individual stimulus components in detection and discrimination tasks. Responses to stimulus samples are compared to stimulus parameters to determine the details of the decision rule. Often, a linear model is assumed and it is of interest to determine the relative contribution of different stimulus elements to the decision. Here, biases in estimated relative weights are considered for the case where the decision variable is given by D = (sigma(alphaiXn(i))k)m and the stimulus components, the Xi, are normally distributed, of equal variance, and mutually independent. The alphai are the "true" combination weights, and n, k, and m are positive reals. The method used to estimate relative weights is the correlation coefficient between the Xi and the observer's responses. Estimates of relative alphai do not depend on m but may depend on the mean values of the Xi and the values of n and k (a dependence on the variance, sigmai2, holds even for linear transformations).