Observations on a maximum likelihood method of sequential threshold estimation and a simplified approximation.

Abstract

In sequential methods of measuring thresholds, the stimuli are chosen from trial to trial with the goal that the information from each trial be highly rele­ vant to the objective. The presentation of many stim­ uli far above or far below threshold is a waste of time, because the responses to such stimuli have little bearing on the question of the location of the thresh­ old. Efficiency of threshold estimation (Taylor, 1970) can be formulated in terms of the tradeoff be­ tween the standard error of the threshold estimate and the average number of trials necessary to achieve a given small standard error. A sequential method, such as the well-known PEST (Percentile Estimation by a Sequential Threshold method) of Taylor and Creelman (1967), can be more efficient by a large factor than one of the classical methods, such as con­ stant stimuli. However, even among the various se­ quential methods that are in use, there are fairly large differences in efficiency. For example, the number of trials needed to achieve a given standard error using a maximum likelihood method (Pentland, 1980; Shelton, Picardi, & Green, 1982) is, in some cases, as small as half those needed with PEST and some vari­ ations (Findlay, 1978) of the PEST method. In the maximum likelihood method of sequential threshold estimation, a new estimate of the threshold is obtained after every trial. That estimate is then used on the following trial to determine the stimulus to be presented. In the simplest cases, the stimulus value is always taken equal to the threshold estimate itself. Thus, a powerful statistical estimation method is used to concentrate stimuli densely in the region of the threshold, to maximize the relevance of the data. That seems to explain the superior efficiency. Although the maximum likelihood method was conceived and used for theoretical purposes early in the development of sequential threshold methods (Smith, 1961; Wetherill, 1963), it has not been very popular as a practical scheme for measuring thresh­ olds. This is somewhat curious, since it seems to be more efficient than more popular ones. Some hints toward an explanation may be inferred from comThis paper was written while the author was on professional leave from Cleveland State University during part of 1984. The ideas stem from research begun while the author was supported by