A cautionary note on the use of the adaptive up-down method

Up-down staircases with equal sizes for the steps up and down are widely used to estimate detection and discrimination thresholds in psychoacoustics, but the conventional average-of-reversals estimator does not converge on its presumed percent point in Yes-No tasks or in two-alternative forced-choice detection tasks. The particular percent point of convergence is partly determined by the relative size of the steps with respect to the spread (inverse of slope) of the underlying psychometric function. In particular, threshold is increasingly underestimated as the spread of the psychometric function decreases. This characteristic may have serious consequences when thresholds estimated via up-down staircases are compared across conditions in which the spread of the psychometric function varies, because then these thresholds do not represent comparable measures of performance. This paper documents the misbehavior of the average-of-reversals estimator under up-down rules and types of forced-choice task that are in common use in psychoacoustics but which have not been studied before in simulations. It is also shown that a relatively simple modification of the up-down design (namely, using steps up and down of different size and in a certain ratio depending only on the task and the up-down rule being used) stabilizes the performance of these staircases.