Forced-choice staircases with fixed step sizes: asymptotic and small-sample properties

Visual detection and discrimination thresholds are often measured using adaptive staircases, and most studies use transformed (or weighted) up/down methods with fixed step sizes--in the spirit of Wetherill and Levitt (Br J Mathemat Statist Psychol 1965;18:1-10) or Kaernbach (Percept Psychophys 1991;49:227-229)--instead of changing step size at each trial in accordance with best-placement rules--in the spirit of Watson and Pelli (Percept Psychophys 1983;47:87-91). It is generally assumed that a fixed-step-size (FSS) staircase converges on the stimulus level at which a correct response occurs with the probabilities derived by Wetherill and Levitt or Kaernbach, but this has never been proved rigorously. This work used simulation techniques to determine the asymptotic and small-sample convergence of FSS staircases as a function of such parameters as the up/down rule, the size of the steps up or down, the starting stimulus level, or the spread of the psychometric function. The results showed that the asymptotic convergence of FSS staircases depends much more on the sizes of the steps than it does on the up/down rule. Yet, if the size delta+ of a step up differs from the size delta- of a step down in a way that the ratio delta-/delta+ is constant at a specific value that changes with up/down rule, then convergence percent-correct is unaffected by the absolute sizes of the steps. For use with the popular one-, two-, three- and four-down/one-up rules, these ratios must respectively be set at 0.2845, 0.5488, 0.7393 and 0.8415, rendering staircases that converge on the 77.85%-, 80.35%-, 83.15%- and 85.84%-correct points. Wetherill and Levitt's transformed up/down rules--which require delta-/delta+ = 1--and the general version of Kaernbach's weighted up/down rule--which allows any delta-/delta+ ratio--fail to reach their presumed targets. The small-sample study showed that, even with the optimal settings, short FSS staircases (up to 20 reversals in length) are subject to some bias, and their precision is less than reasonable, but their characteristics improve when the size delta+ of a step up is larger than half the spread of the psychometric function. Practical recommendations are given for the design of efficient and trustworthy FSS staircases.