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.

MacDonald and Gardner reported the results of a comparative study of two post hoc cellwise tests in 3 X 4 contingency tables under the independence and homogeneity models. Based on their results, they advised against the use of standardized residuals and in favor of adjusted residuals. Here the authors show that the comparison was biased in favor of adjusted residuals because of a failure to consider the nonunit variance of standardized residuals. The authors define a moment-corrected standardized residual that overcomes this bias and present the results of a thorough study including two-way tables of all dimensions between 2 X 2 and 8 X 12 that aimed at comparing moment-corrected standardized residuals with adjusted residuals. Across the entire set of table dimensions included in this study, the results reveal that both residuals yield essentially the same pat-tern of cell-by-cell and experimentwise Type I error rates when the data come from variables with uniform marginal distributions. When the data come from variables with peaked marginal distributions, adjusted residuals behave minimally better than moment-corrected residuals.

Interest in the use of adaptive staircase methods in clinical practice is increasing, but time limitations require that they be based on yes-no trials. The psychometric properties of yes-no staircases with fixed step sizes (FSS staircases) in small-sample situations have never been studied in depth. As a result, information is lacking as to what is the optimal setup for an FSS staircase. To determine this optimal setup, we used simulation techniques to study the asymptotic and small-sample convergence of yes-no FSS staircases as a function of the up/down rule, the size of the steps up or down, the starting stimulus level, the spread of the psychometric function, and the lapsing rate. Our results indicate that yes-no FSS staircases with steps up and down of the same size are unstable because with these settings, the staircases yield different results across variations in irrelevant parameters such as the spread of the psychometric function or the starting level. Our study also identified settings with which the properties of estimates are unaffected by these factors. With these optimal settings, yes-no FSS staircases can provide very quick and accurate estimates in 7 to 8 trials. Practical recommendations are given to get the best out of yes-no FSS staircases.