García-Pérez M A
Departamento de Metodología, Facultad de Psicología, Universidad Complutense, Madrid, Spain.
Vision Res. 1998 Jun;38(12):1861-81. doi: 10.1016/s0042-6989(97)00340-4.
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.
视觉检测和辨别阈值通常使用自适应阶梯法进行测量，并且大多数研究采用具有固定步长的变换（或加权）上/下方法——秉承韦瑟里尔和莱维特（《英国数学与统计心理学杂志》1965年；18:1 - 10）或凯尔恩巴赫（《感知与心理物理学》1991年；49:227 - 229）的精神——而不是根据最佳放置规则在每次试验时改变步长——秉承沃森和佩利（《感知与心理物理学》1983年；47:87 - 91）的精神。一般认为，固定步长（FSS）阶梯法会收敛到在韦瑟里尔和莱维特或凯尔恩巴赫推导的概率下出现正确反应的刺激水平，但这从未得到严格证明。这项工作使用模拟技术来确定FSS阶梯法的渐近收敛和小样本收敛情况，它是诸如上/下规则、向上或向下步长的大小、起始刺激水平或心理测量函数的离散度等参数的函数。结果表明，FSS阶梯法的渐近收敛更多地取决于步长的大小，而不是上/下规则。然而，如果向上步长的大小δ + 与向下步长的大小δ - 不同，使得比率δ - /δ + 在随上/下规则变化的特定值处保持恒定，那么正确反应百分比的收敛不受步长绝对大小的影响。对于流行的一、二、三、四向下/一向上规则的使用，这些比率必须分别设置为0.2845、0.5488、0.7393和0.8415，从而使阶梯法收敛到77.85%、80.35%、83.15%和85.84%正确的点。韦瑟里尔和莱维特的变换上/下规则——要求δ - /δ + = 1——以及凯尔恩巴赫加权上/下规则的一般版本——允许任何δ - /δ + 比率——未能达到其假定目标。小样本研究表明，即使在最佳设置下，短的FSS阶梯法（长度最多20次反转）也存在一些偏差，并且其精度不尽人意，但当向上步长的大小δ + 大于心理测量函数离散度的一半时，其特性会有所改善。针对高效且可靠的FSS阶梯法的设计给出了实际建议。
