Computation of psychophysical thresholds using the probit technique

Probit analysis is a statistical technique that can be used for the computation of psychophysical thresholds, although it was originally developed for the analysis of pharmacological and toxicological data. Probit analysis may be applied to psychophysical data if the method of constant stimuli has been used for data collection. A computer program written in APPLESOFT BASIC that performs probit analyses on such psychophysical data is presented and explained.     

Extracting thresholds from noisy psychophysical data

Psychophysical studies with infants or with patients often are unable to use pilot data, training, or large numbers of trials. To evaluate threshold estimates under these conditions, computer simulations of experiments with small numbers of trials were performed by using psychometric functions based on a model of two types of noise:stimulus-related noise (affecting slope) andextraneous noise (affecting upper asymptote). Threshold estimates were biased and imprecise when extraneous noise was high, as were the estimates of extraneous noise. Strategies were developed for rejecting data sets as too noisy for unbiased and precise threshold estimation; these strategies were most successful when extraneous noise was low for most of the data sets. An analysis of 1,026 data sets from visual function tests of infants and toddlers showed that extraneous noise is often considerable, that experimental paradigms can be developed that minimize extraneous noise, and that data analysis that does not consider the effects of extraneous noise may underestimate test-retest reliability and overestimate interocular differences.     

A comparison of two psychophysical methods using animals

A discrimination box containing two levers with a light above each was used to train eight rats to press beneath the brighter light for a milk reinforcer. The brighter light was held constant and the comparison light was varied to produce 12 brightness differences. The animals were run under two experimental methods: the block method in which each brightness level comparison was presented for a block of 11 contiguous trials, and the staircase method in which the sequence of brightness comparisons was determined by the correctness of the response on the preceding comparison. The block method produced a smaller differential brightness threshold and a larger change in discrimination performance for stimulus magnitude changes than did the staircase method.     

A comparison of three adaptive psychophysical procedures using inexperienced listeners

Thresholds for a masked 1-kHz tone were obtained in single sessions from 60 inexperienced listeners. A two-alternative forced-choice procedure was used in conjunction with one of three adaptive psychophysical techniques. These techniques comprised two staircase techniques targeting either 70.7% or 79.4% correct detection (Staircase-71 and Staircase-79), and parameter estimation by sequential testing (PEST), targeting 80% correct detection. Listeners were provided with a rationale for maintaining concentration at weak signal levels. Similar threshold values were obtained from the Staircase-79 and PEST groups in equal numbers of trials. The degree of oscillation in the level of the signal around the value finally chosen as the threshold was comparable for both staircase techniques. Subsequent fixed-level testing did not provide a true indication of the subjects' capabilities. The amount by which percent correct in fixed-level testing differed from expectations based on adaptive testing varied among the techniques. Additional thresholds were obtained in a second session from 30 of the original subjects. Thresholds with both staircase techniques improved by about 1 dB on retest, while thresholds with PEST were constant across sessions. The variability of the data compared well with that from studies involving experienced listeners.     

Extracting thresholds from noisy psychophysical data.

Psychophysical studies with infants or with patients often are unable to use pilot data, training, or large numbers of trials. To evaluate threshold estimates under these conditions, computer simulations of experiments with small numbers of trials were performed by using psychometric functions based on a model of two types of noise: stimulus-related noise (affecting slope) and extraneous noise (affecting upper asymptote). Threshold estimates were biased and imprecise when extraneous noise was high, as were the estimates of extraneous noise. Strategies were developed for rejecting data sets as too noisy for unbiased and precise threshold estimation; these strategies were most successful when extraneous noise was low for most of the data sets. An analysis of 1,026 data sets from visual function tests of infants and toddlers showed that extraneous noise is often considerable, that experimental paradigms can be developed that minimize extraneous noise, and that data analysis that does not consider the effects of extraneous noise may underestimate test-retest reliability and overestimate interocular differences.     

Differential thresholds for limb movement measured using adaptive techniques

Differential thresholds for limb movement were measured in 10 subjects, using the transformed up-down procedure. Subjects were required to indicate which of two random displacement perturbations delivered to their forearms had the larger standard deviation (SD). TheSD of the reference signal was fixed for each experimental condition at one of seven values ranging from 0.05 to 3.2 mm. TheSD of the other signal varied depending on the subject’s response. Using this procedure, the differential threshold for limb movement was calculated to be 8%, which is very similar to the thresholds estimated previously for changes in limb position (9%) and force (7%). The sensitivity of the human proprioceptive system to changes in limb displacement was much greater than anticipated, with subjects being able to resolve a 5-μm difference between two perturbations delivered to their arms.     

Microcomputer-based estimation of psychophysical thresholds: The Best PEST

A new, maximally efficient technique for measuring psychophysical thresholds (Pentland, 1980) has been implemented on the microcomputer. This PEST (parameter estimation by sequential testing) technique is the most efficient sequential parameter estimation technique possible, given that the form of the psychometric function is known. The technique is similar to but faster and more accurate than other staircase procedures and may be applied whenever staircase techniques are applicable. The “Best PEST” is easily implemented on the micro-computer; a BASIC program for the Apple II which does so is presented. The Best PEST is compared with other staircase procedures, including one recently implemented on a micro-computer (Corwin, Kintz, & Beaty, 1979).     

Differential thresholds for limb movement measured using adaptive techniques.

Differential thresholds for limb movement were measured in 10 subjects, using the transformed up-down procedure. Subjects were required to indicate which of two random displacement perturbations delivered to their forearms had the larger standard deviation (SD). The SD of the reference signal was fixed for each experimental condition at one of seven values ranging from 0.05 to 3.2 mm. THe SD of the other signal varied depending on the subject's response. Using this procedure, the differential threshold for limb movement was calculated to be 8%, which is very similar to the thresholds estimated previously for changes in limb position (9%) and force (7%). The sensitivity of the human proprioceptive system to changes in limb displacement was much greater than anticipated, with subjects being able to resolve a 5-microns difference between two perturbations delivered to their arms.     

An adaptive psychophysical method for subject classification

In psychophysical experiments, one’s goal is usually to measure some continuous parameter hypothesized to determine the statistical properties of a subject’s responses. Methods are well developed that adaptively manipulate stimulus characteristics in such a way that the reliability of the parameter estimate is maximized. However, such methods are inapplicable in situations in which the goal is to assign subjects to discrete categories, rather than to measure a continuous parameter. This paper introduces a technique that is directly applicable to efficient categorization and that adaptively manipulates stimulus characteristics in such a way that the information obtained from each trial is maximized. This technique is based on the principle of minimum estimated expected entropy, whereby stimulus parameters on each trial are chosen in order to minimize the estimated expected entropy of the a posteriori probability distribution that expresses how likely a subject is to belong to each of a group of mutually exclusive categories. A sample implementation of the technique—the classification of infant subjects according to their audiograms—is then described and evaluated via computer simulation.     

An efficient adaptive procedure for psychophysical discrimination experiments

We present an adaptive procedure to conduct psychophysical discrimination experiments. In a discrimination experiment, an observer senses (sees, hears, feels, etc.) two stimuli (separated in space or time) and is asked to order these stimuli with respect to a particular parameter (say,s). Under the usual assumption of a locally linear internal representation ofs, perturbed by additive Gaussian noise, the probabilityP(s ₂)—of judging test stimuluss ₂ “larger” than a reference stimuluss ₁—is an error function (a cumulative normal distribution). Such an error function,Erf [(s ₂μ)/σ], is parametrized by two parameters: μand σ. The parameter μ is the value for whichP(s ₂) = 50% and is related to possible bias effects in the internal representation of the observer. The parameter σ is √2 times the standard deviation of the noise distribution and is generally called thediscrimination threshold. In this paper, we present (1) an algorithm to estimate μ and σ given the data generated by such an observer, (2) an analysis of the efficiency of a stimulus presentation with respect to the estimation of μ and σ , and (3) a method that controls the specific choice of stimulus values during an experiment so that an optimal estimation of μ and σ is obtained.     

Adaptive psychophysical methods for nonmonotonic psychometric functions

Many psychophysical tasks in current use render nonmonotonic psychometric functions; these include the oddball task, the temporal generalization task, the binary synchrony judgment task, and other forms of the same–different task. Other tasks allow for ternary responses and render three psychometric functions, one of which is also nonmonotonic, like the ternary synchrony judgment task or the unforced choice task. In all of these cases, data are usually collected with the inefficient method of constant stimuli (MOCS), because extant adaptive methods are only applicable when the psychometric function is monotonic. This article develops stimulus placement criteria for adaptive methods designed for use with nonmonotonic psychometric functions or with ternary tasks. The methods are transformations of conventional up–down rules. Simulations under three alternative psychophysical tasks prove the validity of these methods, their superiority to MOCS, and the accuracy with which they recover direct estimates of the parameters determining the psychometric functions, as well as estimates of derived quantities such as the point of subjective equality or the difference limen. Practical recommendations and worked-out examples are provided to illustrate how to use these adaptive methods in empirical research.     

The step method: a new adaptive psychophysical procedure

A new adaptive psychophysical method, the step method, is introduced. Simulations show the method to be less biased and more efficient than constant stimuli or Pentland's adaptive method for fewer than 40 trials. An experiment using discrimination of dot number, however, failed to find any differences among the three methods in either bias or efficiency.     

Comparison of cupulometric and psychophysical thresholds for perception of rotation and the oculogyral illusion

The purpose of this study was to compare thresholds for angular acceleration derived by subjective cupulometry and by a staircase method. Thresholds for the perception of rotation and the oculogyral illusion were determined for 10 Os who were rotated about their vertical axis. The cupulometric thresholds were significantly higher, more variable, and not predictable from the staircase thresholds. Furthermore, cupulometry failed to distinguish between the thresholds for the perception of rotation and the oculogyral illusion, although both indicators functioned according to the prediction of the underlying linear model. Individual differences supported the general conclusion that cupulometric thresholds bear no relationship to the sensory threshold derived in a classical psychophysical manner.     

Fast and accurate measurement of taste and smell thresholds using a maximum-likelihood adaptive staircase procedure

This paper evaluates the use of a maximum-likelihood adaptive staircase psychophysical procedure (ML-PEST), originally developed in vision and audition, for measuring detection thresholds in gustation and olfaction. The basis for the psychophysical measurement of thresholds with the ML-PEST procedure is developed. Then, two experiments and four simulations are reported. In the first experiment, ML-PEST was compared with the Wetherill and Levitt up-down staircase method and with the Cain ascending method of limits in the measurement of butyl alcohol thresholds. The four Monte Carlo simulations compared the three psychophysical procedures. In the second experiment, the test-retest reliability of ML-PEST for measuring NaCl and butyl alcohol thresholds was assessed. The results indicate that the ML-PEST method gives reliable and precise threshold measurements. Its ability to detect malingerers shows considerable promise. It is recommended for use in clinical testing.     

Fast and accurate measurement of taste and smell thresholds using a maximum-likelihood adaptive staircase procedure.

This paper evaluates the use of a maximum-likelihood adaptive staircase psychophysical procedure (ML-PEST), originally developed in vision and audition, for measuring detection thresholds in gustation and olfaction. The basis for the psychophysical measurement of thresholds with the ML-PEST procedure is developed. Then, two experiments and four simulations are reported. In the first experiment, ML-PEST was compared with the Wetherill and Levitt up-down staircase method and with the Cain ascending method of limits in the measurement of butyl alcohol thresholds. The four Monte Carlo simulations compared the three psychophysical procedures. In the second experiment, the test-retest reliability of MLPEST for measuring NaCl and butyl alcohol thresholds was assessed. The results indicate that the ML-PEST method gives reliable and precise threshold measurements. Its ability to detect malingerers shows considerable promise. It is recommended for use in clinical testing.     

A comparative study of scales constructed by three psychophysical methods

A comparison is made between the scales constructed by the Method of Paired Comparison, Rank Order, and the Method of Successive Intervals. Application of the three psychophysical methods to handwriting specimens and to nationality preferences results in mutually linear scales. Choice of scaling methods becomes, then, a matter of practical convenience rather than of relative validity.The writer is very much indebted to Professor L. L. Thurstone for the suggestion of this problem and for supervision in carrying out the study.