On the form and location of the Psychometric Bisection Function for time

doi:10.1016/0022-2496(81)90035-3

Journal of Mathematical Psychology

Volume 24, Issue 1, August 1981, Pages 58-87

https://doi.org/10.1016/0022-2496(81)90035-3 Get rights and content

Abstract

The Psychometric Bisection Function for time relates the discriminability of intermediate duration stimuli to a short and long training duration. Bisection Functions for animals (R. M. Church & M. Z. Deluty, Journal of Experimental Psychology: Animal Behavior Processes, 1977, 3, 216–228) confirm Weber's Law and also show indifference between short and long reports at the geometric mean of the training durations. Two discrimination processes are studied which, in combination with different constructions of the subjective time scale, result in Bisection Functions which differ in form and location. The two discrimination processes use a likelihood ratio rule or a similarity rule to compare intermediate durations to the training durations. These rules in combination with two different constructions of the subjective time scale result in four models which conform to Weber's Law. For one of the scales subjective time is a power function of real time with the scalar property on variance (Scalar Timing). For the other, subjective time is a logarithmic function of real time with constant variance (Log Timing). Both Log and Scalar Timing assume normality on the subjective scale. Only three of these models also entail the geometric mean at the indifference point. The exception is Scalar Timing with the likelihood ratio discrimination rule. This model entails indifference at approximately the harmonic mean of the training stimuli. Variants of the remaining three models differ theoretically but alternatives are difficult to discriminate empirically. A contrast is provided by a Poisson Timing subjective scale in which variance increases directly with the mean. This scale results in indifference at the geometric mean for both discrimination rules but violates Weber's Law in both cases.

References (21)

J.C. Falmagne et al.
Conjoint Weber Laws and additivity
Journal of Mathematical Psychology
(1979)
G.J. Iverson
Conditions under which Thurstone Case III representations for binary choice probabilities are also Fechnerian
Journal of Mathematical Psychology
(1979)
J. Aitchison et al.
The lognormal distribution
(1969)
L.A. Aroian
The probability function of the product of two normally distributed random variables
Annals of Mathematical Statistics
(1947)
W.H. Baum
On two types of deviation from the matching law: Bias and undermatching
Journal of the Experimental Analysis of Behavior
(1974)
A.C. Catania
Reinforcement schedules and psychophysical judgments
R.M. Church et al.
Bisection of temporal intervals
Journal of Experimental Psychology: Animal Behavior Processes
(1977)
C.C. Crais
On the frequency function of XY
Annals of Mathematical Statistics
(1936)
J.P. Egan
Signal detection theory and ROC analysis
(1975)
W. Feller

There are more references available in the full text version of this article.

Cited by (153)

A bias-free test of human temporal bisection: Evidence against bisection at the arithmetic mean
2024, Cognition
The temporal bisection procedure has been used to assess theories of time perception. A problem with the procedure for measuring the perceived midpoint of two durations is that the spacing of probe durations affects the length of the bisection point. Linear spacing results in longer bisection points closer to the arithmetic mean of the durations than logarithmic spacing. In three experiments, the influence of probe duration distribution was avoided by presenting a single probe duration of either the arithmetic or geometric mean of the trained durations. It was found that the number of participants that categorised the arithmetic mean as long was significantly larger than those that categorised it as short. The number of participants that categorised the geometric mean as either short or long did not significantly differ. This was true for trained durations of 0.4 s vs. 1.6 s (Experiments 1–3), 0.2 s vs. 3.2 s (Experiment 2) and 0.4 s vs. 6.4 s (Experiment 3). In Experiment 4, the probe trial distribution effect was replicated with logarithmic and linearly distributed probe durations, demonstrating that bisection occurs close to the arithmetic mean with linearly spaced probe durations. The results provide evidence against bisection at the arithmetic mean when probe spacing bias is avoided and, instead, the results are consistent with logarithmic encoding of time, or a comparison rule based on relative rather than absolute differences.
Effects of differential probabilities of reinforcement on human timing
2020, Behavioural Processes
We investigated how differential payoffs affect the temporal discrimination of humans. In a temporal bisection task, participants learned to make one response after a short sample and another after a long sample. When presented with a range of intermediate samples, the proportion of responses fitted well a Gaussian-like distribution function characterized by a location (bias), a scale (sensitivity) parameter, and two asymptote (discrimination) parameters. In Experiment 1, when one response yielded more reinforcers than the other, parameters were unaltered, but overall responses increased for the response producing higher payoffs. In Experiment 2, we used a video game to track motion during the sample and participants learned to approach the “short” response location at sample onset and remain there before departing to the “long” location on long trials. Departure times were shorter when “long” choices produced higher payoffs than “short” and matched well the shifted psychometric functions. However, on some trials, subjects were biased for short, returning to the short side after having departed towards long. Evidence was found for effects of differential payoffs on response bias, but discrimination and sensitivity did not change consistently. These results favor a behavioral account of timing processes.
A systematic exploration of temporal bisection models across sub- and supra-second duration ranges
2020, Journal of Mathematical Psychology
An integral component to the validity of timing models is their ability to accurately fit behavioral data from detection and discrimination tasks such as the temporal bisection procedure. Two of the most prominent timing models are the Sample Known Exactly (SKE), based on scalar timing theory, and the pseudo-logistic model (PLM). Recently, evidence accumulation models based on drift–diffusionprocesses (DDM) have been utilized for modeling temporal bisection data. Currently, there is no standard by which timing behavioral data are fit, resulting in the implementation of both theoretical and atheoretical models, such as generalized logistic functions (L4P). As differences in timing behavior have been shown across sub- and supra-second durations, a comparative evaluation of these 4 different types of models (SKE, PLM, L4P, and DDM) was conducted to assess each model’s ability to capture these differences using timing data from the temporal bisection procedure. Psychometric functions from rats, trained on a bisection task using sub-sec (200 ms vs. 800 ms.) and supra-sec (2 s vs. 8 s) conditions, were fit with each of the four models. Using Akaike Information Criterion (AIC) we demonstrate that theoretical models vastly outperformed the L4P across both duration ranges, Furthermore, significant differences existed in key parameters between L4P and PLM. Three DDM models were analyzed with varying degrees of freedom, which showed that allowing for non-decision time, drift rate, and starting point to vary across signal durations outperform simpler models ( $Δ_{i}$ $\geq$ 10) that only allowed for variation in drift rate or drift rate and starting point. These models explained variance in both choice and reaction time data. While we were unable to directly compare timing and temporal decision models the results from both demonstrate the need for a shift toward theoretically based models such as the SKE, PLM, or DDM which provide greater parsimony as well as provide for greater qualitative analysis and interpretation of the data.
Approaching subjective interval timing with a non-Gaussian perspective
2018, Journal of Mathematical Psychology
Perceiving time intervals is an essential ability of many animals, whose psychophysical properties have yet to be fully understood. A common theoretical approach is to consider that internal representations of time intervals are reflected in probability distribution functions. Depending on the mechanism proposed for interval timing inverse Gaussian and log-normal probability distributions are candidate distributions to represent internal representations of time. In this article, we show that these two distributions approximate each other under the assumptions of mean accuracy and scalar timing when considering experimentally-relevant Weber fractions. Afterward, we show that both distributions may be used in the description of the temporal bisection task, predicting bisection times approximately at the geometric mean of reference time intervals for the experimental range of Weber fractions. Taken together these results suggest that the log-normal and the inverse Gaussian, when adapted to model subjective time intervals, are experimentally indistinguishable, and so are the models that use them as benchmarks.
The effect of the long anchor duration on performance in the temporal bisection procedure
2017, Behavioural Processes
The present study investigated how the value of the long anchor duration influences the bisection point (BP) in the temporal bisection procedure. The ratio similarity rule (Gibbon, 1981, On the form and location of the psychometric bisection function for time. Journal of Mathematical Psychology, 24(1), 58–87) hypothesizes that the location of the BP is determined by a ratio comparison between the short and long anchor durations. The Short/no short hypothesis (Machado and Keen, 2003, Temporal discrimination in a long operant chamber. Behavioural Processes, 62(1), 157–182) indicates that the location of the BP is influenced mainly by the short duration. Two experiments with pigeons manipulated the long anchor duration while fixing the short anchor duration in a series of anchor pairs. The results showed that the location of the BP depended on the initial training condition, a primacy effect. When the initial training condition used anchors of 3- vs. 9-s, a relatively strong influence of the long anchor duration was observed as predicted by the ratio similarity rule. On the other hand, when anchors were 3- vs. 27-s in the initial training condition, the effect of the long duration was relatively small, conforming more closely to the Short/no short hypothesis. Keypecking to the sample key during a trial, sample-cue responding, was also observed. The functions relating sample-cue responding to elapsed trial time also indicated the primacy effect.
The effects of payoff manipulations on temporal bisection performance
2016, Acta Psychologica
There is growing evidence that alterations in reward rates modify timing behavior demonstrating the role of motivational factors in interval timing behavior. This study aimed to investigate the effects of manipulations of rewards and penalties on temporal bisection performance in humans. Participants were trained to classify experienced time intervals as short or long based on the reference durations. Two groups of participants were tested under three different bias conditions in which either the relative reward magnitude or penalty associated with correct or incorrect categorizations of short and long reference durations was manipulated. Participants adapted their choice behavior (i.e., psychometric functions shifted) based on these payoff manipulations in directions predicted by reward maximization. The signal detection theory-based analysis of the data revealed that payoff contingencies affected the response bias parameter (B″) without altering participants' sensitivity (A′) to temporal distances. Finally, the response time (RT) analysis showed that short categorization RTs increased, whereas long categorization RTs decreased as a function of stimulus durations. However, overall RTs did not exhibit any modulation in response to payoff manipulations. Taken together, this study provides additional support for the effects of motivational variables on temporal decision-making.

View all citing articles on Scopus

View full text

On the form and location of the Psychometric Bisection Function for time

Abstract

Journal of Mathematical Psychology

Journal of Mathematical Psychology

The lognormal distribution

The probability function of the product of two normally distributed random variables

Annals of Mathematical Statistics

On two types of deviation from the matching law: Bias and undermatching

Journal of the Experimental Analysis of Behavior

Reinforcement schedules and psychophysical judgments

Bisection of temporal intervals

Journal of Experimental Psychology: Animal Behavior Processes

On the frequency function of XY

Annals of Mathematical Statistics

Signal detection theory and ROC analysis