Vierordt's The Experimental Study of the Time Sense (1868) and its legacy

The paper discusses the results from Vierordt's 1868 book Der Zeitsinn nach Versuchen [The Experimental Study of the Time Sense]. Illustrations of “Vierordt's Law”, the proposition that short durations are judged as longer than they really are, whereas long durations are judged as shorter, with an “indifference point” in between, are provided, mainly from reproduction experiments where Vierordt and his students or colleagues served as experimental participants. Other work from Der Zeitsinn including time discrimination and categorical timing procedures is also presented. Some subsequent research on Vierordt's Law and the “indifference point” is discussed with respect to some issues in contemporary timing theory.     

Bayes,time perception,and relativity: The central role of hopelessness

Time judgement and time experience are distinct elements of time perception. It is known that time experience tends to be slow, or dilated, when depressed, but there is less certainty or clarity concerning how depression affects time judgement. Here, we use a Bayesian Prediction Error Minimisation (PEM) framework called ‘distrusting the present’ as an explanatory and predictive model of both aspects of time perception. An interval production task was designed to probe and modulate the relationship between time perception and depression. Results showed that hopelessness, a symptom of severe depression, was associated with the ordering of interval lengths, reduced overall error, and dilated time experience. We propose that ‘distrusting the future’ is accompanied by ‘trusting the present’, leading to the experiences of time dilation when depressed or hopeless. Evidence was also found to support a relative difference model of how hopelessness dilates, and arousal accelerates, the rate of experienced time.     

Differences in delay, not ratios, control choice in concurrent chains

In two experiments, pigeons were trained with concurrent-chains schedules, wherein responding to equal initial links measured preference between variable-interval terminal links. Absolute terminal-link duration was varied by keeping constant the difference between the terminal-link delays and forcing their ratio to change. Delay-reduction theory scales value relative to a common temporal context and requires that delay differences control choice. Thus, preference should remain invariant. Most competing accounts, including the matching law and a strong form of Weber's law, require that preference vary with the delay ratio. Experiment 1 employed standard concurrent chains, in which terminal-link position and color were confounded. Although average preference remained constant, individual preferences were highly variable and inconsistent, possibly due to carryover of position biases across conditions. In an attempt to reduce variability, Experiment 2 used a modified concurrent-chains procedure. Preference at different terminal-link durations was assessed simultaneously to prevent order effects, and terminal-link position was alternated randomly across trials to minimize the impact of position biases. In Experiment 2, both individual and mean preferences showed the constant-difference invariance. Overall, choice was controlled by terminal-link differences, not ratios.     

On the classic and modern theories of matching

Classic matching theory, which is based on Herrnstein's (1961) original matching equation and includes the well-known quantitative law of effect, is almost certainly false. The theory is logically inconsistent with known experimental findings, and experiments have shown that its central constant-k assumption is not tenable. Modern matching theory, which is based on the power function version of the original matching equation, remains tenable, although it has not been discussed or studied extensively. The modern theory is logically consistent with known experimental findings, it predicts the fact and details of the violation of the classic theory's constant-k assumption, and it accurately describes at least some data that are inconsistent with the classic theory.     

Assessing preference for reinforcers using demand curves, work-rate functions, and expansion paths

Much research has focused on the effects of environmental variability on foraging decisions. However, the general pattern of preference for variability in delay to reward and aversion to variability in amount of reward remains unexplained a either a mechanistic or a functional level. Starlings' preferences between a fixed and a variable option were studied in two treatments, A and D. The fixed option was the same in both treatments (20-s fixed-interval delay, five units food). In Treatment A the variable option gave two equiprobable amounts of food (20-s delay, three or seven units) and in D it gave two equiprobable delays to food (2.5-s or 60.5-s delays, five units). In both treatments the programmed ratio [amount/(intertrial interval+latency+delay)] in the fixed option equaled the arithmetic mean of the two possible ratios in the variable option (ITI = 40 s, latency = 1 s). The variable option was strongly preferred in Treatment D and was weakly avoided in Treatment A. These results are discussed in the light of two theoretical models, a form of constrained rate maximization and a version of scalar expectancy theory. The latter accommodates more of the data and is based on independently verifiable assumptions, including Weber's law.     

Feedback functions, optimization, and the relation of response rate to reinforcer rate

The present experiment arranged a series of inverted U-shaped feedback functions relating reinforcer rate to response rate to test whether responding was consistent with an optimization account or with a one-to-one relation of response rate to reinforcer rate such as linear system theory's rate equation or Herrnstein's hyperbola. Reinforcer rate was arranged according to a quadratic equation with a maximum at a unique response rate. The experiment consisted of two phases, during which 6 Long Evans rats lever pressed for food. In the first phase of the experiment, the rats responded on six fixed-interval-plus-quadratic-feedback schedules, and in the second phase the rats responded on three variable-interval-plus-quadratic-feedback schedules. Responding in both phases was inconsistent with a one-to-one relation of response rate to reinforcer rate. Instead, different response rates were obtained at equivalent reinforcer rates. Responding did vary directly with the vertex of the feedback function in both phases, a finding consistent with optimization of reinforcer rate. The present results suggest that the feedback function relating reinforcer rate to response rate imposed by a reinforcement schedule can be an important determinant of behavior. Furthermore, the present experiment illustrates the benefit of arranging feedback functions to investigate assumptions about the variables that control schedule performance.     

The role of associative strength in children's false memory illusions

The effects of associative strength on rates of 7- and 11-year-old children's true and false memories were examined when category and Deese-Roediger-McDermott (DRM) lists were used to cue the same critical lure. Backward associative strength (BAS) was varied such that the category and DRM lists had the same strength (DRM=category), DRM lists had more BAS (DRM>category), or category lists had more BAS (DRM<category). If BAS drives children's false memories then BAS, not the type of relation across items in a list, should determine false memory production. The results confirmed this prediction using both recall and recognition measures: (1) both true and false memories increased with age, (2) true memory was better for category than DRM lists but there were no differences for false memory, and (3) at all ages, false memories varied predictably with changes in BAS but were unaffected by list-type manipulations. These findings are discussed in the context of models of false memory development.     

Numerical approximation of the observed information matrix with Oakes' identity

An efficient and accurate numerical approximation methodology useful for obtaining the observed information matrix and subsequent asymptotic covariance matrix when fitting models with the EM algorithm is presented. The numerical approximation approach is compared to existing algorithms intended for the same purpose, and the computational benefits and accuracy of this new approach are highlighted. Instructive and real-world examples are included to demonstrate the methodology concretely, properties of the estimator are discussed in detail, and a Monte Carlo simulation study is included to investigate the behaviour of a multi-parameter item response theory model using three competing finite-difference algorithms.     

Confirmation of linear system theory prediction: Changes in Herrnstein's k as a function of changes in reinforcer magnitude

Eight human subjects pressed a lever on a range of variable-interval schedules for 0.25¢ to 35.0¢ per reinforcement. Herrnstein's hyperbola described seven of the eight subjects' response-rate data well. For all subjects, the y-asymptote of the hyperbola increased with increasing reinforcer magnitude and its reciprocal was a linear function of the reciprocal of reinforcer magnitude. These results confirm predictions made by linear system theory; they contradict formal properties of Herrnstein's account and of six other mathematical accounts of single-alternative responding.