Hick's Law in a Stochastic Race Model with Speed-Accuracy Tradeoff

We present an analytic solution for a race model of n stochastic accumulators for multiple choice reaction time. We show that to maintain a constant level of accuracy, the response criterion needs to be increased approximately logarithmically with n, to compensate for the increase with n in the likelihood that an incorrect alternative will be most active after any fixed amount of time accumulating information. Assuming that participants monitor and maintain a constant level of performance can then explain the logarithmic dependency of the response latency on n as specified by Hick's law. Moreover, we show that for short time intervals, the Shannon information that observers extract from a stimulus, is predicted to increase linearly with processing time.     

Context effects in multi-alternative decision making: empirical data and a Bayesian model

For decisions between many alternatives, the benchmark result is Hick's Law: that response time increases log-linearly with the number of choice alternatives. Even when Hick's Law is observed for response times, divergent results have been observed for error rates-sometimes error rates increase with the number of choice alternatives, and sometimes they are constant. We provide evidence from two experiments that error rates are mostly independent of the number of choice alternatives, unless context effects induce participants to trade speed for accuracy across conditions. Error rate data have previously been used to discriminate between competing theoretical accounts of Hick's Law, and our results question the validity of those conclusions. We show that a previously dismissed optimal observer model might provide a parsimonious account of both response time and error rate data. The model suggests that people approximate Bayesian inference in multi-alternative choice, except for some perceptual limitations.     

Internalization of the thin and muscular body ideal and disordered eating in adolescence: the mediation effects of body esteem

This study investigates body esteem factors (weight-esteem and appearance-esteem) as mediators of the relationship between ‘internalization of the ideal body figure’ and disordered eating behaviors (restrained, emotional and external eating) in a community sample of adolescent males (n = 810) and females (n = 1137) from the Ontario Research on Eating and Adolescent Lifestyles (REAL) study. Mediation models were examined using a bootstrapping approach to test indirect effects and indirect contrasts. In males, weight-esteem partially mediated the relationship between muscular ideal and restrained eating; appearance-esteem partially mediated effects in the emotional and external eating regressions. In females, both weight-esteem and appearance-esteem partially mediated the relationship between thin ideal and all three forms of disordered eating; weight-esteem was a stronger mediator for restrained eating, and appearance-esteem a stronger mediator for emotional and external eating. Body esteem is important to consider for prevention and treatment of disordered eating in both genders.     

Hick's law revisited

In 1952, Hick proposed his ‘information theory’, that in making choice reactions the subject gains information at a constant rate such that: Mean reaction time = K log₂ (N + 1).     

Two-generator semigroups of binary relations

We study upper bounds on the size of the semigroups generated two randomly chosen n × n Boolean matrices having exactly N one entries. In general, as $\text{N}\text{n}$ increases the semigroups tend to be smaller. If N is unrestricted, or is a function which is at least [$\text{((2 + ?)n}{}^3\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$] the probability tends to one that the size of the semigroup is exactly 3. If N is a function which is at least [(r + 1 + ?)n log n] the probability tends to one that the size of the semigroup will be no more than $\text{2}{}^{\mathrm{<mtext>(</mtext><mtext>(n?1)</mtext><mtext>r</mtext><mtext>)+1</mtext>}}\text{? 1}$. However, if N is any function w(n) such that $\text{w(n)}\text{n}{}^2\text{→ 0}$ and w(n) > n, the average size of the resulting semigroups will be at least . This phenomenon is caused by some semigroups of extremely large size. It is thought that w(n) > n and $\text{w(n)}\text{n}{}^2\text{→ 0}$ hold for those matrices usually encountered in sociology and psychology.     

Footbridge usage in high-traffic flow highways: The intersection of safety and security in pedestrian decision-making​

In the Decade of Action for Road Safety, the vulnerability of pedestrians to severe outcomes from road crash is well-recognised. This study explored the factors that are influential in pedestrians’ decisions to cross a high-traffic highway in Barranquilla, Colombia, by using a footbridge over the highway, or by cutting across the highway in close proximity to a footbridge. Participants were recruited at each crossing (n = 105 footbridge, n = 105 ground; n = 117 males; M(SD) age = 23.28(5.98) years) and completed a paper survey exploring demographics, highway crossing behaviour, and attitudes regarding the safety and security of the crossing. Despite the majority of participants reporting the ground crossing was dangerous, and that the footbridge crossing was safe, one third of participants never or rarely (approximately 0% and 25% of the time) used the footbridge to cross the highway. Logistic regression revealed that the decision regarding the highway crossing mode (footbridge or ground) was predicted by (a) the frequency in which the footbridge is crossed, (b) the perception of footbridge security in relation to crime, (c) the perception of footbridge safety about traffic conflicts in general, (d) the proximity of the footbridge to the highway ground crossing, and (e) if the subject had experienced an injury during a previous highway crossing. The findings extend our understanding of pedestrian behaviour in crossing highways, not only in relation to the use of footbridges which separate both road user groups, but also ground crossings which place the pedestrians at significant risk especially those in areas with high traffic flow density. Moreover, the findings highlight the complexity involved in improving pedestrian road safety, particularly as the provision of an engineering solution (a footbridge) in close proximity to a high-risk zone (highway) is clearly not the only solution in the much larger road safety system.     

Nearest neighbor analysis of point processes: Applications to multidimensional scaling

A new approach for evaluating spatial statistical models based on the (random) number 0 ≤ N(i, n) ≤ n of points whose nearest neighbor is i in an ensemble of n + 1 points is discussed. The second moment of N(i, n) offers a measure of the centrality of the ensemble. The asymptotic distribution of N(i, n) and the expected degree of centrality for several spatial and nonspatial point processes is described. The use of centrality as a diagnostic statistic for multidimensional scaling is explored.     

Comparing the predictive powers of alternative multiple regression models

Mean squared error of prediction is used as the criterion for determining which of two multiple regression models (not necessarily nested) is more predictive. We show that an unrestricted (or true) model witht parameters should be chosen over a restricted (or misspecified) model withm parameters if (P _t ² ?P _m ² )>(1?P _t ² )(t?m)/n, whereP _t ² andP _m ² are the population coefficients of determination of the unrestricted and restricted models, respectively, andn is the sample size. The left-hand side of the above inequality represents the squared bias in prediction by using the restricted model, and the right-hand side gives the reduction in variance of prediction error by using the restricted model. Thus, model choice amounts to the classical statistical tradeoff of bias against variance. In practical applications, we recommend thatP ² be estimated by adjustedR ² . Our recommendation is equivalent to performing theF-test for model comparison, and using a critical value of 2?(m/n); that is, ifF>2?(m/n), the unrestricted model is recommended; otherwise, the restricted model is recommended.     

A counterexample to Fishburn's conjecture on finite linear qualitative probability

Kraft, Pratt and Seidenberg (Ann. Math. Statist. 30 (1959) 408) provided an infinite set of axioms which, when taken together with de Finetti's axiom, gives a necessary and sufficient set of “cancellation” conditions for representability of an ordering relation on subsets of a set by an order-preserving probability measure. Fishburn (1996) defined f(n) to be the smallest positive integer k such that every comparative probability ordering on an n-element set which satisfies the cancellation conditions C₄,…,C_k is representable. By the work of Kraft, Pratt, and Seidenberg (1959) and Fishburn (J. Math. Psychol. 40 (1996) 64; J. Combin. Design 5 (1997) 353), it is known that n-1?f(n)?n+1 for all n?5. Also Fishburn proved that f(5)=4, and conjectured that f(n)=n-1 for all n?5. In this paper we confirm that f(6)=5, but give counter-examples to Fishburn's conjecture for n=7, showing that f(7)?7. We summarise, correct and extend many of the known results on this topic, including the notion of “almost representability”, and offer an amended version of Fishburn's conjecture.     

Matrices with a given number of violations of Regular Minimality

A row (or column) of an n×n matrix complies with Regular Minimality (RM) if it has a unique minimum entry which is also a unique minimum entry in its column (respectively, row). The number of violations of RM in a matrix is defined as the number of rows (equivalently, columns) that do not comply with RM. We derive a formula for the proportion of n×n matrices with a given number of violations of RM among all n×n matrices with no tied entries. The proportion of matrices with no more than a given number of violations can be treated as the p-value of a permutation test whose null hypothesis states that all permutations of the entries of a matrix without ties are equiprobable, and the alternative hypothesis states that RM violations occur with lower probability than predicted by the null hypothesis. A matrix with ties is treated as being represented by all matrices without ties that have the same set of strict inequalities among their entries.     

On the reciprocity of proximity relations

The degree of reciprocity of a proximity order is the proportion, P(1), of elements for which the closest neighbor relation is symmetric, and the R value of each element is its rank in the proximity order from its closest neighbor. Assuming a random sampling of points, we show that Euclidean n-spaces produce a very high degree of reciprocity, $\text{P(1) ≥}\text{1}\text{2}$, and correspondingly low R values, E(R) ≤ 2, for all n. The same bounds also apply to homogeneous graphs, in which the same number of edges meet at every node. Much less reciprocity and higher R values, however, can be attained in finite tree models and in the contrast model in which the “distance” between objects is a linear function of the numbers of their common and distinctive features.     

Multilinear representations for ordinal utility functions

An ordinal utility function u over two attributes X₁, X₂ is additive if there exists a strictly monotonic function ϕ such that ϕ(u) = v₁(x₂) + v₂(x₂) for some functions v₁, v₂. Here we consider the class of ordinal utility functions over n attributes for which each pair of attributes is additive, but not necessarily separable, for any fixed levels of the remaining attributes. We show that while this class is more general than those that are ordinally additive, the assessment task is of the same order of difficulty, and involves a hierarchy of multilinear rather than additive decompositions.     

Estimation of careless error and lucky guess probabilities for dichotomous test items: A psychometric application of a biometric latent class model with random effects

Medical research has extensively dealt with the estimation of the accuracy (sensitivity and specificity) of a diagnostic test for screening individuals. In this paper we apply the biometric latent class model with random effects by Qu, Tan, and Kutner [(1996). Random effects models in latent class analysis for evaluating accuracy of diagnostic tests. Biometrics, 52, 797-810] to estimate the response error (careless error and lucky guess) probabilities for dichotomous test items in the psychometric theory of knowledge spaces. The approach is illustrated with simulated data. In particular, we extend this approach to give a generalization of the basic local independence model in knowledge space theory. This allows for local dependence among the indicators given the knowledge state of an examinee and/or for the incorporation of covariates.     

On the robustness of maximum likelihood scaling for violations of the error model

A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of thex ² test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.     

Utility of extension of functional equations—when possible

Jean-Claude Falmagne observed in 1981 [On a recurrent misuse of classical functional equation result. Journal of Mathematical Psychology, 23, 190-193] that, even under regularity assumptions, not all solutions of the functional equation k(s+t)=k(s)+k(t), important in many fields, also in the theory of choice, are of the form k(s)=Cs. This is certainly so when the domain of the equation (the set of (s,t) for which the equation is satisfied) is finite. We mention an example showing that this can happen even on some infinite, open, connected sets (open regions). The more general equations k(s+t)=?(s)+n(t) and k(s+t)=m(s)n(t), called Pexider equations, have been completely solved on R². In case they are assumed valid only on an open region, they have been extended to R² and solved that way (the latter if k is not constant). In this paper their common generalization

k(s+t)=?(s)+m(s)n(t)     

Individual differences and the all-or-none vs incremental learning controversy

This paper provides an analysis of the ability of the all-or-none and incremental learning frameworks augmented with individual difference assumptions to account for several standard all-or-none learning demonstrations. The paper allows the one element model (OEM) and the linear operator model (LOM) to have a completely arbitrary individual difference distribution. Among the results are: 1. the LOM can not account for either the Rock replacement studies or Bower's constancy of the error given error curve. 2. The OEM can account for Rock's and Bower's results only if there are no individual differences. 3. No independent responding scheme can jointly predict Rock's and Bower's results. 4. A beta distribution on the learning rate allows the LOM to achieve a close fit to at most one of Rock's and Bower's results. 5. An RT⁽ⁿ⁾ experiment can not rule out an independent responding scheme for any n, n ≥ 1.     

The time course of perceptual choice: the leaky, competing accumulator model

The time course of perceptual choice is discussed in a model of gradual, leaky, stochastic, and competitive information accumulation in nonlinear decision units. Special cases of the model match a classical diffusion process, but leakage and competition work together to address several challenges to existing diffusion, random walk, and accumulator models. The model accounts for data from choice tasks using both time-controlled (e.g., response signal) and standard reaction time paradigms and its adequacy compares favorably with other approaches. A new paradigm that controls the time of arrival of information supporting different choice alternatives provides further support. The model captures choice behavior regardless of the number of alternatives, accounting for the log-linear relation between reaction time and number of alternatives (Hick's law) and explains a complex pattern of visual and contextual priming in visual word identification.     

The general psychophysical differential equation: A comparison of three specifications

The general psychophysical differential equation, dy/dx = W₂(y)/W₁(x), with the solution y = f(x), where x and y are subjective variables and W₁ and W₂ their subjective Weber functions, is (a) compared with a corresponding functional equation, and (b) studied from a stochastic point of view by error calculus, Methods for evaluating and handling divergences are proposed and illustrated for a number of combinations of Weber functions. It is shown that either the differential: and the functional equations have the same solution or the difference between the solutions is negligible compared to empirical scatter. The error calculus gives the same result: either no error at all or a negligible one.