A Bayesian psychophysical model for angular variables

Bayesian theories of perception provide a link between observed response distributions and theoretical constructs from Bayesian decision theory. Using Bayesian psychophysics we derive response distributions for two cases, one based on a normal distribution and one on a von Mises distribution for angular variables. Interestingly, where the theoretical response distribution is always unimodal in the case of normal distributions, it can become bimodal in the angular setting in the case when prior and likelihood are about equally strong.     

Distributional expectations and the induction of category structure

Previous research on how categories are learned from observation of exemplars has largely ignored the possible role of prior expectations concerning how exemplars will be distributed. The experiments reported here explored this issue by presenting subjects with category-learning tasks in which the distributions of exemplars defining the categories were varied. In Experiments 1 and 2 the distributional form of a category was found to affect speed of learning. Learning was faster when a category's distribution was normal than when it was multimodal. Also, subjects in the early stages of learning a multimodal category responded as if it were unimodal. These results suggested that subjects enter category-learning tasks with expectations of unimodal, possibly normal, distributions of exemplars. Experiments 3 and 4 attempted to manipulate subjects' prior expectations by varying the distribution of exemplars in the first of two consecutive category-learning tasks. Learning a multimodal category was influenced by the shape of a previously learned distribution and was facilitated when the earlier distribution was either multimodal or skewed, rather than normal. These results are interpreted as support for a dual-process model of category learning that incorporates the effects of prior expectations concerning exemplar distributions.     

Bimodal familiarization re-sensitizes 12-month-old infants to other-race faces

Perceptual narrowing in the domain of face perception typically begins to reduce infants’ sensitivity to differences distinguishing other-race faces from approximately 6 months of age. The present study investigated whether it is possible to re-sensitize Caucasian 12-month-old infants to other-race Asian faces through statistical learning by familiarizing them with different statistical distributions of these faces. The familiarization faces were created by generating a morphed continuum from one Asian face identity to another. In the unimodal condition, infants were familiarized with a frequency distribution wherein they saw the midpoint face of the morphed continuum the most frequently. In the bimodal condition, infants were familiarized with a frequency distribution wherein they saw faces closer to the endpoints of the morphed continuum the most frequently. After familiarization, infants were tested on their discrimination of the two original Asian faces. The infants’ looking times during the test indicated that infants in the bimodal condition could discriminate between the two faces, while infants in the unimodal condition could not. These findings therefore suggest that 12-month-old Caucasian infants could be re-sensitized to Asian faces by familiarizing them with a bimodal frequency distribution of such faces.     

Testing for developmental continuity or discontinuity: Class inclusion and reversal shifts

Often the data in cognitive development consist of continuous cross-sectional increases in the proportion of children who respond in one of two ways to a diagnostic task. Such developmental trends are, however, ambiguous with reference to the course of individual, longitudinal development because they could be the result of either saltatory or gradual ontogenetic change. The present research tested a nonlongitudinal procedure for discriminating between these two alternatives. This procedure involves the administration of several equivalent versions of the diagnostic task at weekly intervals to children within a given age level in order to obtain a frequency distribution of the number of “successes” produced by each child. A bimodal distribution is interpreted as indicating saltatory ontogenetic change, while a unimodal distribution is taken to indicate gradual change. This method was applied to Piaget's class-inclusion problem and to the optional-shift task by administering four versions of each task to a range of different age-level samples. The optional-shift task produced clearly unimodal, bell-shaped distributions at all tested age levels, results which were interpreted to indicate a gradual ontogenetic increase in the probability of making reversal shifts. On the other hand, the frequency distributions for the class-inclusion task, at the appropriate age levels, were distinctly bimodal, and therefore consistent with a discontinuous ontogenetic transition. The capacity of the repeated-measures method to produce reliable within-task results and to differentiate between tasks was thus demonstrated.     

Should metacognition be measured by logistic regression?

Are logistic regression slopes suitable to quantify metacognitive sensitivity, i.e. the efficiency with which subjective reports differentiate between correct and incorrect task responses? We analytically show that logistic regression slopes are independent from rating criteria in one specific model of metacognition, which assumes (i) that rating decisions are based on sensory evidence generated independently of the sensory evidence used for primary task responses and (ii) that the distributions of evidence are logistic. Given a hierarchical model of metacognition, logistic regression slopes depend on rating criteria. According to all considered models, regression slopes depend on the primary task criterion. A reanalysis of previous data revealed that massive numbers of trials are required to distinguish between hierarchical and independent models with tolerable accuracy. It is argued that researchers who wish to use logistic regression as measure of metacognitive sensitivity need to control the primary task criterion and rating criteria.     

Judgment and Response Processes across Two Knowledge Domains

Wallsten and Gonzalez-Vallejo (1994) developed the Stochastic Judgment Model to account for true-false judgment and response processes in a single well-defined knowledge domain. This paper generalizes the model to a four-category rating task that encompasses two knowledge domains simultaneously. It then applies the model to an experiment in which Ph.D. students in history and English literature rated confidence in the truth of statements in both domains, and also decided which statement within a pair consisting of one from each domain was more likely true. Constrained versions of the general model fit the rating data very well and accurately predicted the pair-comparison (PC) choices. The results suggest that (a) the mean distance between the true and false statement distributions of confidence was greater in the better known domain; (b) judged confidence variability is greater in the domain of greater knowledge; while simultaneously (c) criterion variability is constant across domains; (d) the extreme response criteria are located symmetrically around the central one; which (e) is located to yield the usual bias to call statements true. Finally, cross-domain PC choices were very well predicted by assuming that respondents judged only the statement in the single domain they knew better and not well predicted by the more common assumption that they compare their levels of confidence in the two statements. Implications for the underlying cognitive processes are discussed including the effects of expertise.     

Combining probability values from independent permutation tests: a discrete analog of Fisher's classical method

Permutation tests are based on all possible arrangements of observed data sets. Consequently, such tests yield exact probability values obtained from discrete probability distributions. An exact nondirectional method to combine independent probability values that obey discrete probability distributions is introduced. The exact method is the discrete analog to Fisher's classical method for combining probability values from independent continuous probability distributions. If the combination of probability values includes even one probability value that obeys a sparse discrete probability distribution, then Fisher's classical method may be grossly inadequate.     

Why aren’t all deep superior colliculus neurons multisensory? A Bayes’ ratio analysis

Multisensory neurons in the deep superior colliculus (SC) show response enhancement to cross-modal stimuli that coincide in time and space. However, multisensory SC neurons respond to unimodal input as well. It is thus legitimate to ask why not all deep SC neurons are multisensory or, at least, develop multisensory behavior during an organism's maturation. The novel answer given here derives from a signal detection theory perspective. A Bayes' ratio model of multisensory enhancement is suggested. It holds that deep SC neurons operate under the Bayes' ratio rule, which guarantees optimal performance-that is, it maximizes the probability of target detection while minimizing the false alarm rate. It is shown that optimal performance of multisensory neurons vis-à-vis cross-modal stimuli implies, at the same time, that modality-specific neurons will outperform multisensory neurons in processing unimodal targets. Thus, only the existence of both multisensory and modality-specific neurons allows optimal performance when targets of one or several modalities may occur.     

Amplitude discrimination of sinusoids and narrow-band noise with Rayleigh properties

The ability of human observers to discriminate differences in the amplitude of sinusoids and narrow-band noises was measured by the rating method of detection theory. Although each sinusoid (always 1000 Hz) was presented at a fixed amplitude, its amplitude on any trial was drawn from one of two Rayleigh probability distributions that differed in mean amplitude: a signal distribution and a noise distribution. Similarly, the amplitudes of the narrow-band noises were distributed as the Rayleigh distribution by virtue of the reciprocal relation between their bandwidth (100 Hz centered on 1000 Hz) and duration (10 msec). The obtained psychometric functions showing the area under the ROC as a function of signal-to-noise ratio were similar for both kinds of signals and were displaced, on average, about 4 dB from an ideal observer’s function. The slopes of the obtained functions were similar to those of an ideal observer using 1 degree of freedom—half the number available in Rayleigh noise.     

Amplitude discrimination of sinusoids and narrow-band noise with Rayleigh properties.

The ability of human observers to discriminate differences in the amplitude of sinusoids and narrow-band noises was measured by the rating method of detection theory. Although each sinusoid (always 1000 Hz) was presented at a fixed amplitude, its amplitude on any trial was drawn from one of two Rayleigh probability distributions that differed in mean amplitude: a signal distribution and a noise distribution. Similarly, the amplitudes of the narrow-band noises were distributed as the Rayleigh distribution by virtue of the reciprocal relation between their bandwidth (100 Hz centered on 1000 Hz) and duration (10 msec). The obtained psychometric functions showing the area under the ROC as a function of signal-to-noise ratio were similar for both kinds of signals and were displaced, on average, about 4 dB from an ideal observer's function. The slopes of the obtained functions were similar to those of an ideal observer using 1 degree of freedom--half the number available in Rayleigh noise.     

How to Generate Non-normal Data for Simulation of Structural Equation Models

A procedure for generating non-normal data for simulation of structural equation models is proposed. A simple transformation of univariate random variables is used for the generation of data on latent and error variables under some restrictions for the elements of the covariance matrices for these variables. Data on the observed variables is then computed from latent and error variables according to the model. It is shown that by controlling univariate skewness and kurtosis on pre-specified random latent and error variables, observed variables can be made to have a relatively wide range of univariate skewness and kurtosis characteristics according to the pre-specified model. Univariate distributions are used for the generation of data which enables a user to choose from a large number of different distributions. The use of the proposed procedure is illustrated for two different structural equation models and it is shown how PRELIS can be used to generate the data.     

Effects of reward distribution and performance feedback on competitive responding

To implement competitive contingencies, one must select a distribution of unequal rewards and a schedule of feedback for competitors regarding one another's performance. This study investigated three bases for distributing rewards and two performance feedback conditions. Pairs of college students competed over a series of 2-min contests in which the competitive response was a knob pull. A sum of money was divided using a proportional distribution or one of two fixed reward distributions. In the proportional distribution, a subject's proportion of the sum was his or her proportion of the total number of responses. The two fixed distributions were divisions of 100%/0% or 67%/33%. Also, in every contest either subject could make a response that would end the contest prematurely and give both subjects the same amount—a sum equal to 33% of the competitive total. In the two feedback conditions, cumulative responses by each subject were either shown to both subjects during the contest or were not shown. The proportional distribution was clearly superior to either of the fixed distributions in number of responses produced across contests. The proportional distribution with feedback produced the largest number of competitive responses, and the 100%/0% distribution without feedback produced the smallest number. Differences among distributions typically emerged only during later blocks of contests. Fixed distributions of rewards often produced decelerating rates of responding, with losing competitors ending the contests before they were completed. Response-rate decreases were greatest for pairs in which the 2 subjects differed most in their response rates and proportion of wins. The presence of feedback had a small effect, increasing responding for some pairs in the 100%/0% distribution. Performance patterns were interpreted in terms of the consequences arranged for the individual participants by the reward distributions and differences in performance between competitors.     

A note on the sampling properties of the Vincentizing (quantile averaging) procedure

To assess the effect of a manipulation on a response time distribution, psychologists often use Vincentizing or quantile averaging to construct group or “average” distributions. We provide a theorem characterizing the large sample properties of the averaged quantiles when the individual RT distributions all belong to the same location-scale family. We then apply the theorem to estimating parameters for the quantile-averaged distributions. From the theorem, it is shown that parameters of the group distribution can be estimated by generalized least squares. This method provides accurate estimates of standard errors of parameters and can therefore be used in formal inference. The method is benchmarked in a small simulation study against both a maximum likelihood method and an ordinary least-squares method. Generalized least squares essentially is the only method based on the averaged quantiles that is both unbiased and provides accurate estimates of parameter standard errors. It is also proved that for location-scale families, performing generalized least squares on quantile averages is formally equivalent to averaging parameter estimates from generalized least squares performed on individuals. A limitation on the method is that individual RT distributions must be members of the same location-scale family.     

Brain responses reveal that infants' face discrimination is guided by statistical learning from distributional information

Infants' perception of faces becomes attuned to the environment during the first year of life. However, the mechanisms that underpin perceptual narrowing for faces are only poorly understood. Considering the developmental similarities seen in perceptual narrowing for faces and speech and the role that statistical learning has been shown to play for speech, the current study examined whether and how learning from distributional information impacts face identity discrimination. We familiarized 6.5‐month‐old infants with exemplars of female faces taken from a morphed continuum going from one identity to another. Using event‐related brain potentials (ERPs), we show that only infants who were familiarized with a bimodal frequency distribution, but not infants familiarized with a unimodal frequency distribution, discriminated between identities. These results are the first to demonstrate the influence of probabilistic information on infants' face identity discrimination, suggesting that statistical learning contributes to perceptual attunement for both faces and language.     

Multivariate analysis with linearizable regressions

We study the class of multivariate distributions in which all bivariate regressions can be linearized by separate transformation of each of the variables. This class seems more realistic than the multivariate normal or the elliptical distributions, and at the same time its study allows us to combine the results from multivariate analysis with optimal scaling and classical multivariate analysis. In particular a two-stage procedure which first scales the variables optimally, and then fits a simultaneous equations model, is studied in detail and is shown to have some desirable properties.     

A note on the use of rank-ordered logit models for ordered response categories

Models for rankings have been shown to produce more efficient estimators than comparable models for first/top choices. The discussions and applications of these models typically only consider unordered alternatives. But these models can be usefully adapted to the case where a respondent ranks a set of ordered alternatives that are ordered response categories. This paper proposes eliciting a rank order that is consistent with the ordering of the response categories, and then modelling the observed rankings using a variant of the rank ordered logit model where the distribution of rankings has been truncated to the set of admissible rankings. This results in lower standard errors in comparison to when only a single top category is selected by the respondents. And the restrictions on the set of admissible rankings reduces the number of decisions needed to be made by respondents in comparison to ranking a set of unordered alternatives. Simulation studies and application examples featuring models based on a stereotype regression model and a rating scale item response model are provided to demonstrate the utility of this approach.     

Estimating averages from distributions of tone durations

We examined whether estimating average duration was influenced by the distribution peak location. We presented participants with samples of various tone durations and then presented comparison tone durations. Participants judged whether each comparison duration was longer than the average sample duration. Estimates of the averages were inferred from the psychophysical functions. The durations were sampled from three distributions: one positively skewed, one symmetric, and one negatively skewed. In Experiment 1, every participant was presented with every distribution. Estimates of the averages were unbiased for the symmetric distribution but were biased toward the long tail of each skewed distribution. This would occur if participants combined the sample to be judged with the previous, irrelevant samples, or with the comparison durations. In Experiment 2, each participant was presented with samples from only one of the distributions. Estimates of the averages were still biased toward the long tails of the skewed distributions. This would occur if participants combined the sample to be judged with the comparison durations, which were the same for the three distributions. In Experiment 3, each participant was presented with only one distribution, and each distribution was tested with its own comparison durations, selected as percentiles from the distribution. The estimates were accurate for the smallest population mean (positively skewed distribution) but underestimated the larger means. These results could be explained by subjective shortening of the durations in memory, with a simple equation from scalar timing theory. This equation correctly predicted two results: The estimated averages were a linear function of the stimulus means, and the variances were a linear function of the squared stimulus means. Neither prediction was dependent on the skewness of the stimulus durations.     

The Effect of Individual Differences in Factor Loadings on the Standard Factor Model

It is shown that the population-covariance matrix of a heterogeneous factor model may be indistinguishable from that of a standard factor model and that the standard likelihood-ratio goodness-of-fit statistic has but little power in detecting loading heterogeneity. The relation between loading heterogeneity and factor score reliability is studied and it is recommended that non-normality of the test-score distributions be tested to use factor scores with more confidence. Substantive justifications for the model assumptions and model-based methods to test specific hypotheses about the loading distribution, are discussed.     

第二语言水平对双语者语言抑制能力的影响——来自英语–汉语单通道双语者和英语–美国手语双通道双语者的证据

采用同形异义词干扰任务考察第二语言水平对英语–汉语单通道双语者和英语–美国手语双通道双语者语言抑制能力的影响。结果发现:(1)高水平英语–汉语单通道双语者的语言抑制能力较强,但低水平英语–汉语单通道双语者与英语单语者的语言抑制能力没有显著差异,说明少量的双语经验不足以导致双语认知优势;(2)不同水平的英语–美国手语双通道双语者的语言抑制能力差异不显著。所以如此,与英语–美国手语双通道双语者不存在口语和手语的双语表征加工竞争有关。整个研究表明,双语认知优势效应与双语者的二语水平以及通道经验有关。