The use of decomposition in probability assessments of continuous variables

In this study, decomposition is used as a tool for the assessment of continuous probability distributions. The goal of using decomposition is to obtain better calibrated probability distributions by expressing a variable as a known function of several component variables. Three target quantities were used in the study. Each was assessed holistically and using two different decompositions. Thus, each subject provided three distributions for each of the three target quantities. The recomposed assessments were compared to holistic assessments. The distributions obtained using decomposition were found to be much better calibrated than those obtained holistically. Two methods of aggregating distributions from multiple subjects were also examined. One involves aggregating (averaging) distributions before recomposing while the second method involves recomposing and then averaging distributions for the target variable. The second method was found to be slightly better, although both showed better calibration than was found in the individual assessments.     

DRL interresponse-time distributions: quantification by peak deviation analysis.

Peak deviation analysis is a quantitative technique for characterizing interresponse-time distributions that result from training on differential-reinforcement-of-low-rate schedules of reinforcement. It compares each rat's obtained interresponse-time distribution to the corresponding negative exponential distribution that would have occurred if the rat had emitted the same number of responses randomly in time, at the same rate. The comparison of the obtained distributions with corresponding negative exponential distributions provides the basis for computing three standardized metrics (burst ratio, peak location, and peak area) that quantitatively characterize the profile of the obtained interresponse-time distributions. In Experiment 1 peak deviation analysis quantitatively described the difference between the interresponse-time distributions of rats trained on variable-interval 300-s and differential-reinforcement-of-low-rate 72-s schedules of reinforcement. In Experiment 2 peak deviation analysis differentiated between the effects of the psychomotor stimulant d-amphetamine, the anxiolytic compound chlordiazepoxide, and the antidepressant desipramine. The results suggest that peak deviation analysis of interresponse-time distributions may provide a useful behavioral assay system for characterizing the effects of drugs.     

Two-sample models with monotonic likelihood ratios for ordinal regression

Signal detection experiments with human observers frequently generate ordinal data which are evaluated using receiver operating characteristic (ROC) methods. These methods may include regressing a continuous two-distribution model to the data set. Because we assume that human observers will not systematically select observations absent the signal, this model should have a monotonic likelihood ratio between the distributions. This paper gives a general method for constructing pairs of distributions that have monotonic likelihood ratios and a possibly large number flexible parameters. It suggests two specific simple parametric forms of monotonic likelihood ratios, constructs new distributions with those likelihood ratios using other standard distributions, and performs ordinal regression with those new distributions to model some example data from the literature.     

Coefficient of variation calculated from the range for skewed distributions

In this research a coefficient of variation (CVS(high.low)) is developed that is calculated from the highest and lowest values in a set of data for samples from skewed distributions. A correction factor is determined such that CVS(high-low) is a dose estimate of the population coefficient of variation when sampling from three skewed chi-squared distributions and three skewed empirical distributions. The empirical distributions are from "real-world" data sets in psychology and education.     

QMPE: estimating Lognormal, Wald, and Weibull RT distributions with a parameter-dependent lower bound.

We describe and test quantile maximum probability estimator (QMPE), an open-source ANSI Fortran 90 program for response time distribution estimation. QMPE enables users to estimate parameters for the ex-Gaussian and Gumbel (1958) distributions, along with three "shifted" distributions (i.e., distributions with a parameter-dependent lower bound): the Lognormal, Wald, and Weibul distributions. Estimation can be performed using either the standard continuous maximum likelihood (CML) method or quantile maximum probability (QMP; Heathcote & Brown, in press). We review the properties of each distribution and the theoretical evidence showing that CML estimates fail for some cases with shifted distributions, whereas QMP estimates do not. In cases in which CML does not fail, a Monte Carlo investigation showed that QMP estimates were usually as good, and in some cases better, than CML estimates. However, the Monte Carlo study also uncovered problems that can occur with both CML and QMP estimates, particularly when samples are small and skew is low, highlighting the difficulties of estimating distributions with parameter-dependent lower bounds.     

Considerations in psychometric modeling of response time

Semiparametric models express a set of distributions of event times in terms of (a) a single parameter which varies across distributions and (b) a single function which does not vary across distributions and which has an unspecified form. These models appear to be attractive alternatives to parametric models of response times in psychometrics. However, our use of such models may require incorporating additional functions which do not vary across distributions and may require expressing the models in terms of the joint distribution of response class and response time.     

The Localization of Pitch Perception on the Basilar Membrane

Durations of four categories of perceptions in the binocular rivalry of 3.9° bright contours were considered. Distributions of durations of perceptions of each of the rivalry categories and of all rivalry reports together were tested against five standard distributions. The empirical distributions showed no tendency to fit any of the standard theoretical distributions well. The results suggest that speculations concerning the underlying mechanisms of binocular rivalry based on observations of the durations of rivalry reports may not be useful until the distributions of durations are satisfactorily described.     

Learning Continuous Probability Distributions with Symmetric Diffusion Networks

In this article we present symmetric diffusion networks, a family of networks that instantiate the principles of continuous, stochastic, adaptive and interactive propagation of information. Using methods of Markovion diffusion theory, we formalize the activation dynamics of these networks and then show that they can be trained to reproduce entire multivariate probability distributions on their outputs using the contrastive Hebbion learning rule (CHL). We show that CHL performs gradient descent on an error function that captures differences between desired and obtained continuous multivariate probability distributions. This allows the learning algorithm to go beyond expected values of output units and to approximate complete probability distributions on continuous multivariate activation spaces. We argue that learning continuous distributions is an important task underlying a variety of real-life situations that were beyond the scope of previous connectionist networks. Deterministic networks, like back propagation, cannot learn this task because they are limited to learning average values of independent output units. Previous stochastic connectionist networks could learn probability distributions but they were limited to discrete variables. Simulations show that symmetric diffusion networks can be trained with the CHL rule to approximate discrete and continuous probability distributions of various types.     

QMPE: Estimating Lognormal, Wald, and Weibull RT distributions with a parameter-dependent lower bound

We describe and test quantile maximum probability estimator (QMPE), an open-source ANSI Fortran 90 program for response time distribution estimation.¹ QMPE enables users to estimate parameters for the ex-Gaussian and Gumbel (1958) distributions, along with three “shifted” distributions (i.e., distributions with a parameter-dependent lower bound): the Lognormal, Wald, and Weibull distributions. Estimation can be performed using either the standard continuous maximum likelihood (CML) method or quantile maximum probability (QMP; Heathcote & Brown, in press). We review the properties of each distribution and the theoretical evidence showing that CML estimates fail for some cases with shifted distributions, whereas QMP estimates do not. In cases in which CML does not fail, a Monte Carlo investigation showed that QMP estimates were usually as good, and in some cases better, than CML estimates. However, the Monte Carlo study also uncovered problems that can occur with both CML and QMP estimates, particularly when samples are small and skew is low, highlighting the difficulties of estimating distributions with parameter-dependent lower bounds.     

Intrinsic and extrinsic contributions to heavy tails in visual foraging

Eyes move over visual scenes to gather visual information. Studies have found heavy-tailed distributions in measures of eye movements during visual search, which raises questions about whether these distributions are pervasive to eye movements, and whether they arise from intrinsic or extrinsic factors. Three different measures of eye movement trajectories were examined during visual foraging of complex images, and all three were found to exhibit heavy tails: Spatial clustering of eye movements followed a power law distribution, saccade length distributions were lognormally distributed, and the speeds of slow, small amplitude movements occurring during fixations followed a 1/f spectral power law relation. Images were varied to test whether the spatial clustering of visual scene information is responsible for heavy tails in eye movements. Spatial clustering of eye movements and saccade length distributions were found to vary with image type and task demands, but no such effects were found for eye movement speeds during fixations. Results showed that heavy-tailed distributions are general and intrinsic to visual foraging, but some of them become aligned with visual stimuli when required by task demands. The potentially adaptive value of heavy-tailed distributions in visual foraging is discussed.     

A proof for Rhiel's range estimator of the coefficient of variation for skewed distributions

In this research study is proof that the coefficient of variation (CV(high-low)) calculated from the highest and lowest values in a set of data is applicable to specific skewed distributions with varying means and standard deviations. Earlier Rhiel provided values for d(n), the standardized mean range, and a(n), an adjustment for bias in the range estimator of micro. These values are used in estimating the coefficient of variation from the range for skewed distributions. The d(n) and an values were specified for specific skewed distributions with a fixed mean and standard deviation. In this proof it is shown that the d(n) and an values are applicable for the specific skewed distributions when the mean and standard deviation can take on differing values. This will give the researcher confidence in using this statistic for skewed distributions regardless of the mean and standard deviation.     

The shape of self-evaluation: Implicit theories of intelligence and judgments of intellectual ability

Self-report distributions of self-evaluations are proposed to convey information beyond unidimensional (e.g., Likert-type) measures. Two studies tested the hypothesis that the shape of a distribution-type measure of self-evaluation of intellectual ability, as well as the central tendency and variability, is a meaningful indicator of individual differences. Specifically, one correlational study showed that measures of central tendency, variability, and skew were uniquely associated with self-ratings of ability level, self-certainty, and implicit theories of intelligence, respectively. An experiment explored the finding that incremental theorists (Dweck, 1999) reported more negatively skewed distributions than entity theorists. Only incremental theorists who wrote essays about recent intellectual growth created negatively skewed distributions; entity theorists did not. Evidence supports the hypotheses that self-report distributions are multiply informative and idiographic measures of self-evaluation, that negative skew on intellectual ability distributions can represent perceptions of growth, and that incremental theorists typically take this perspective when evaluating their own intellectual ability.     

The mirror effect and mixture signal detection theory

The mirror effect for word frequency refers to the finding that low-frequency words have higher hit rates and lower false alarm rates than high-frequency words. This result is typically interpreted in terms of conventional signal detection theory (SDT), in which case it indicates that the order of the underlying old item distributions mirrors the order of the new item distributions. However, when viewed in terms of a mixture version of SDT, the order of hits and false alarms does not necessarily imply the same order in the underlying distributions because of possible effects of mixing. A reversal in underlying distributions did not appear for fits of mixture SDT models to data from 4 experiments.     

The structure of response rate

Interresponse time distributions of the terminal rate under fixed-interval and fixed-ratio schedules were examined, using data from three rats in each case. By means of a sequential analysis, the overall interresponse time distributions were separated into orderly components. Consideration of the component distributions suggested that multiple determinants of rate act in succession, not simultaneously, and that probability of reinforcement has an important effect on the probability of occurrence of interresponse times.     

A statistical test for the significance of a coefficient of reproducibility

Under the assumption that responses to different dichotomous items are statistically independent, exact distributions for coefficients of reproducibility, CR, are derived. These distributions are useful in testing whether an observed coefficient of reproducibility differs significantly from chance expectations and whether further scaling manipulations are warranted. The effects of purification on sampling distributions of errors are calculated to demonstrate a relationship between scaling operations and expected chance values of the CR.     

Analyses of multinomial mixture distributions: new tests for stochastic models of cognition and action

Mixture distributions are formed from a weighted linear combination of 2 or more underlying basis distributions [g(x) = sigma j alpha j fj(x); sigma alpha j = 1]. They arise frequently in stochastic models of perception, cognition, and action in which a finite number of discrete internal states are entered probabilistically over a series of trials. This article reviews various distributional properties that have been examined to test for the presence of mixture distributions. A new multinomial maximum likelihood mixture (MMLM) analysis is discussed for estimating the mixing probabilities alpha j and the basis distributions fj(x) of a hypothesized mixture distribution. The analysis also generates a maximum likelihood goodness-of-fit statistic for testing various mixture hypotheses. Stochastic computer simulations characterize the statistical power of such tests under representative conditions. Two empirical studies of mental processes hypothesized to involve mixture distributions are summarized to illustrate applications of the MMLM analysis.     

Application of a variable criterion model to auditory reaction time as a function of the type of catch trial

A variable criterion model has been applied to data from an RT experiment by LaBerge in which catch trial stimuli varied in similarity to a single auditory RT signal. Estimates of a sensory recruitment function and the parameters of hypothetical criterion distributions were obtained by a scaling solution. Auditory recruitment was an exponential growth function, and the criterion distributions were normal. In combination, these estimates successfully predicted the empirical RT distributions.     

A test of the effectiveness of the differential-reinforcement-of-low-rate schedule

Pigeons and rats were used in a yoked-control design that equated the reinforcement distributions of differential-reinforcement-of-low-rate and variable-interval schedules. Both a between-subjects design and a within-subjects design found response rate higher for the variable-interval schedule than for the differential-reinforcement-of-low-rate schedule, thus demonstrating the effectiveness of the differential-reinforcement-of-low-rate contingency. The interresponse-time distributions were unimodal for all subjects under the variable-interval schedule and bimodal for pigeons under the differential-reinforcement-of-low-rate schedule. The interresponse-time distributions for rats under the differential-reinforcement-of-low-rate schedule were also bimodal in three of four cases but the height of the modes at the shorter interresponse times were small in both absolute value and in relation to the height of the modes at the shorter interresponse times of the pigeons' distributions.