Determination of joint distributions from marginal distributions in dichotomous systems

In many areas of research in which the experimenter deals with a set of dichotomous dimensions it is often desirable to specify them in terms of marginal distributions and some conditional distributions. This paper develops a set of systematic procedures for determination of joint distributions with an arbitrary number of dimensions. Most of these procedures require only addition and/or subtraction. An extension of the procedures permits easy determination of covariance and correlation matrices.This research was supported in part by National Institutes of Health grant No. MH-12541. Part of this work was done at Stanford University during work sessions of the 1967 Summer Conference on Mathematical Models in Perception and Learning. The author is indebted to Richard S. Bogartz for many helpful comments.     

Perception of social distributions

Accurate representation of the distribution of social attitudes and behaviors can guide effective social behavior and is often essential for correct inferences. We examined the accuracy of people's beliefs about the distributions of a large number of attitudinal and behavioral dimensions. In two studies we measured actual attitudes and behaviors in a student population, and we assessed beliefs by asking subjects to estimate the distribution of 100 students on these dimensions. We examined the accuracy of subjects' perceptions of the means, standard deviations, and distribution shapes. Subjects showed a number of systematic biases, including overestimation of dispersion and overestimation of the means of behavioral distributions and a false consensus bias, but their overall accuracy was impressive.     

Estimating latent distributions

Consider vectors of item responses obtained from a sample of subjects from a population in which ability is distributed with densityg(), where the are unknown parameters. Assuming the responses depend on through a fully specified item response model, this paper presents maximum likelihood equations for the estimation of the population parameters directly from the observed responses; i.e., without estimating an ability parameter for each subject. Also provided are asymptotic standard errors and tests of fit, computing approximations, and details of four special cases: a non-parametric approximation, a normal solution, a resolution of normal components, and a beta-binomial solution.The author would like to thank R. Darrell Bock for his comments, suggestions, and encouragement during the course of this work.     

Comparing latent distributions

We consider the problem of comparingm latent population distributions when the observed values are scores on a test battery with binary items. The latent densities are assumed to be normal densities, and we consider a test for equality of the means as well as a test equality of the variances. In addition, we consider a longitudinal model, where the test battery has been applied to the same individuals at different points in time. This model allows for correlations between the latent variable at different time points, and methods are discussed for estimating the correlation coefficient.This work was supported in part by a grant from the Danish Social Science Research Council.     

The survival of the survival lottery

ABSTRACT In his paper 'The Survival Lottery'John Harris suggested that there could be situations where the rational thing to do would be to kill a healthy person and harvest his organs for transplantation, thereby saving several lives at the cost of one. Anne Maclean claims that such a proposal, far from being rational, does not qualify as a moral proposal at all since what it suggests is 'plain murder'. I argue that she is correct to claim that the proposal is not uniquely rational and that doctors could quite rationally reject it, but that she overreaches herself when she holds it not to be a moral proposal at all.     

Skewness and asymmetry of distributions

The ‘skewness’ of a distribution, a poorly-defined term, is conventionally deemed to be invariant under linear transformations. A comparison is made of three criteria of it: the sign of odd central moments; the several relationships of the mean, the median and the mode; and asymmetry proper which is the set of ratios of the probability densities of all pairs of points equidistant above and below some arbitrary point, usually the principal mode. Some useful general relationships are discussed. The skeness of convolutions is briefly discussed. A class of distributions is identified for which the skewness of the minimum ofk competing processes is independent ofk. It has importance in the parametric study of survivorship.     

Visual discrimination of spectral distributions

Human observers seem to be able to use different features that classify materials with a large degree of accuracy. In this paper, we look at human perception of statistical properties of the spectral distribution in a scene. We investigated whether human observers can discriminate just as accurately between coloured textures that have a spectral distribution due either to shading only or to both shading and specular reflectance as between uniform colours. Thresholds for the discrimination of coloured textures are about 15 times as high as thresholds for the discrimination of uniform colours, provided there is a sharp transition between the two colours. However, the coloured texture thresholds are only 1.5 times higher when we introduce a gradual transition between the two colours. There are also distinct qualitative differences in discrimination thresholds for different base colours. These differences cannot be predicted from discrimination thresholds for uniform colours. Human observers are surprisingly good at discriminating between a material edge and a shadow edge in complex scenes. Statistical differences in the orientation of the colour distributions in colour space might be used to accomplish this. In a second experiment we investigated how well observers can discriminate between two linear distributions in colour space that have the same base colour but different orientations. When we vary the line-length in R, G, B space, thresholds cannot be predicted completely by the conservation of the average distance between the two distributions. This means that observers use not only the maximum colour difference in the stimulus to do the task, but other cues are also involved.     

Simulating multivariate nonnormal distributions

A method for generating multivariate nonnormal distributions with specified intercorrelations and marginal means, variances, skews, and kurtoses is proposed. As an example, the method is applied to the generation of simulated scores on three psychological tests administered to a single group of individuals.     

Rasch-representable reaction time distributions

This article investigates properties of a representation based on the Rasch test model for reaction times (RT) that was proposed by Micko. Necessary and sufficient conditions for a set of RT distributions to be Rasch-representable are derived. It is shown that independent serial and independent parallel processing models cannot be reconciled with the representation. However, random extreme models compatible with the Reasch-representation exist that assume RT is determined by the longest or he shortest processing time of a random number of independent paraloel channels. Nonparametric properties of Rasch-representable distributions are derived that can be used for testing the model and for estimating its parameters. Conditions are presented for Rasch-representable distributions to form a scale family. Finally, Rasch-represent-able distributions are characterized interms of their hazard functions.For helpful discussions, we are grateful to Hans Irtel, Christoph Micko, Hartmann Scheiblechner, and Hans-Henning Schultz.     

On simulating non-normal distributions

Six different algorithms to generate widely different non-normal distributions are reviewed. These algorithms are compared in terms of speed, simplicity and generality of the technique. The advantages and disadvantages of using these algorithms are briefly discussed.The author would like to thank an anonymous referee for his comments and suggestions which lead to the inclusion of the section A Comparative Study.     

Symmetric limiting distributions of response probabilities

Necessary and sufficient conditions for the symmetry of the limiting distributions of response probabilities are derived for the Bush and Mosteller two experimenter-controlled events learning model [Bush and Mosteller, 1955].     

Prior distributions on symmetric groups

Fully and partially ranked data arise in a variety of contexts. From a Bayesian perspective, attention has focused on distance-based models; in particular, the Mallows model and extensions thereof. In this paper, a class of prior distributions, the Binary Tree, is developed on the symmetric group. The attractive features of the class are: it provides a closed-form solution to the posterior distribution; and a simple way to interpret the parameters of the prior distribution. The advantages of the proposed method are illustrated by comparing it to metric-based models using data analyzed by other researchers in this context.This revised article was published online in August 2005 with the PDF paginated correctly.     

Individual distributions under ordinal measurement

For ordinal measurement the concept of an individual propensity distribution is developed. For any given individual the mean of this distribution is his true score, for which estimation procedures are discussed. Two measures of individual dispersion are considered and their distributions derived in the null case. These measures are shown to be counterparts at the individual level of Kendall's tau and Spearman's rho. Estimation of the two dispersion measures from sample data is investigated, and the relation of these estimates to the variance of the individual propensity distribution is derived.     

The sampling distributions of Gaussian ROC statistics

For a discrimination experiment, a plot of the hit rate against the false-alarm rate--the ROC curve--summarizes performance across a range of confidence levels. In many content areas, ROCs are well described by a normal-distribution model and the z-transformed hit and false-alarm rates are approximately linearly related. We examined the sampling distributions of three parameters of this model when applied to a ratings procedure: the area under the ROC (Az), the normalized difference between the means of the underlying signal and noise distributions (da), and the slope of the ROC on z-coordinates (s). Statistical bias (the degree to which the mean of the sampling distribution differed from the true value) was trivial for Az, small but noticeable for da, and substantial for s. Variability of the sampling distributions decreased with the number of trials and was also affected by the number of response categories available to the participant and by the overall sensitivity level. Figures in the article and tables available on line can be used to construct confidence intervals around ROC statistics and to test statistical hypotheses.     

On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes. The authors thank an associate editor and three referees for their constructive comments, which led to an improved version of the paper. This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of Notre Dame faculty research grant.     

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.     

A method for simulating non-normal distributions

A method of introducing a controlled degree of skew and kurtosis for Monte Carlo studies was derived. The form of such a transformation on normal deviates [X N(0, 1)] isY =a +bX +cX ² +dX ³. Analytic and empirical validation of the method is demonstrated.This work was done while the author was at the University of Illinois at Champaign-Urbana.     

Prior distributions for random choice structures

We study various axioms of discrete probabilistic choice, measuring how restrictive they are, both alone and in the presence of other axioms, given a specific class of prior distributions over a complete collection of finite choice probabilities. We do this by using Monte Carlo simulation to compute, for a range of prior distributions, probabilities that various simple and compound axioms hold. For example, the probability of the triangle inequality is usually many orders of magnitude higher than the probability of random utility. While neither the triangle inequality nor weak stochastic transitivity imply the other, the conditional probability that one holds given the other holds is greater than the marginal probability, for all priors in the class we consider. The reciprocal of the prior probability that an axiom holds is an upper bound on the Bayes factor in favor of a restricted model, in which the axiom holds, against an unrestricted model. The relatively high prior probability of the triangle inequality limits the degree of support that data from a single decision maker can provide in its favor. The much lower probability of random utility implies that the Bayes factor in favor of it can be much higher, for suitable data.     

Log-normal nanograin-size distributions in nanostructured composites

A key challenge in the fabrication of superhard nanocomposite films is how to control the distribution of grain sizes in these materials. High-resolution transmission electron microscopy has been used to measure nanograin-size distributions in the Ti–B–N films with various B contents. The results show that the mean grain size decreases with increase of B content and the grain-size distribution conforms to a log-normal function when the hardness approaches a maximum value. The transition from normal to log-normal distributions can be determined by analysis in terms of a minimum information criterion. The origin of a log-normal size distribution probably results from heterogeneity arising from a diffusion-drift process.