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.     

Time Perception: Does It Distinguish ADHD and RD Children in a Clinical Sample?

This study used a double-dissociation design to evaluate whether children with ADHD demonstrated specific deficits relative to children with Reading Disorders. Recent theory suggests that ADHD children have deficits in time perception and working memory, whereas RD children have deficits in phonological decoding. The performance of 113 clinic-referred children aged 6-11 was examined using measures of working memory, phonological processing, and time perception. Respondents completed two time production tasks in which they were to judge when 30-s had elapsed, and another in which they were asked to estimate the duration of the Conners' CPT (CCPT). Time Perception and phonological processing variables were submitted to a 2 x 2 ANCOVA (ADHD vs. RD), covarying for age, SES, IQ, and working memory. Children with ADHD were more likely to overestimate the time taken for the CCPT than children without ADHD, but no group differences were found on the 30-s estimation tasks. Children with RD did not display deficits in time estimation, but showed deficits in auditory phonological processing. The lack of interaction effects supported an "etiological subtype" over the "phenocopy" model of ADHD and RD. No group differences were detected using the CCPT. Although our previous studies did not find an order effect for the Conners' CPT in a 1-hr battery, a fatigue effect was evident with a 1.5-hr battery. The implications for Barkley's behavioral inhibition theories (R. Barkley, 1997) are discussed.     

Second-order schedules of token reinforcement with pigeons: implications for unit price

Four pigeons were exposed to second-order schedules of token reinforcement, with stimulus lights serving as token reinforcers. Tokens were earned according to a fixed-ratio (token-production) schedule, with the opportunity to exchange tokens for food (exchange period) occurring after a fixed number had been produced (exchange-production ratio). The token-production and exchange-production ratios were manipulated systematically across conditions. Response rates varied inversely with the token-production ratio at each exchange-production ratio. Response rates also varied inversely with the exchange-production ratio at each token-production ratio, particularly at the higher token-production ratios. At higher token-production and exchange-production ratios, response rates increased in token-production segments closer to exchange periods and food. Some conditions were conducted in a closed economy, in which the pigeons earned all their daily ration of food within the session. Relative to comparable open-economy conditions, response rates in the closed economy were less affected by changes in token-production ratio, resulting in higher levels of food intake and body weight. Some of the results are consistent with the economic concept of unit price, a cost-benefit ratio comprised of responses per unit of food delivery, but most are well accounted for by a consideration of the number of responses required to produce exchange periods, without regard to the amount of reinforcement available during those exchange periods.     

Neuropsychological assessment with the Visual Gestalt Test: psychometric properties and differential diagnostic probabilities

The Visual Gestalt Test is a neuropsychological instrument developed for evaluation of learning and memory of visuo-spatial material. A revised strategy of scoring has motivated the present study, where data from 153 normal persons, 99 epilepsy patients, and 24 depressed patients are presented and compared. The Visual Gestalt Test is observed to discriminate between normal and diagnosed groups in several ways. Additionally it is found to discriminate between depressed and brain damage subgroups of patients. Data are presented in order to supplement previously published ways of scoring and norms. Practical guidelines for the clinical applications of the test are suggested as perspectives.     

Archery performance level and repeatability of event-related EMG

The purpose of the current study was to compare the repeatability of electromyographic linear envelopes (LE) of archery groups. Surface electromyography (EMG) signals of musculus flexor digitorum superficialis (MFDS) and extensor digitorum (MED) of 23 participants (seven skilled, six beginner archers and ten non-archers) were recorded during archery shooting. Two-second periods (clicker falls at first second) of 12 shots' EMG data were recorded, full-wave rectified and filtered (60 ms moving-average filter) for each participant's drawing arm. Repeatability was investigated by using a statistical criterion, variance ratio (VR). Archers' performances were evaluated in terms of FITA scores. The results showed that FITA scores were significantly correlated to the VRs of MFDS and MED. EMG LEs were more repeatable among archers than non-archers. Therefore, we inferred that VRs of MFDS and MED might be important variables for (a) assessing shooting techniques, (b) evaluation of archers' progress, and (c) selection of talented archers.     

Predicting the relative efficacy of three presentation methods for assessing preferences of persons with developmental disabilities

Choices were presented to 9 individuals with developmental disabilities using a two-choice format. Each pair of items, selected based on prior preference assessment, was presented to each participant in three conditions (actual items, pictures of the items, and spoken-name presentation) using a reversal design. The evaluation was conducted using food items, and was then repeated using nonfood items. The participants were also given a test to measure their skills on discrimination tasks ranging in difficulty from simple to conditional discriminations. The participants' abilities to make consistent choices with food and nonfood items were predicted, with 94% accuracy, by their discrimination skills. The findings suggest that presentation methods can affect the accuracy of a choice assessment, and that the systematic assessment of basic discrimination skills can be used to predict the effectiveness of different presentation methods in this population.     

On Nonequivalence of Several Procedures of Structural Equation Modeling

The normal theory based maximum likelihood procedure is widely used in structural equation modeling. Three alternatives are: the normal theory based generalized least squares, the normal theory based iteratively reweighted least squares, and the asymptotically distribution-free procedure. When data are normally distributed and the model structure is correctly specified, the four procedures are asymptotically equivalent. However, this equivalence is often used when models are not correctly specified. This short paper clarifies conditions under which these procedures are not asymptotically equivalent. Analytical results indicate that, when a model is not correct, two factors contribute to the nonequivalence of the different procedures. One is that the estimated covariance matrices by different procedures are different, the other is that they use different scales to measure the distance between the sample covariance matrix and the estimated covariance matrix. The results are illustrated using real as well as simulated data. Implication of the results to model fit indices is also discussed using the comparative fit index as an example. The work described in this paper was supported by a grant from the Research Grants Council of Hong Kong Special Administrative Region (Project No. CUHK 4170/99M) and by NSF grant DMS04-37167.     

Labor supply and consumption of food in a closed economy under a range of fixed- and random-ratio schedules: tests of unit price

The behavioral economic concept of unit price predicts that consumption and response output (labor supply) are determined by the unit price at which a good is available regardless of the value of the cost and benefit components of the unit price ratio. Experiment 1 assessed 4 pigeons' consumption and response output at a range of unit prices. In one condition, food was available according to a range of fixed-ratio schedules, whereas in the other condition, food was available according to a range of random-ratio schedules. Consistent with unit price predictions, consumption and response output were approximately equivalent across schedule types within the lower range of unit prices. However, at Unit Prices 64 (ratio value = 192) and greater, considerably more consumption and response output were observed in the random-ratio condition. Experiment 2 replicated these findings with 4 pigeons using the rapid demand curve assay procedure that is commonly used in the behavioral economics literature. Findings are integrated with two mathematical models of behavior under variable reinforcer delays.     

Bayesian modeling of measurement error in predictor variables using item response theory

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.This paper is part of the dissertation by Fox (2001) that won the 2002 Psychometric Society Dissertation Award.     

Classical Test Theory as a first-order Item Response Theory: Application to true-score prediction from a possibly nonparallel test

We give an account of Classical Test Theory (CTT) in terms of the more fundamental ideas of Item Response Theory (IRT). This approach views classical test theory as a very general version of IRT, and the commonly used IRT models as detailed elaborations of CTT for special purposes. We then use this approach to CTT to derive some general results regarding the prediction of the true-score of a test from an observed score on that test as well from an observed score on a different test. This leads us to a new view of linking tests that were not developed to be linked to each other. In addition we propose true-score prediction analogues of the Dorans and Holland measures of the population sensitivity of test linking functions. We illustrate the accuracy of the first-order theory using simulated data from the Rasch model, and illustrate the effect of population differences using a set of real data.This research is collaborative in every respect and the order of authorship is alphabetical. It was begun when both authors were on the faculty of the Graduate School of Education at the University of California, Berkeley.We would like to thank both Neil Dorans, Skip Livingston and two anonymous referees for many suggestions that have greatly improved this paper.