A Dynamic Model of Cognitive Growth in a Population: Spatial Tasks and Conservation

A dynamic model of cognitive growth is developed that is applicable to cross-sectional studies of growth in a population, and that predicts the distribution of scores as a function of age. This model modifies the theory of P. van Geert (1991, Psychological Review, 98, 3-53) for the cognitive growth of an individual under limited resources, by taking into account the effect of schooling or training. The modified theory provides an explanation for the puzzling effect reported by C. J. Brainerd (1977, Psychological Bulletin, 84, 919-939), that the gain in scores obtained in concept training experiments is independent of the initial state of the learner. In developing the modified theory, it was found that the assumption of delayed feedback is unnecessary. To apply the model to populations, it was assumed that the ability parameters that appear in the growth function are distributed normally in the population. A distribution function G(s; t) was derived that predicts the distributions of scores s at age t in terms of a small number of parameters. Although the abilities are assumed to be distributed normally, the score distributions are often bimodal. The model was applied to the studies of H. Thomas and A. Lohaus (1993, Monographs of the Society for Research in Child Development, 58, 1-169) on the growth of the concepts of verticality and horizontality, and to the study of P. M. Bentler (1970, Perceptual and Motor Skills, 31, 855-859) on the growth of the concept of conservation. In these applications the score distributions are clearly bimodal, but there is no evidence of abrupt transitions or catastrophes. This shows that the presence of a "catastrophe flag" (bimodal distributions) does not necessarily imply the presence of a catastrophe. Copyright 1999 Academic Press.     

Random effects and extended generalized partial credit models

In this paper it is shown that under the random effects generalized partial credit model for the measurement of a single latent variable by a set of polytomously scored items, the joint marginal probability distribution of the item scores has a closed-form expression in terms of item category location parameters, parameters that characterize the distribution of the latent variable in the subpopulation of examinees with a zero score on all items, and item-scaling parameters. Due to this closed-form expression, all parameters of the random effects generalized partial credit model can be estimated using marginal maximum likelihood estimation without assuming a particular distribution of the latent variable in the population of examinees and without using numerical integration. Also due to this closed-form expression, new special cases of the random effects generalized partial credit model can be identified. In addition to these new special cases, a slightly more general model than the random effects generalized partial credit model is presented. This slightly more general model is called the extended generalized partial credit model. Attention is paid to maximum likelihood estimation of the parameters of the extended generalized partial credit model and to assessing the goodness of fit of the model using generalized likelihood ratio tests. Attention is also paid to person parameter estimation under the random effects generalized partial credit model. It is shown that expected a posteriori estimates can be obtained for all possible score patterns. A simulation study is carried out to show the usefulness of the proposed models compared to the standard models that assume normality of the latent variable in the population of examinees. In an empirical example, some of the procedures proposed are demonstrated.     

Personality and density distributions of behavior,emotions, and situations

Whole Trait Theory defines personality as a density distribution of one’s momentary behavior, complete with all of its parameters (e.g., mean, SD, skew, kurtosis). Two questions regarding these parameters remain largely unexamined: (1) are individual differences in these parameters stable? And (2) do scores on standard personality tests correspond to these parameters? The current study (N = 209) employed an experience sampling design (N_obs ≈ 8300) to examine the stability of density distribution parameters and the relationship between standard personality test scores and density distribution parameters of 10 behaviors/emotions and 8 situation characteristics. Results showed that, (a) individual differences in density distribution parameters are moderately stable and (b) at the bivariate level, personality was associated with numerous distribution parameters for a number of behaviors/emotions, and situations. However, when the appropriate statistical controls were taken into account, these associations diminished. While individual differences in density distribution parameters we moderately stable, standard personality measures rarely correspond to any other parameters of density distributions once the mean of the density distribution is known. Emotionality and eXtraversion appear as exceptions to this general pattern. These results imply that both theory and measurement in personality should be cognizant of within-person variability in behavior.     

Individual differences in components of reaction time distributions and their relations to working memory and intelligence

The authors bring together approaches from cognitive and individual differences psychology to model characteristics of reaction time distributions beyond measures of central tendency. Ex-Gaussian distributions and a diffusion model approach are used to describe individuals' reaction time data. The authors identified common latent factors for each of the 3 ex-Gaussian parameters and for 3 parameters central to the diffusion model using structural equation modeling for a battery of choice reaction tasks. These factors had differential relations to criterion constructs. Parameters reflecting the tail of the distribution (i.e., tau in the ex-Gaussian and drift rate in the diffusion model) were the strongest unique predictors of working memory, reasoning, and psychometric speed. Theories of controlled attention and binding are discussed as potential theoretical explanations.     

Bayesian inference for finite mixtures of generalized linear models with random effects

We present an hierarchical Bayes approach to modeling parameter heterogeneity in generalized linear models. The model assumes that there are relevant subpopulations and that within each subpopulation the individual-level regression coefficients have a multivariate normal distribution. However, class membership is not known a priori, so the heterogeneity in the regression coefficients becomes a finite mixture of normal distributions. This approach combines the flexibility of semiparametric, latent class models that assume common parameters for each sub-population and the parsimony of random effects models that assume normal distributions for the regression parameters. The number of subpopulations is selected to maximize the posterior probability of the model being true. Simulations are presented which document the performance of the methodology for synthetic data with known heterogeneity and number of sub-populations. An application is presented concerning preferences for various aspects of personal computers.     

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.     

Generalized latent trait models

In this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model. A unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented. We discuss in addition the scoring of individuals on the latent dimensions. The general framework presented allows, not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions. The approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly.     

An algorithm to generate discrete probability distributions: Binomial,hypergeometric, negative binomial,inverse hypergeometric,and Poisson

A recursion procedure for generating point probability values, cumulative probability values, and interval probability values of discrete distributions is described. Algorithms and associated FORTRAN functions for five representative discrete probability distributions are presented : binomial, hypergeometric, negative binomial, inverse hypergeometric, and Poisson. The joint use of recursion and a small arbitrary initial value ensures computational efficiency.     

Optimal Designs for the Rasch Model

In this paper, optimal designs will be derived for estimating the ability parameters of the Rasch model when difficulty parameters are known. It is well established that a design is locally D-optimal if the ability and difficulty coincide. But locally optimal designs require that the ability parameters to be estimated are known. To attenuate this very restrictive assumption, prior knowledge on the ability parameter may be incorporated within a Bayesian approach. Several symmetric weight distributions, e.g., uniform, normal and logistic distributions, will be considered. Furthermore, maximin efficient designs are developed where the minimal efficiency is maximized over a specified range of ability parameters.     

Fitting the psychometric function

A constrained generalized maximum likelihood routine for fitting psychometric functions is proposed, which determines optimum values for the complete parameter set--that is, threshold and slope--as well as for guessing and lapsing probability. The constraints are realized by Bayesian prior distributions for each of these parameters. The fit itself results from maximizing the posterior distribution of the parameter values by a multidimensional simplex method. We present results from extensive Monte Carlo simulations by which we can approximate bias and variability of the estimated parameters of simulated psychometric functions. Furthermore, we have tested the routine with data gathered in real sessions of psychophysical experimenting.     

Extensions of Rasch's multiplicative poisson model

Consideration will be given to a model developed by Rasch that assumes scores observed on some types of attainment tests can be regarded as realizations of a Poisson process. The parameter of the Poisson distribution is assumed to be a product of two other parameters, one pertaining to the ability of the subject and a second pertaining to the difficulty of the test. Rasch's model is expanded by assuming a prior distribution, with fixed but unknown parameters, for the subject parameters. The test parameters are considered fixed. Secondly, it will be shown how additional between- and within-subjects factors can be incorporated. Methods for testing the fit and estimating the parameters of the model will be discussed, and illustrated by empirical examples.     

A theoretical distribution for mental test scores

The negative hypergeometric distribution of raw scores on mental tests is derived from certain assumptions relating to test theory. This result is checked empirically in a number of examples. Further derivations lead to the bivariate distribution of parallel tests which is also verified with actual data. The bivariate distribution of raw score and true score is also derived from a further assumption. This distribution is used to set confidence limits for true scores for persons with a given raw score.This work was supported in part by contract Nonr-2752(00) between the Office of Naval Research and Educational Testing Service. Reproduction in whole or in part for any purpose of the United States Government is permitted.     

四参数Logistic模型和传统模型对被试作答拟合能力的比较研究

针对测验中高能力被试答错容易试题的睡眠现象,可使用四参数Logistic模型分析数据。研究选取了来自心理测验和成就测验的实际数据,分别采用传统模型和四参数Logistic模型进行拟合,对不同模型的拟合指标及参数估计结果进行比较。结果表明,四参数Logistic模型能够提高拟合程度,增强估计结果的准确性,有效纠正高能力被试能力被低估的现象。建议在必要时使用四参数Logistic模型进行数据分析。     

Some contributions to efficient statistics in structural models: Specification and estimation of moment structures

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.     

Bootstrap-Calibrated Interval Estimates for Latent Variable Scores in Item Response Theory

In most item response theory applications, model parameters need to be first calibrated from sample data. Latent variable (LV) scores calculated using estimated parameters are thus subject to sampling error inherited from the calibration stage. In this article, we propose a resampling-based method, namely bootstrap calibration (BC), to reduce the impact of the carryover sampling error on the interval estimates of LV scores. BC modifies the quantile of the plug-in posterior, i.e., the posterior distribution of the LV evaluated at the estimated model parameters, to better match the corresponding quantile of the true posterior, i.e., the posterior distribution evaluated at the true model parameters, over repeated sampling of calibration data. Furthermore, to achieve better coverage of the fixed true LV score, we explore the use of BC in conjunction with Jeffreys’ prior. We investigate the finite-sample performance of BC via Monte Carlo simulations and apply it to two empirical data examples.     

The contagious Poisson distribution of Rorschach scores

Rorschach scores show frequency distributions that are neither normal nor Poisson. Their generating equation may be a formula for a “contagious Poisson” distribution developed by Coleman for use in social psychology. Data from the Harvard Student Study and from a sample of residents and interns both show good fit to the Coleman formula for nearly all Rorschach variables. Scores associated with abnormal mental states do not show good fit, however. The mathematical model may represent contagion once a type of response has occurred, or individual differences in response proneness, or perhaps an emission process. The corresponding urn model is one in which when one black ball is drawn, two are replaced.     

Alternative Approaches to Addressing Non-Normal Distributions in the Application of IRT Models to Personality Measures

It is generally assumed that the latent trait is normally distributed in the population when estimating logistic item response theory (IRT) model parameters. This assumption requires that the latent trait be fully continuous and the population homogenous (i.e., not a mixture). When this normality assumption is violated, models are misspecified, and item and person parameter estimates are inaccurate. When normality cannot be assumed, it might be appropriate to consider alternative modeling approaches: (a) a zero-inflated mixture, (b) a log-logistic, (c) a Ramsay curve, or (d) a heteroskedastic-skew model. The first 2 models were developed to address modeling problems associated with so-called quasi-continuous or unipolar constructs, which apply only to a subset of the population, or are meaningful at one end of the continuum only. The second 2 models were developed to address non-normal latent trait distributions and violations of homogeneity of error variance, respectively. To introduce these alternative IRT models and illustrate their strengths and weaknesses, we performed real data application comparing results to those from a graded response model. We review both statistical and theoretical challenges in applying these models and choosing among them. Future applications of these and other alternative models (e.g., unfolding, diffusion) are needed to advance understanding about model choice in particular situations.     

On appropriate procedures for combining probability distributions within the same family

This article considers procedures for combining individual probability distributions that belong to some “family” into a “group” probability distribution that belongs to the same family. The procedures considered are Vincentizing, in which quantiles are averaged across distributions; generalized Vincentizing, in which the quantiles are transformed before averaging; and pooling based on the distribution function or the probability density function. Some of these results are applied to models of reaction time in psychological experiments.     

Response styles in the assessment of anger expression

This study demonstrates how mixture distribution item response models can be used to detect different response styles in the clinical assessment of anger expression. Analyses of 3 subscales of the State-Trait Anger Expression Inventory in a clinical sample of 4,497 patients revealed that there are different response styles that manifest themselves in 2- and 3-class solutions. These solutions are robust across subsamples. Response styles reflect both psychologically meaningful biases (i.e., social desirability) and nonmeaningful response category preferences. Person parameters that correct for class membership (and thus, for response styles) are computed and compared with raw scores. The implications of these results for research on clinical assessment are discussed.     

A Dynamic Model for Rule Induction Tasks

In this paper, a model for performance on rule induction tasks (e.g., items on intelligence tests) is developed. The model simultaneously specifies distributions for response times and response accuracies on an item-by-item basis. It is dynamic in the sense that it can be used to specify and test different ways of learning throughout a test. Three versions of the general model (i.e., with three different learning rules) are described and the fit of these versions is investigated in two datasets on solving number series. The results indicate that one of these versions (one of the learning rules) is better at accounting for the data.