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.     

Empirical bayes estimation of coefficients in the general linear model from data of deficient rank

Empirical Bayes methods are shown to provide a practical alternative to standard least squares methods in fitting high dimensional models to sparse data. An example concerning prediction bias in educational testing is presented as an illustration.The authors would like to thank the referees for several useful comments.The analysis of the data discussed in this report was part of a study funded jointly by the Graduate Management Admission Council and Educational Testing Service.     

Do We “do”?

A normative framework for modeling causal and counterfactual reasoning has been proposed by Spirtes, Glymour, and Scheines (1993; cf. Pearl, 2000). The framework takes as fundamental that reasoning from observation and intervention differ. Intervention includes actual manipulation as well as counterfactual manipulation of a model via thought. To represent intervention, Pearl employed the do operator that simplifies the structure of a causal model by disconnecting an intervened-on variable from its normal causes. Construing the do operator as a psychological function affords predictions about how people reason when asked counterfactual questions about causal relations that we refer to as undoing, a family of effects that derive from the claim that intervened-on variables become independent of their normal causes. Six studies support the prediction for causal (A causes B) arguments but not consistently for parallel conditional (if A then B) ones. Two of the studies show that effects are treated as diagnostic when their values are observed but nondiagnostic when they are intervened on. These results cannot be explained by theories that do not distinguish interventions from other sorts of events.     

Marginal maximum likelihood estimation for a psychometric model of discontinuous development

Item response theory models posit latent variables to account for regularities in students' performances on test items. Wilson's “Saltus” model extends the ideas of IRT to development that occurs in stages, where expected changes can be discontinuous, show different patterns for different types of items, or even exhibit reversals in probabilities of success on certain tasks. Examples include Piagetian stages of psychological development and Siegler's rule-based learning. This paper derives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing approximation based on the EM algorithm. For individual examinees, empirical Bayes probabilities of learning-stage are given, along with proficiency parameter estimates conditional on stage membership. The MML solution is illustrated with simulated data and an example from the domain of mixed number subtraction. The authors' names appear in alphabetical order. We would like to thank Karen Draney for computer programming, Kikumi Tatsuoka for allowing us to use the mixed-number subtraction data, and Eric Bradlow, Chan Dayton, Kikumi Tatsuoka, and four anonymous referees for helpful suggestions. The first author's work was supported by Contract No. N00014-88-K-0304, R&T 4421552, from the Cognitive Sciences Program, Cognitive and Neural Sciences Division, Office of Naval Research, and by the Program Research Planning Council of Educational Testing Service. The second author's work was supported by a National Academy of Education Spencer Fellowship and by a Junior Faculty Research Grant from the Committee on Research, University of California at Berkeley. A copy of the Saltus computer program can be obtained from the second author.     

General estimators for the reliability of qualitative data

We study a proportional reduction in loss (PRL) measure for the reliability of categorical data and consider the general case in which each ofN judges assigns a subject to one ofK categories. This measure has been shown to be equivalent to a measure proposed by Perreault and Leigh for a special case when there are two equally competent judges, and the correct category has a uniform prior distribution. We consider a general framework where the correct category is assumed to have an arbitrary prior distribution, and where classification probabilities vary by correct category, judge, and category of classification. In this setting, we consider PRL reliability measures based on two estimators of the correct category—the empirical Bayes estimator and an estimator based on the judges' consensus choice. We also discuss four important special cases of the general model and study several types of lower bounds for PRL reliability.Bruce Cooil is Associate Professor of Statistics, and Roland T. Rust is Professor and area head for Marketing, Owen Graduate School of Management, Vanderbilt University. The authors thank three anonymous reviewers and an Associate Editor for their helpful comments and suggestions. This work was supported in part by the Dean's Fund for Faculty Research of the Owen Graduate School of Management, Vanderbilt University.     

Estimating latent distributions in recurrent choice data

This paper introduces a flexible class of stochastic mixture models for the analysis and interpretation of individual differences in recurrent choice and other types of count data. These choice models are derived by specifying elements of the choice process at the individual level. Probability distributions are introduced to describe variations in the choice process among individuals and to obtain a representation of the aggregate choice behavior. Due to the explicit consideration of random effect sources, the choice models are parsimonious and readily interpretable. An easy to implement EM algorithm is presented for parameter estimation. Two applications illustrate the proposed approach.     

A comparison of three simple test theory models

In very simple test theory models such as the Rasch model, a single parameter is used to represent the ability of any examinee or the difficulty of any item. Simple models such as these provide very important points of departure for more detailed modeling when a substantial amount of data are available, and are themselves of real practical value for small or even medium samples. They can also serve a normative role in test design.As an alternative to the Rasch model, or the Rasch model with a correction for guessing, a simple model is introduced which characterizes strength of response in terms of the ratio of ability and difficulty parameters rather than their difference. This model provides a natural account of guessing, and has other useful things to contribute as well. It also offers an alternative to the Rasch model with the usual correction for guessing. The three models are compared in terms of statistical properties and fits to actual data. The goal of the paper is to widen the range of minimal models available to test analysts.This research was supported by grant AP320 from the Natural Sciences and Engineering Research Council of Canada. The author is grateful for discussions with M. Abrahamowicz, I. Molenaar, D. Thissen, and H. Wainer.     

Relationship between social anxiety and perceived trustworthiness

Four different patterns of biased ratings of facial expressions of emotions have been found in socially anxious participants: higher negative ratings of (1) negative, (2) neutral, and (3) positive facial expressions than nonanxious controls. As a fourth pattern, some studies have found no group differences in ratings of facial expressions of emotion. However, these studies usually employed valence and arousal ratings that arguably may be less able to reflect processing of social information. We examined the relationship between social anxiety and face ratings for perceived trustworthiness given that trustworthiness is an inherently socially relevant construct. Improving on earlier analytical strategies, we evaluated the four previously found result patterns using a Bayesian approach. Ninety-eight undergraduates rated 198 face stimuli on perceived trustworthiness. Subsequently, participants completed social anxiety questionnaires to assess the severity of social fears. Bayesian modeling indicated that the probability that social anxiety did not influence judgments of trustworthiness had at least three times more empirical support in our sample than assuming any kind of negative interpretation bias in social anxiety. We concluded that the deviant interpretation of facial trustworthiness is not a relevant aspect in social anxiety.