A hierarchical bayesian statistical framework for response time distributions

This paper provides a statistical framework for estimating higher-order characteristics of the response time distribution, such as the scale (variability) and shape. Consideration of these higher order characteristics often provides for more rigorous theory development in cognitive and perceptual psychology (e.g., Luce, 1986). RT distribution for a single participant depends on certain participant characteristics, which in turn can be thought of as arising from a distribution of latent variables. The present work focuses on the three-parameter Weibull distribution, with parameters for shape, scale, and shift (initial value). Bayesian estimation in a hierarchical framework is conceptually straightforward. Parameter estimates, both for participant quantities and population parameters, are obtained through Markov Chain Monte Carlo methods. The methods are illustrated with an application to response time data in an absolute identification task. The behavior of the Bayes estimates are compared to maximum likelihood (ML) estimates through Monte Carlo simulations. For small sample size, there is an occasional tendency for the ML estimates to be unreasonably extreme. In contrast, by borrowing strength across participants, Bayes estimation shrinks extreme estimates. The results are that the Bayes estimators are more accurate than the corresponding ML estimators.We are grateful to Michael Stadler who allowed us use of his data. This research is supported by (a) National Science Foundation Grant SES-0095919 to J. Rouder, D. Sun, and P. Speckman, (b) University of Missouri Research Board Grant 00-77 to J. Rouder, (c) National Science Foundation grant DMS-9972598 to Sun and Speckman, and (d) a grant from the Missouri Department of Conservation to D. Sun.     

Testing indirect effect with a complete or incomplete dichotomous mediator

Past methodological research on mediation analysis mainly focused on situations where all variables were complete and continuous. When issues of categorical data occur combined with missing data, more methodological considerations are involved. Specifically, appropriate decisions need to be made on estimation methods of the indirect effects and on confidence intervals for testing the indirect effects with accommodations of missing data. We compare strategies that address these issues based on a model with a dichotomous mediator, aiming to provide guidelines for researchers facing such challenges in practice.     

Observed-score equating as a test assembly problem

A set of linear conditions on item response functions is derived that guarantees identical observed-score distributions on two test forms. The conditions can be added as constraints to a linear programming model for test assembly that assembles a new test form to have an observed-score distribution optimally equated to the distribution on an old form. For a well-designed item pool and items fitting the IRT model, use of the model results into observed-score pre-equating and prevents the necessity ofpost hoc equating by a conventional observed-score equating method. An empirical example illustrates the use of the model for an item pool from the Law School Admission Test.The authors are most indebted to Norman D. Verhelst for suggesting Proposition 4 and its proof, to the Law School Admission Council (LSAC) for making available the data set, and to Wim M. M. Tielen for his computational assistance.     

Tests for linear trend in the smallest eigenvalues of the correlation matrix

A test for linear trend among a set of eigenvalues of a correlation matrix is developed. As a technical implementation of Cattell's scree test, this is a generalization of Anderson's test for the equality of eigenvalues, and extends Bentler and Yuan's work on linear trends in eigenvalues of a covariance matrix. The power of minimumx ² and maximum likelihood ratio tests are compared. Examples show that the linear trend hypothesis is more realistic than the standard hypothesis of equality of eigenvalues, and that the hypothesis is compatible with standard decisions on the number of factors or components to retain in data analysis.This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017. The assistance of Maia Berkane and several anonymous reviewers is gratefully acknowledged.     

A rasch model for continuous ratings

A unidimensional latent trait model for continuous ratings is developed. This model is an extension of Andrich's rating formulation which assumes that the response process at latent thresholds is governed by the dichotomous Rasch model. Item characteristic functions and information functions are used to illustrate that the model takes ceiling and floor effects into account. Both the dichotomous Rasch model and a linear model with normally distributed error can be derived as limiting cases. The separability of the structural and incidental parameters is demonstrated and a procedure for estimating the parameters is outlined.     

A model for repeat trial product tests

A non-forced choice model is developed that describes subject behavior on repeat trial discrimination tests of the pick 1 ofk form. The model is developed from the Dirichlet distribution, and it allows for the derivation of individual true scores and of sampling properties for various constructs of interest. These results permit the analysis and comparison of test designs. The model is applied to issues such as forced vs. non-forced choice formats, the best number of alternatives at a choice point, and the selection of expert panels.     

The mixed model for multivariate repeated measures: validity conditions and an approximate test

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate correction factor is derived. The correction factor generalizes Box's and can be used to construct an adjusted MMM test.I am grateful to an anonymous referee for carefully attending to the mathematical details of this paper.     

Modelling non‐normal data: The relationship between the skew‐normal factor model and the quadratic factor model

Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew‐normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew‐normal factor are equivalent to those under a quadratic model up to third‐order moments. The reverse only holds if the quadratic loadings are equal to each other and within certain bounds. We illustrate that observed data which follow any skew‐normal factor model can be so well approximated with the quadratic factor model that the models are empirically indistinguishable, and that the reverse does not hold in general. The choice between the two models to account for deviations of normality is illustrated by an empirical example from clinical psychology.     

The effect of reinforcement differences on choice and response distribution during stimulus compounding

In Experiments I and II, rats were trained to respond on one lever during light and another during tone. The absence of tone and light controlled response cessation. In the multiple schedule of Experiment I, all reinforcements were received for responding in tone or light; in the chain schedule of Experiment II, all reinforcements were received in no tone + no light for not responding. Experiment I subjects, for which tone and light were associated with response and reinforcement increase, responded significantly more to tone-plus-light than to tone or light alone (additive summation). Experiment II subjects, for which tone and light were associated with response increase and reinforcement decrease, responded comparably to tone, light, and tone + light. Thus, additive summation was observed when stimulus-response and stimulus-reinforcer associations in tone and light were both positive, but not when they were conflicting. All subjects in both experiments responded predominantly on the light-correlated lever during tone + light, even when light intensity was reduced in testing. Furthermore, when a light was presented to a subject engaged in tone-associated responding, all subjects immediately switched the locus of responding to the light-correlated lever. No change in locus occurred when a tone was presented to a subject engaged in light-associated responding, irrespective of the stimulus-reinforcer association conditioned to tone. The light-lever preference in tone + light indicates that the heightened responding observed in Experiment I was not the summation of tone-associated behavior with light-associated behavior. Rather, it appears to be the result of a facilitation of one operant (light-associated responding) by the reinforcement-associated cue for the other.