Fitting Psychometric Models with Methods Based on Automatic Differentiation

Quantitative psychology is concerned with the development and application of mathematical models in the behavioral sciences. Over time, models have become more complex, a consequence of the increasing complexity of research designs and experimental data, which is also a consequence of the utility of mathematical models in the science. As models have become more elaborate, the problems of estimating them have become increasingly challenging. This paper gives an introduction to a computing tool called automatic differentiation that is useful in calculating derivatives needed to estimate a model. As its name implies, automatic differentiation works in a routine way to produce derivatives accurately and quickly. Because so many features of model development require derivatives, the method has considerable potential in psychometric work. This paper reviews several examples to demonstrate how the methodology can be applied. From the Presidential Address delivered at the 70th Annual Meeting of the Psychometric Society, Tilburg University, The Netherlands, July 5–8, 2005.     

“Seeing Conversion Whole”: Testing a Model of Religious Conversion

Rambo (1993) theorized that religious conversion consists of a process involving seven dimensions, labeled context, crisis, quest, encounter, interaction, commitment, and consequences. To test the Rambo model, a new measure, the 97-item Adult Religious Conversion Experiences Questionnaire (ARCEQ), was developed, revised on the basis of feedback from a focus group, and administered by mail to 110 adult volunteers who self-identified as having experienced conversion. Reliability analysis of the subscales of the ARCEQ resulted in five (crisis, quest, encounter, commitment, and consequences) meeting the criterion ( > .80). A sixth (interaction) achieved a reliability of .76 and was retained, but the seventh (context) was dropped. Principal components factor analysis with Varimax rotation produced factors labeled Redemptive Love, Zealous Dedication, Dysphoric Need, Openness to Uncertainty, Extrinsic Crisis, and Experiential Learning, which could readily be interpreted in relation to Rambo's model. Subsidiary analyses showed a number of significant correlations between demographic characteristics of the sample and differences in religious conversion experience.     

Motivating operations and terms to describe them: some further refinements

Over the past decade, behavior analysts have increasingly used the term establishing operation (EO) to refer to environmental events that influence the behavioral effects of operant consequences. Nonetheless, some elements of current terminology regarding EOs may interfere with applied behavior analysts' efforts to predict, control, describe, and understand behavior. The present paper (a) describes how the current conceptualization of the EO is in need of revision, (b) suggests alternative terms, including the generic term motivating operation (MO), and (c) provides examples of MOs and their behavioral effects using articles from the applied behavior analysis literature.     

A person-fit index for polytomous rasch models,latent class models,and their mixture generalizations

A normally distributed person-fit index is proposed for detecting aberrant response patterns in latent class models and mixture distribution IRT models for dichotomous and polytomous data.This article extends previous work on the null distribution of person-fit indices for the dichotomous Rasch model to a number of models for categorical data. A comparison of two different approaches to handle the skewness of the person-fit index distribution is included.Major parts of this paper were written while the first author worked at the Institute for Science Education, Kiel, Germany. Any opinions expressed in this paper are those of the authors and not necessarily of Educational Testing Service. The results presented in this paper were improved by valuable comments from J. Rost, K. Yamamoto, N.D. Verhelst, E. Bedrick and two anonymous reviewers.     

A note on the equivalence of the graded response model and the sequential model

This paper concerns items that consist of several item steps to be responded to sequentially. The item scoreX is defined as the number of correct responses until the first failure. Samejima's graded response model states that each steph=1,...,m is characterized by a parameterb _h , and, for a subject with ability, Pr(Xh; )=F(–b _h ). Tutz's general sequential model associates with each step a parameterdh, and it states that Pr(Xh;)= _r ^=1h G(–d _r ). Tutz's (1991, 1997) conjectures that the models are equivalent if and only ifF(x)=G(x) is an extreme value distribution. This paper presents a proof for this conjecture.     

Effects of practice and task constraints on stiffness and friction functions in biological movements

A dynamical model of the movements of the platform of a ski-simulator was derived from experimental data, using the graphical and statistical methods developed by Beek and Beek (Beek, P. J., & Beek, W. J. (1988). Human Movement Science, 7, 301–342). The data were collected in an experiment in which both amplitude and practice were manipulated. The data were filtered and further reduced to normalised cycles that were averaged within and across subjects. Graphical methods were applied to these averaged normalised cycles to determine the stiffness and friction terms to be included in the model. The relative contribution of each term was assessed by means of multiple regression. The model, which included cubic and quintic Duffing terms and one or two dissipative Van der Pol terms, accounted on average for 99.2% of the variance. The exact parameter setting of the model differed considerably across subjects. For one subject, a qualitatively different model, including Rayleigh instead of Van der Pol terms, provided a better account of the data. Systematic changes of the coefficients in the model, related to amplitude and the duration of practice, were evident.     

Common and Unique Factors Associated with DSM-IV-TR Internalizing Disorders in Children

With the fifth edition of the Diagnostic and Statistical Manual of Mental Disorders (American Psychiatric Association. Diagnostic and statistical manual of mental disorders DSM-IV Fourth Edition-Text Revision. Author, Washington, DC. 2000) ahead, decisions will be made about the future of taxonomic conceptualizations. This study examined the factor structure of items from three internalizing disorders (Social Phobia, Generalized Anxiety Disorder, and Depression) on the Anxiety Disorders Interview Schedule for DSM-IV-Child Version (Silverman, W. K., & Albano, A. M. Anxiety disorders interview schedule for children for DSM-IV, child and parent versions. Psychological Corporation, San Antonio, 1996). Two-, three-, and four-factor models emerged in an exploratory factor analysis. Confirmatory factor analysis provided additional empirical support for the four-factor model over the two- or three-factor models. Implications for the structure of the DSM-V taxonomy in children and adolescents are discussed.     

Automated verbal credibility assessment of intentions: The model statement technique and predictive modeling

Recently, verbal credibility assessment has been extended to the detection of deceptive intentions, the use of a model statement, and predictive modeling. The current investigation combines these 3 elements to detect deceptive intentions on a large scale. Participants read a model statement and wrote a truthful or deceptive statement about their planned weekend activities (Experiment 1). With the use of linguistic features for machine learning, more than 80% of the participants were classified correctly. Exploratory analyses suggested that liars included more person and location references than truth‐tellers. Experiment 2 examined whether these findings replicated on independent‐sample data. The classification accuracies remained well above chance level but dropped to 63%. Experiment 2 corroborated the finding that liars' statements are richer in location and person references than truth‐tellers' statements. Together, these findings suggest that liars may over‐prepare their statements. Predictive modeling shows promise as an automated veracity assessment approach but needs validation on independent data.     

Hierarchical classes: Model and data analysis

A discrete, categorical model and a corresponding data-analysis method are presented for two-way two-mode (objects × attributes) data arrays with 0, 1 entries. The model contains the following two basic components: a set-theoretical formulation of the relations among objects and attributes; a Boolean decomposition of the matrix. The set-theoretical formulation defines a subset of the possible decompositions as consistent with it. A general method for graphically representing the set-theoretical decomposition is described. The data-analysis algorithm, dubbed HICLAS, aims at recovering the underlying structure in a data matrix by minimizing the discrepancies between the data and the recovered structure. HICLAS is evaluated with a simulation study and two empirical applications.This research was supported in part by a grant from the Belgian NSF (NFWO) to Paul De Boeck and in part by NSF Grant BNS-83-01027 to Seymour Rosenberg. We thank Iven Van Mechelen for clarifying several aspects of the Boolean algebraic formulation of the model and Phipps Arabie for his comments on an earlier draft.