Abstract
Higher-order latent traits are proposed for specifying the joint distribution of binary attributes in models for cognitive diagnosis. This approach results in a parsimonious model for the joint distribution of a high-dimensional attribute vector that is natural in many situations when specific cognitive information is sought but a less informative item response model would be a reasonable alternative. This approach stems from viewing the attributes as the specific knowledge required for examination performance, and modeling these attributes as arising from a broadly-defined latent trait resembling theϑ of item response models. In this way a relatively simple model for the joint distribution of the attributes results, which is based on a plausible model for the relationship between general aptitude and specific knowledge. Markov chain Monte Carlo algorithms for parameter estimation are given for selected response distributions, and simulation results are presented to examine the performance of the algorithm as well as the sensitivity of classification to model misspecification. An analysis of fraction subtraction data is provided as an example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Best, N.G., Cowles, M.K., & Vines, S.K. (1995). CODA: Convergence diagnosis and output analysis software for Gibbs sampling output (Version 0.30). [Computer software]. Cambridge: MRC Biostatistics Unit.
Bock, R.D., & Aitken, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.Psychometrika, 46, 443–459.
Casella, G., & George, E.I. (1992). Explaining the Gibbs sampler.The American Statistician, 46, 167–174.
Chib, S., & Greenberg, E. (1995). Understanding the Metropolis-Hastings algorithm.The American Statistician, 49, 327–335.
DiBello, L.V., Stout, W.F., & Roussos, L.A. (1995). Unified cognitive/psychometric diagnostic assessment likelihood-based classification techniques. In P.D. Nichols, S.F. Chipman, & R.L. Brennan,Cognitively diagnostic assessment (pp. 361–389). Hillsdale, NJ: Erlbaum.
Doignon, J.P., & Falmagne, J.C. (1999).Knowledge spaces. New York: Springer-Verlag.
Doornik, J.A. (2002). Object-oriented matrix programming using Ox (Version 3.1). [Computer software]. London: Timberlake Consultants Press.
Draney, K.L., Pirolli, P., & Wilson, M. (1995). A measurement model for complex cognitive skill. In P.D. Nichols, S.F. Chipman, R.L. Brennan (Eds.),Cognitively diagnostic assessment (pp. 103–125). Hillsdale, NJ: Erlbaum.
Embretson, S. (1984). A general latent trait model for response processes.Psychometrika, 49, 175–186.
Embretson, S. (1997). Multicomponent response models. In W.J. van der Linden & R.K. Hambleton (Eds.),Handbook of modern item response theory (pp. 305–321). New York: Springer-Verlag.
Everitt, B. S. (1998).The Cambridge dictionary of statistics. Cambridge, UK: Cambridge University Press.
Fischer, G. H. (1995). The linear logistic test model. In G. H. Fischer & I. W. Molenaar (Eds.),Rasch models: Foundations, recent developments, and applications (pp. 131–155) New York: Springer-Verlag.
Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images.IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
Gelman, A., & Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences (with discussion).Statistical Science, 7, 457–511.
Haertel, E.H. (1989). Using restricted latent class models to map the skill structure of achievement items.Journal of Educational Measurement, 26, 333–352.
Hagenaars, J.A. (1990).Categorical longitudinal data: Loglinear panel, trend, and cohort analysis. Newbury Park, CA: Sage.
Hagenaars, J.A. (1993).Loglinear models with latent variables. Newbury Park, CA: Sage.
Hartz, S. (2002).A Bayesian framework for the Unified Model for assessing cognitive abilities: Blending theory with practicality. Unpublished doctoral dissertation.
Junker, B.W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory.Applied Psychological Measurement, 25, 258–272.
Macready, G.B., & Dayton, C. M. (1977). The use of probabilistic models in the assessment of mastery.Journal of Educational Statistics, 33, 379–416.
Maris, E. (1999). Estimating multiple classification latent class models.Psychometrika, 64, 187–212.
Mislevy, R.J. (1996). Test theory reconceived.Journal of Educational Measurement, 33, 379–416.
Muthén, B. (1978). Contribution to factor analysis of binary variables.Psychometrika, 43, 551–560.
Patz, R.J., & Junker, B.W. (1999a). A straightforward approach to Markov chain Monte Carlo methods for item response theory.Journal of Educational and Behavioral Statistics, 24, 146–178.
Patz, R.J., & Junker, B.W. (1999b). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses.Journal of Educational and Behavioral Statistics, 24, 342–366.
Raftery, A.E. (1995). Bayesian model selection in social research.Sociological Methodology, 25, 111–163.
Raftery, A.E. (1996). Hypothesis testing and model selection. In R. W. Gilks, S. Richardson, & D. J. Spiegelhalter (Eds.),Markov chain Monte Carlo in practice (pp. 163–187). London: Chapman & Hall.
Reckase, M. (1997). A linear logistic multidimensional model for dichotomous item response data. In W.J. van der Linden & R.K. Hambleton (Eds.),Handbook of modern item response theory (pp. 271–286). New York: Springer-Verlag.
Tatsuoka, C. (2002). Data-analytic methods for latent partially ordered classification models.Journal of the Royal Statistical Society Series C (Applied Statistics), 51, 337–350.
Tatsuoka, K. (1985). A probabilistic model for diagnosing misconceptions in the pattern classification approach.Journal of Educational Statistics, 12, 55–73.
Tatsuoka, K. (1990). Toward an integration of item-response theory and cognitive error diagnosis. In N. Frederiksen, R. Glaser, A. Lesgold, & Safto, M. (Eds.),Monitoring skills and knowledge acquisition (pp. 453–488). Hillsdale, NJ: Erlbaum.
Tatsuoka, K. (1995). Architecture of knowledge structures and cognitive diagnosis: A statistical pattern recognition and classification approach. In P.D. Nichols, S.F. Chipman, & R.L. Brennan (Eds.),Cognitively diagnostic assessment (pp. 327–359). Hillsdale, NJ: Erlbaum.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was funded by National Institute of Health grant R01 CA81068. We would like to thank William Stout and Sarah Hartz for many useful discussions, three anonymous reviewers for helpful comments and suggestions, and Kikumi Tatsuoka and Curtis Tatsuoka for generously sharing data.
Rights and permissions
About this article
Cite this article
de la Torre, J., Douglas, J.A. Higher-order latent trait models for cognitive diagnosis. Psychometrika 69, 333–353 (2004). https://doi.org/10.1007/BF02295640
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02295640