Hostname: page-component-7b7567c8c-2rsb6 Total loading time: 0 Render date: 2026-11-13T03:03:11.340Z Has data issue: false hasContentIssue false
  • English
  • Français

A flexible latent trait model for response times in tests

Published online by Cambridge University Press:  01 January 2025

Jochen Ranger and
Jorg-Tobias Kuhn
Jochen Ranger*
Affiliation:
University of Giessen
Jorg-Tobias Kuhn
Affiliation:
University of Munster
*
Requests for reprints should be sent to Jochen Ranger, University of Giessen, Giessen, Germany. E-mail: Jochen.M.Ranger@psychol.uni-giessen.de
Get accessRights & Permissions [Opens in a new window]

Abstract

Latent trait models for response times in tests have become popular recently. One challenge for response time modeling is the fact that the distribution of response times can differ considerably even in similar tests. In order to reduce the need for tailor-made models, a model is proposed that unifies two popular approaches to response time modeling: Proportional hazard models and the accelerated failure time model with log–normally distributed response times. This is accomplished by resorting to discrete time. The categorization of response time allows the formulation of a response time model within the framework of generalized linear models by using a flexible link function. Item parameters of the proposed model can be estimated with marginal maximum likelihood estimation. Applicability of the proposed approach is demonstrated with a simulation study and an empirical application. Additionally, means for the evaluation of model fit are suggested.

Keywords

response timeproportional hazard modelaccelerated failure time model

Information

Type
Article
Information
Psychometrika , Volume 77 , Issue 1 , January 2012, pp. 31 - 47
Copyright
Copyright © 2011 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aranda-Ordaz, F.J. (1981). On two families of transformations to additivity for binary response data. Biometrika, 68, 357–363.CrossRefGoogle Scholar
Bartholomew, D.Knott, M. (1999). Latent variable models and factor analysis, London: Arnold.Google Scholar
Berger, M. (1997). Optimal designs for latent variable models: a review. In Rost, J.Langeheine, R. (Eds.), Applications of latent trait and latent class models in the social sciences (pp. 71–79). Münster: Waxmann.Google Scholar
Berger, M. (1998). Optimal design of tests with dichotomous and polytomous items. Applied Psychological Measurement, 22, 248–258.CrossRefGoogle Scholar
Borkenau, P.Ostendorf, F. (1993). NEO-Fünf-Faktoren Inventar (NEO-FFI) nach Costa und McCrae, Göttingen: Hogrefe.Google Scholar
Bos, C. (2002). A comparison of marginal likelihood computation methods (Tinbergen Institute Discussion Paper No. TI2002-084/4). Amsterdam: Vrije Universiteit.CrossRefGoogle Scholar
Bradburn, M., Clark, T., Love, S., Altman, D. (2003). Survival analysis Part II: Multivariate data analysis: an introduction to concepts and methods. British Journal of Cancer, 89, 431–436.CrossRefGoogle Scholar
Cowan, N., Elliott, E.M., Saults, J.S., Morey, C.C., Mattox, S., Hismjatullina, A., et al. (2005). On the capacity of attention: its estimation and its role in working memory and cognitive aptitudes. Cognitive Psychology, 51, 42–100.CrossRefGoogle ScholarPubMed
Cox, D. (1972). Regression models and life-tables. Journal of the Royal Statistical Society, B, 34, 187–220.CrossRefGoogle Scholar
Czado, C. (1994). Parametric link modification of both tails in binary regression. Statistical Papers, 35, 189–201.CrossRefGoogle Scholar
DeMars, C. (2005). Type I error rates for Parscale’s fit index. Educational and Psychological Measurement, 65, 42–50.CrossRefGoogle Scholar
Dempster, A., Laird, N., Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B, 39, 1–38.CrossRefGoogle Scholar
Doksum, K. (1987). An extension of partial likelihood methods for proportional hazard models to general transformation models. The Annals of Statistics, 15, 325–345.CrossRefGoogle Scholar
Doksum, K., Gasko, M. (1990). On a correspondence between models in binary regression analysis and in survival analysis. International Statistical Review, 58, 243–252.CrossRefGoogle Scholar
Douglas, J., Kosorok, M., Chewing, B. (1999). A latent variable model for discrete multivariate psychometric waiting times. Psychometrika, 64, 69–82.CrossRefGoogle Scholar
Evans, M., Swartz, T. (1995). Methods for approximating integrals in statistics with special emphasis on Bayesian integration problems. Statistical Science, 10, 254–272.CrossRefGoogle Scholar
Eysenck, H., Wilson, C., Jackson, C. (1998). Eysenck Personality Profiler (EPP-D), Frankfurt: Swets.Google Scholar
Fleming, T., Lin, D. (2000). Survival analysis in clinical trials: past developments and future directions. Biometrics, 56, 971–983.CrossRefGoogle ScholarPubMed
Furneaux, W. (1952). Some speed, error and difficulty relationships within a problem-solving situation. Nature, 170, 37–38.CrossRefGoogle Scholar
Heath, J.W., Fu, M.C., Jank, W. (2009). New global optimization algorithms for model-based clustering. Computational Statistics & Data Analysis, 53, 3999–4017.CrossRefGoogle Scholar
Heinzmann, D. (2008). A filtered polynomial approach to density estimation. Computational Statistics, 23, 343–360.CrossRefGoogle Scholar
Kang, T., Chen, T. (2008). Performance of the generalized S−X 2 item fit index for polytomous IRT models. Journal of Educational Measurement, 45, 391–406.CrossRefGoogle Scholar
Klein Entink, R., van der Linden, W., Fox, J. (2009). A Box–Cox normal model for response times. British Journal of Mathematical and Statistical Psychology, 62, 621–640.CrossRefGoogle ScholarPubMed
Luck, S.J., Vogel, E.K. (1997). The capacity of visual working memory for features and conjunctions. Nature, 390, 279–281.CrossRefGoogle ScholarPubMed
Maris, E. (1993). Additive and multiplicative models for gamma distributed random variables, and their application as psychometric models for response times. Psychometrika, 58, 445–469.CrossRefGoogle Scholar
Marubini, E., Valsecchi, M. (1995). Analysing survival data from clinical trials and observational studies, Chichester: Wiley.Google Scholar
Maydeu-Olivares, A., Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713–732.CrossRefGoogle Scholar
McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society, B, 42, 109–142.CrossRefGoogle Scholar
Meng, X., Rubin, D. (1993). Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika, 80, 267–278.CrossRefGoogle Scholar
Micko, H. (1969). A psychological scale for reaction time measurement. Acta Psychologica, 30, 324–335.CrossRefGoogle Scholar
Moran, P. (1971). Maximum-likelihood estimation in non-standard conditions. Mathematical Proceedings of the Cambridge Philosophical Society, 70, 441–451.CrossRefGoogle Scholar
Muraki, E., Bock, R.D. (1997). Parscale: IRT item analysis and test scoring for rating-scale data, Chicago: Scientific Software [Computer software].Google Scholar
Nelder, J., Mead, R. (1965). A simplex method for function minimization. The Computer Journal, 7, 308–313.CrossRefGoogle Scholar
Nettleton, D. (1999). Convergence properties of the EM algorithm in constrained parameter spaces. The Canadian Journal of Statistics, 27, 639–648.CrossRefGoogle Scholar
Orchard, T., Woodbury, M. (1972). A missing information principle: theory and applications. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 1, 697–715.Google Scholar
Orlando, M., Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. Applied Psychological Measurement, 24, 50–64.CrossRefGoogle Scholar
Parner, E. (1997). Inference in semiparametric frailty models. Unpublished doctoral dissertation, University of Aarhus, Arhus, Denmark.Google Scholar
Pregibon, D. (1980). Goodness of link tests for generalized linear models. Journal of the Royal Statistical Society, Series C, 29, 15–24.Google Scholar
Ramsay, J. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.CrossRefGoogle Scholar
Rubin, D. (1976). Inference and missing data. Biometrika, 63, 581–592.CrossRefGoogle Scholar
Salavei, V. (2006). Logistic approximation to the normal: the KL rational. Psychometrika, 71, 763–767.Google Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17, 1–100.Google Scholar
Scheiblechner, H. (1979). Specifically objective stochastic latency mechanisms. Journal of Mathematical Psychology, 19, 19–38.CrossRefGoogle Scholar
Schilling, S., Bock, R. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70, 533–555.CrossRefGoogle Scholar
Schnipke, D., Scrams, D. (2002). Exploring issues of examinee behavior: insights gaines from response-time analyses. In Mills, C., Potenza, M., Fremer, J., Ward, W. (Eds.), Computer-based testing: building the foundation for future assessments (pp. 237–266). Mahwah: Lawrence Erlbaum.Google Scholar
Self, S., Liang, K. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82, 605–610.CrossRefGoogle Scholar
Stroud, A. (1971). Approximate calculation of multiple integrals, Englewood Cliffs: Prentice-Hall.Google Scholar
Therneau, T., Grambsch, P. (2000). Modeling survival data: extending the Cox model, New York: Springer.CrossRefGoogle Scholar
van Breukelen, G. (1995). Psychometric and information processing properties of selected response time models. Psychometrika, 60, 95–113.CrossRefGoogle Scholar
van Breukelen, G. (1997). Separability of item and person parameters in response time models. Psychometrika, 62, 525–544.CrossRefGoogle Scholar
van der Linden, W. (2006). A lognormal model for response times on test items. Journal of Educational and Behavioral Statistics, 31, 181–204.CrossRefGoogle Scholar
van der Linden, W. (2009). Conceptual issues in response-time modeling. Journal of Educational Measurement, 46, 247–272.CrossRefGoogle Scholar
van der Linden, W., Klein Entink, R., Fox, J. (2010). IRT parameter estimation with response times as collateral information. Applied Psychological Measurement, 34, 327–347.CrossRefGoogle Scholar
van der Maas, H., Wagenmakers, E. (2005). A psychometric analysis of chess expertise. American Journal of Psychology, 118, 29–60.CrossRefGoogle ScholarPubMed
Vorberg, D., Schwarz, W. (1990). Rasch-representable reaction time distributions. Psychometrika, 55, 617–632.CrossRefGoogle Scholar
Wenger, M., Gibson, B. (2004). Using hazard functions to assess changes in processing capacity in an attentional cuing paradigm. Journal of Experimental Psychology, 30, 708–719.Google Scholar
Woods, C. (2007). Ramsay curve IRT for Likert-type data. Applied Psychological Measurement, 31, 195–212.CrossRefGoogle Scholar
Wu, C.F.J. (1983). On the convergence properties of the EM algorithm. The Annals of Statistics, 11, 95–103.CrossRefGoogle Scholar