Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-07T17:48:13.459Z Has data issue: false hasContentIssue false
  • English
  • Français

Response Mixture Modeling: Accounting for Heterogeneity in Item Characteristics across Response Times

Published online by Cambridge University Press:  01 January 2025

Dylan Molenaar  and
Paul de Boeck
Dylan Molenaar*
Affiliation:
Psychological Methods, Department of Psychology, University of Amsterdam
Paul de Boeck
Affiliation:
Ohio State University
*
Correspondence should be made to Dylan Molenaar, Psychological Methods, Department of Psychology, University of Amsterdam, PO box 15906, 1001 NK Amsterdam, The Netherlands. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In item response theory modeling of responses and response times, it is commonly assumed that the item responses have the same characteristics across the response times. However, heterogeneity might arise in the data if subjects resort to different response processes when solving the test items. These differences may be within-subject effects, that is, a subject might use a certain process on some of the items and a different process with different item characteristics on the other items. If the probability of using one process over the other process depends on the subject’s response time, within-subject heterogeneity of the item characteristics across the response times arises. In this paper, the method of response mixture modeling is presented to account for such heterogeneity. Contrary to traditional mixture modeling where the full response vectors are classified, response mixture modeling involves classification of the individual elements in the response vector. In a simulation study, the response mixture model is shown to be viable in terms of parameter recovery. In addition, the response mixture model is applied to a real dataset to illustrate its use in investigating within-subject heterogeneity in the item characteristics across response times.

Keywords

item response theoryresponse time modelingmixture modeling
Type
Original Paper
Information
Psychometrika , Volume 83 , Issue 2 , June 2018 , pp. 279 - 297
Copyright
Copyright © 2018 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.)

Footnotes

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11336-017-9602-9) contains supplementary material, which is available to authorized users.

References

Bacci, S., Pandolfi, S., Pennoni, F., (2014). A comparison of some criteria for states selection in the latent Markov model for longitudinal data, Advances in Data Analysis and Classification, 8(2), 125–145.CrossRefGoogle Scholar
Bartholomew, D.J., Knott, M., Moustaki, I., (2011). Latent variable models and factor analysis: An unified approach (Vol. 904).Hoboken:Wiley.CrossRefGoogle Scholar
Birnbaum, A., (1968). Some latent trait models and their use in inferring an examinee’s ability. In Lord, E.M., Novick, M.R.(Eds.), Statistical theories of mental test scores (chap) (pp.17–20) Reading, MA:Addison Wesley.Google Scholar
Bock, R.D., Aitkin, M., (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm, Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bolsinova, M., Maris, G., (2016). A test for conditional independence between response time and accuracy, British Journal of Mathematical and Statistical Psychology, 69(1), 62–79.CrossRefGoogle ScholarPubMed
Bolsinova, M., Tijmstra, J., (2016). Posterior predictive checks for conditional independence between response time and accuracy, Journal of Educational and Behavioral Statistics, 41(2), 123–145.CrossRefGoogle Scholar
Bolsinova, M., de Boeck, P., & Tijmstra, J., (in press). Modelling conditional dependence between response time and accuracy. Psychometrika.Google Scholar
Bolsinova, M., Tijmstra, J., Molenaar, D., De Boeck, P., (2017). Conditional dependence between response time and accuracy: An overview of its possible sources and directions for distinguishing between them, Frontiers in Psychology, 8, 202.CrossRefGoogle ScholarPubMed
Carpenter, P.A., Just, M.A., Shell, P., (1990). What one intelligence test measures: A theoretical account of the processing in the Raven Progressive Matrices Test, Psychological Review, 97, 404–431.CrossRefGoogle Scholar
Celeux, G., Forbes, F., Robert, C.P., Titterington, D.M., (2006). Deviance information criteria for missing data models, Bayesian Analysis, 1(4), 651–673.CrossRefGoogle Scholar
Cohen, J., (1983). The cost of dichotomization, Applied Psychological Measurement, 7, 249–253.CrossRefGoogle Scholar
De Boeck, P., Chen, H., Davison, M., (2017). Spontaneous and imposed speed of cognitive test responses, British Journal of Mathematical and Statistical Psychology, 70(2), 225–237.CrossRefGoogle ScholarPubMed
De Boeck, P., Partchev, I., (2012). IRTrees: Tree-based item response models of the GLMM family, Journal of Statistical Software, 48(1), 1–28.CrossRefGoogle Scholar
DiTrapani, J., Jeon, M., De Boeck, P., Partchev, I., (2016). Attempting to differentiate fast and slow intelligence: Using generalized item response trees to examine the role of speed on intelligence tests, Intelligence, 56, 82–92.CrossRefGoogle Scholar
Dolan, C.V., Colom, R., Abad, F.J., Wicherts, J.M., Hessen, D.J., van de Sluis, S., (2006). Multi- group covariance and mean structure modeling of the relationship between the WAIS-III common factors and sex and educational attainment in Spain, Intelligence, 34, 193–210.CrossRefGoogle Scholar
Dolan, C.V., van der Maas, H.L., (1998). Fitting multivariage normal finite mixtures subject to structural equation modeling, Psychometrika, 63(3), 227–253.CrossRefGoogle Scholar
Ericsson, K.A., Staszewski, J.J., (1989). Skilled memory and expertise: Mechanisms of exceptional performance. In Klahr, D., Kotovsky, K.(Eds.), Complex information processing: The impact of Herbert A. Simon. NY:Hillsdale: Erlbaum.Google Scholar
Ferrando, P.J., Lorenzo-Seva, U., (2007). An item response theory model for incorporating response time data in binary personality items, Applied Psychological Measurement, 31, 525–543.CrossRefGoogle Scholar
Ferrando, P.J., Lorenzo-Seva, U., (2007). A measurement model for Likert responses that incorporates response time, Multivariate Behavioral Research, 42, 675–706.CrossRefGoogle Scholar
Fox, J.P., Klein Entink, R., Linden, W., (2007). Modeling of responses and response times with the package cirt, Journal of Statistical Software, 20(7), 1–14.CrossRefGoogle Scholar
Grabner, R.H., Ansari, D., Koschutnig, K., Reishofer, G., Ebner, F., Neuper, C., (2009). To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving, Neuropsychologia, 47(2), 604–608.CrossRefGoogle ScholarPubMed
Gelman, A., Rubin, D.B., (1992). Inference from iterative simulation using multiple sequences, Statistical Science, 7, 457–511.CrossRefGoogle Scholar
Gilks, W. R., Richardson, S., & Spiegelhalter, D. J. (1996). Introducing markov chain monte carlo. In: Markov chain Monte Carlo in practice (pp. 1–19). US: Springer..Google Scholar
Goldhammer, F., Naumann, J., Stelter, A., Tóth, K., Rölke, H., Klieme, E., (2014). The time on task effect in reading and problem solving is moderated by task difficulty and skill: Insights From a computer-based large-scale assessment, Journal of Educational Psychology, 106, 608–626.CrossRefGoogle Scholar
Hall, D.B., (2000). Zero-inflated Poisson and binomial regression with random effects: A case study, Biometrics, 56(4), 1030–1039.CrossRefGoogle ScholarPubMed
Jansen, B.R., Van der Maas, H.L., (2001). Evidence for the phase transition from Rule I to Rule II on the balance scale task, Developmental Review, 21(4), 450–494.CrossRefGoogle Scholar
Jeon, M., De Boeck, P., (2016). A generalized item response tree model for psychological assessments, Behavior Research Methods, 48(3), 1070–1085.CrossRefGoogle ScholarPubMed
Larson, G.E., Alderton, D.L., (1990). Reaction time variability and intelligence: A “worst performance” analysis of individual differences, Intelligence, 14(3), 309–325.CrossRefGoogle Scholar
Lubke, G.H., Muthén, B., (2005). Investigating population heterogeneity with factor mixture models, Psychological Methods, 10(1), 21.CrossRefGoogle ScholarPubMed
Luce, R.D., (1986). Response times. New York:Oxford University Press.Google Scholar
MacCallum, R.C., Zhang, S., Preacher, K.J., Rucker, D.D., (2002). On the practice of dichotomization of quantitative variables, Psychological Methods, 7, 19–40.CrossRefGoogle ScholarPubMed
McLeod, L., Lewis, C., Thissen, D., (2003). A Bayesian method for the detection of item preknowledge in computerized adaptive testing, Applied Psychological Measurement, 27(2), 121–137.CrossRefGoogle Scholar
Mellenbergh, G.J., (1994). Generalized linear item response theory, Psychological Bulletin, 115, 300–300.CrossRefGoogle Scholar
Min, Y., Agresti, A., (2005). Random effect models for repeated measures of zero-inflated count data, Statistical Modelling, 5(1), 1–19.CrossRefGoogle Scholar
Molenaar, D., Oberski, D., Vermunt, J., De Boeck, P., (2016). Hidden Markov item response theory models for responses and response times, Multivariate Behavioral Research, 51(5), 606–626.CrossRefGoogle ScholarPubMed
Molenaar, D., Tuerlinckx, F., van der Maas, H.L.J., (2015). A bivariate generalized linear item response theory modeling framework to the analysis of responses and response times, Multivariate Behavioral Research, 50(1), 56–74.CrossRefGoogle Scholar
Partchev, I., De Boeck, P., (2012). Can fast and slow intelligence be differentiated?, Intelligence, 40(1), 23–32.CrossRefGoogle Scholar
Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd international workshop on distributed statistical computing (DSC 2003) (pp. 20–22)..Google Scholar
Rabbitt, P., (1979). How old and young subjects monitor and control responses for accuracy and speed, British Journal of Psychology, 70, 305–311.CrossRefGoogle Scholar
Ranger, J., Ortner, T.M., (2011). Assessing personality traits through response latencies using item response theory, Educational and Psychological Measurement, 71(2), 389–406.CrossRefGoogle Scholar
Rummel, R.J., (1970). Applied factor analysis. Evanston:Northwestern University Press.Google Scholar
Schnipke, D.L., Scrams, D.J., (1997). Modeling item response times with a two-state mixture model: A new method of measuring speededness, Journal of Educational Measurement, 34, 213–232.CrossRefGoogle Scholar
Shiffrin, R.M., Schneider, W., (1977). Controlled and automatic human information processing: II. Perceptual learning, automatic attending and a general theory, Psychological Review, 84, 127.CrossRefGoogle Scholar
Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R., (1995). BUGS: Bayesian inference using Gibbs sampling, Version 0.50. Cambridge:MRC Biostatistics Unit.Google Scholar
Spiegelhalter, D.J., Best, NG-, Carlin, B.P., van der Linde, A., (2002). Bayesian measures of model complexity and fit (with discussion), Journal of the Royal Statistical Society Series B, 64, 583–640.CrossRefGoogle Scholar
Stan Development Team. (2015). Stan modeling language users guide and reference manual. Version 2(9).Google Scholar
Siegler, R.S., (1976). Three aspects of cognitive development, Cognitive Psychology, 8, 481–520.CrossRefGoogle Scholar
Thissen, D., (1983). Timed testing: An approach using item response testing. In Weiss, D.J.(Eds.), New horizons in testing: Latent trait theory and computerized adaptive testing. (pp 179–203). New York:Academic Press.Google Scholar
Thomas, A., Hara, B.O., Ligges, U., Sturtz, S., (2006). Making BUGS open, R News, 6, 12–17.Google Scholar
Titterington, D.M., Smith, A.F.M., Makov, U.E., (1985) Statistical analysis of finite mixture distributions. Chicester:Wiley.Google Scholar
Tuerlinckx, F., De Boeck, P., (2005). Two interpretations of the discrimination parameter, Psychometrika, 70(4), 629–650.CrossRefGoogle Scholar
Van Harreveld, F., Wagenmakers, E.J., Van Der Maas, H.L., (2007). The effects of time pressure on chess skill: An investigation into fast and slow processes underlying expert performance, Psychological Research, 71, 591–597.CrossRefGoogle ScholarPubMed
Van der Linden, W.J., (2007). A hierarchical framework for modeling speed and accuracy on test items, Psychometrika, 72, 287–308.CrossRefGoogle Scholar
Van der Linden, W.J., (2009). Conceptual issues in response-time modeling, Journal of Educational Measurement, 46(3), 247–272.CrossRefGoogle Scholar
Van der Linden, W.J., Breithaupt, K., Chuah, S.C., Zhang, Y., (2007). Detecting differential speededness in multistage testing, Journal of Educational Measurement, 44(2), 117–130.CrossRefGoogle Scholar
Van der Linden, W.J., Glas, C.A., (2010). Statistical tests of conditional independence between responses and/or response times on test items, Psychometrika, 75, 120–139.CrossRefGoogle Scholar
Van der Linden, W.J., Guo, F., (2008). Bayesian procedures for identifying aberrant response-time patterns in adaptive testing, Psychometrika, 73(3), 365–384.CrossRefGoogle Scholar
Van der Linden, W.J., Klein Entink, R.H., Fox, J.P., (2010). IRT parameter estimation with response times as collateral information, Applied Psychological Measurement, 34(5), 327–347.CrossRefGoogle Scholar
van der Linden, W.J., van Krimpen-Stoop, E.M., (2003). Using response times to detect aberrant responses in computerized adaptive testing, Psychometrika, 68(2), 251–265.CrossRefGoogle Scholar
van der Maas, H.L., Molenaar, D., Maris, G., Kievit, R.A., Borsboom, D., (2011). Cognitive psychology meets psychometric theory: On the relation between process models for decision making and latent variable models for individual differences, Psychological review, 118(2), 339–356.CrossRefGoogle ScholarPubMed
Wang, C., Xu, G., (2015). A mixture hierarchical model for response times and response accuracy, British Journal of Mathematical and Statistical Psychology, 68, 456–477.CrossRefGoogle ScholarPubMed
Wang, C., Xu, G., & Shang, Z., (2016). A two-stage approach to differentiating normal and aberrant behavior in computer based testing. Psychometrika. https://doi.org/10.1007/s11336-016-9525-x.CrossRefGoogle Scholar
Watanabe, S., (2010). Asymptotic equivalence of Bayes cross validation and widely applicable Information criterion in singular learning theory, Journal of Machine Learning Research, 11, 3571–3594.Google Scholar
Yung, Y.F., (1997). Finite mixtures in confirmatory factor-analysis models, Psychometrika, 62(3), 297–330.CrossRefGoogle Scholar
Supplementary material: File

Molenaar and de Boeck supplementary material

Molenaar and de Boeck supplementary material
Download Molenaar and de Boeck supplementary material(File)
File 6.4 KB