Hostname: page-component-89b8bd64d-7zcd7 Total loading time: 0 Render date: 2026-05-05T14:45:35.422Z Has data issue: false hasContentIssue false
  • English
  • Français

Semiparametric Factor Analysis for Item-Level Response Time Data

Published online by Cambridge University Press:  01 January 2025

Yang Liu Open the ORCID record for Yang Liu [Opens in a new window]  and
Weimeng Wang
Yang Liu*
Affiliation:
University of Maryland
Weimeng Wang
Affiliation:
University of Maryland
*
Correspondence should be made to Yang Liu, Department of Human Development and Quantitative Methodology, University of Maryland, College Park, USA. Email: yliu87@umd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Item-level response time (RT) data can be conveniently collected from computer-based test/survey delivery platforms and have been demonstrated to bear a close relation to a miscellany of cognitive processes and test-taking behaviors. Individual differences in general processing speed can be inferred from item-level RT data using factor analysis. Conventional linear normal factor models make strong parametric assumptions, which sacrifices modeling flexibility for interpretability, and thus are not ideal for describing complex associations between observed RT and the latent speed. In this paper, we propose a semiparametric factor model with minimal parametric assumptions. Specifically, we adopt a functional analysis of variance representation for the log conditional densities of the manifest variables, in which the main effect and interaction functions are approximated by cubic splines. Penalized maximum likelihood estimation of the spline coefficients can be performed by an Expectation-Maximization algorithm, and the penalty weight can be empirically determined by cross-validation. In a simulation study, we compare the semiparametric model with incorrectly and correctly specified parametric factor models with regard to the recovery of data generating mechanism. A real data example is also presented to demonstrate the advantages of the proposed method.

Keywords

Factor analysisConditional density estimationFunctional ANOVACubic splinePenalized maximum likelihoodExpectation–maximization algorithm

Information

Type
Theory and Methods
Information
Psychometrika , Volume 87 , Issue 2: Special Issue on Forecasting with Intensive Longitudinal Data , June 2022 , pp. 666 - 692
Copyright
copyright © 2021 The Author(s) under exclusive licence to 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

Footnotes

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s11336-021-09832-8.

References

Agresti, A. (2003). Categorical data analysis. Wiley.Google Scholar
Aitkin, M. (1999). A general maximum likelihood analysis of variance components in generalized linear models. Biometrics, 55 (11), 117– 128. CrossRefGoogle ScholarPubMed
Alexander, P. A. & The Disciplined Reading and Learning Research Laboratory. (2012). Reading into the future: Competence for the 21st century. Educational Psychologist, 47(4), 259–280.CrossRefGoogle Scholar
Alexander, P. A., Dumas, D., Grossnickle, E. M., List, A., & Firetto, C. M. (2016). Measuring relational reasoning. The Journal of Experimental Education, 84 (1), 119– 151. CrossRefGoogle Scholar
Bartholomew, D., Knott, M., & Moustaki, I. (2011). Latent variable models and factor analysis: A unified approach. Wiley.CrossRefGoogle Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37 (1), 29– 51. CrossRefGoogle Scholar
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46 (4), 443– 459. CrossRefGoogle Scholar
Bollen, K. (1989). Structural equations with latent variables. Wiley.CrossRefGoogle Scholar
Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge University Press.CrossRefGoogle Scholar
Brainard, D. H. (1997). The psychophysics toolbox. Spatial Vision, 10 (4), 433– 436. CrossRefGoogle ScholarPubMed
Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S., & Zhao, L. (2005). Statistical analysis of a telephone call center: A queueing-science perspective. Journal of the American Statistical Association, 100 (469), 36– 50. CrossRefGoogle Scholar
Cai, L. (2010). High-dimensional exploratory item factor analysis by a Metropolis–Hastings Robbins–Monro algorithm. Psychometrika, 75 (13), 3– 57. CrossRefGoogle Scholar
Chambers, J. M., Cleveland, W. S., Kleiner, B., & Tukey, P. A. (1983). Graphical methods for data analysis. Chapman.Google Scholar
Currie, I. D., Durban, M., & Eilers, P. H. (2006). Generalized linear array models with applications to multidimensional. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 682, 259– 280. CrossRefGoogle Scholar
Davis, P., & Polonsky, I. (1964). Numerical interpolation, differentiation and integration. In M. Abramowitz & I. A. Stegun (Eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables. DCNational Bureau of Standards.Google Scholar
De Boeck, P., & Jeon, M. (2019). An overview of models for response times and processes in cognitive. Frontiers in Psychology, 10, 102 CrossRefGoogle ScholarPubMed
De Boor, C. (1978). A practical guide to splines. Springer.CrossRefGoogle Scholar
De Boor, C., & Daniel, J. W. (1974). Splines with nonnegative B-spline coefficients. Mathematics of Computation, 28 (126), 565– 568. Google Scholar
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39 (1), 1– 22. CrossRefGoogle Scholar
Dierckx, P. (1993). Curve and surface fitting with splines. Clarendon.CrossRefGoogle Scholar
Douglas, J. (1997). Joint consistency of nonparametric item characteristic curve and ability estimation. Psychometrika, 6 (21), 7– 28. CrossRefGoogle Scholar
Eilers, P. H., & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11, 89– 102. CrossRefGoogle Scholar
Entink, R. K., van der Linden, W., & Fox, J. P. (2009). A Box–Cox normal model for response times. British Journal of Mathematical and Statistical Psychology, 62 (3), 621– 640. CrossRefGoogle Scholar
Glas, C. A., & van der Linden, W. J. (2010). Marginal likelihood inference for a model for item responses and response times. British Journal of Mathematical and Statistical Psychology, 63 (3), 603– 626. CrossRefGoogle Scholar
Gu, C. & Qiu, C. (1993). Smoothing spline density estimation: Theory. The Annals of Statistics, 217–234.CrossRefGoogle Scholar
Gu, C. (1995). Smoothing spline density estimation: Conditional distribution. Statistica Sinica, 709–726.Google Scholar
Gu, C. (1993). Smoothing spline density estimation: A dimensionless automatic algorithm. Journal of the American Statistical Association, 88 (422), 495– 504. CrossRefGoogle Scholar
Gu, C. (2013). Smoothing spline ANOVA models. Springer.CrossRefGoogle Scholar
Gu, M., & Kong, F. (1998). A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems. Proceedings of the National Academy of Sciences, 95 (13), 7270– 7274. CrossRefGoogle ScholarPubMed
Gu, C., & Wahba, G. (1993). Smoothing spline ANOVA with component-wise Bayesian ‘confidence interval’. Journal of Computational and Graphical Statistics, 2 (1), 97– 117. Google Scholar
Hastie, T., Tibshirani, R., & Friedman, J. (2013). The elements of statistical learning: Data mining, inference, and prediction. Springer.Google Scholar
Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling: A Multidisciplinary Journal, 6 (1), 1– 55. CrossRefGoogle Scholar
Kang, H. A. (2017). Penalized partial likelihood inference of proportional hazards latent trait models. British Journal of Mathematical and Statistical Psychology, 70 (2), 187– 208. CrossRefGoogle ScholarPubMed
Kendall, M. (1955). Rank correlation methods (2nd ed.). Charles Griffin and Co.Google Scholar
Kyllonen, P. C., & Zu, J. (2016). Use of response time for measuring cognitive ability. Journal of Intelligence, 4 (14), 1– 29. CrossRefGoogle Scholar
Lee, Y. H., & Haberman, S. J. (2016). Investigating test-taking behaviors using timing and process data. International Journal of Testing, 16 (3), 240– 267. CrossRefGoogle Scholar
Lee, S. Y., Lu, B., & Song, X. Y. (2008). Semiparametric Bayesian analysis of structural equation models with fixed covariates. Statistics in Medicine, 27 (13), 2341– 2360. CrossRefGoogle ScholarPubMed
Leitenstorfer, F., & Tutz, G. (2007). Generalized monotonic regression based on B-splines with an application to air pollution data. Biostatistics, 8 (3), 654– 673. CrossRefGoogle ScholarPubMed
Liu, Y., Magnus, B. E., & Thissen, D. (2016). Modeling and testing differential item functioning in unidimensional binary item response models with a single continuous covariate: A functional data analysis approach. Psychometrika, 81 (2), 371– 398. CrossRefGoogle ScholarPubMed
MacCallum, R. C. (2003). 2001 presidential address: Working with imperfect models. Multivariate Behavioral Research, 38 (1), 113– 139. CrossRefGoogle ScholarPubMed
MacCallum, R. C., Roznowski, M., & Necowitz, L. B. (1992). Model modifications in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111 (3), 490– 504. CrossRefGoogle ScholarPubMed
Maydeu-Olivares, A. (2017). Assessing the size of model misfit in structural equation models. Psychometrika, 82 (3), 533– 558. CrossRefGoogle Scholar
Molenaar, D., Bolsinova, M., & Vermunt, J. K. (2018). A semi-parametric within-subject mixture approach to the analyses of responses and response times. British Journal of Mathematical and Statistical Psychology, 71 (2), 205– 228. CrossRefGoogle Scholar
Nocedal, J., & Wright, S. (2006). Numerical optimization. Springer.Google Scholar
OECD. (2017). PISA 2015 assessment and analytical framework. https://10.1787/9789264281820-en CrossRefGoogle Scholar
Pya, N., & Wood, S. N. (2015). Shape constrained additive models. Statistics and Computing, 25 (3), 543– 559. CrossRefGoogle Scholar
R Core Team. (2020). R: A language and environment for statistical computing [computer oftware manual], Vienna, Austria. https://www.R-project.org/ Google Scholar
Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56 (4), 611– 630. CrossRefGoogle Scholar
Ramsay, J. O., & Silverman, B. W. (1997). Functional data analysis. Springer.CrossRefGoogle Scholar
Ramsay, J. O., & Winsberg, S. (1991). Maximum marginal likelihood estimation for semiparametric item analysis. Psychometrika, 56 (3), 365– 379. CrossRefGoogle Scholar
Ranger, J., Kuhn, J. T., & Ortner, T. M. (2020). Modeling responses and response times in tests with the hierarchical model and the three-parameter lognormal distribution. Educational and Psychological Measurement, 80 (6), 1059– 1089. CrossRefGoogle ScholarPubMed
Ranger, J., & Ortner, T. (2012). A latent trait model for response times on tests employing the proportional hazards model. British Journal of Mathematical and Statistical Psychology, 65 (2), 334– 349. CrossRefGoogle ScholarPubMed
Ranger, J., & Ortner, T. M. (2013). Response time modeling based on the proportional hazards model. Multivariate Behavioral Research, 48 (4), 503– 533. CrossRefGoogle ScholarPubMed
Ranger, J., & Wolgast, A. (2019). Using response times as collateral information about latent traits in psychological tests. Methodology, 15, 185– 196. CrossRefGoogle Scholar
Rossi, N., Wang, X., & Ramsay, J. O. (2002). Nonparametric item response function estimates with the EM algorithm. Journal of Educational and Behavioral Statistics, 27 (3), 291– 317. CrossRefGoogle Scholar
Rudin, W. (1964). Principles of mathematical analysis. McGraw-Hill.Google Scholar
Schnipke, D. L., & Scrams, D. J. (2002). Exploring issues of examinee behavior: Insights gained from response-time analyses. In C. N. Mills, M. Potenza, J. J. Fremer, & W. Ward (Eds.), Computer-based testing: Building the foundation for future assessments (pp. 237–266). Lawrence Erlbaum Associates.Google Scholar
Shaked, M., & Shanthikumar, J. (2007). Stochastic orders. Springer.CrossRefGoogle Scholar
Sinharay, S., & Johnson, M. S. (2019). The use of item scores and response times to detect examinees who may have benefited from item preknowledge. British Journal of Mathematical and Statistical Psychology. Google ScholarPubMed
Sinharay, S., & van Rijn, P. W. (2020). Assessing fit of the lognormal model for response times. Journal of Educational and Behavioral Statistics, 45 (5), 534– 568. CrossRefGoogle Scholar
Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal, and structural equation models. CRC Press.CrossRefGoogle Scholar
Snow, J. (2012). Qualtrics survey software: Handbook for research professionals. Qualtrics Labs Inc.Google Scholar
Song, X. Y., & Lu, Z. H. (2010). Semiparametric latent variable models with Bayesian P-splines. Journal of Computational and Graphical Statistics, 19 (3), 590– 608. CrossRefGoogle Scholar
Thissen, D., & Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51 (4), 567– 577. CrossRefGoogle Scholar
Thissen, D., & Wainer, H. (2001). Test scoring. Taylor & Francis.CrossRefGoogle Scholar
van der Linden, W. J. (2006). A lognormal model for response times on test items. Journal of Educational and Behavioral Statistics, 31 (2), 181– 204. 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
Wang, C., Fan, Z., Chang, H. H., & Douglas, J. A. (2013). A semiparametric model for jointly analyzing response times and accuracy in computerized testing. Journal of Educational and Behavioral Statistics, 38 (4), 381– 417. CrossRefGoogle Scholar
Wood, S. N. (2004). Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of the American Statistical Association, 99 (467), 673– 686. CrossRefGoogle Scholar
Wu, C. J. (1983). On the convergence properties of the EM algorithm. The Annals of Statistics, 95–103.CrossRefGoogle Scholar
Yalcin, I. & Amemiya, Y. (2001). Nonlinear factor analysis as a statistical method. Statistical Science, 275–294.Google Scholar
Zhang, S., Chen, Y., & Liu, Y. (2020). An improved stochastic EM algorithm for large-scale full-information item factor analysis. British Journal of Mathematical and Statistical Psychology, 73 (1), 44– 71. CrossRefGoogle ScholarPubMed
Zhang, D., & Davidian, M. (2001). Linear mixed models with flexible distributions of random effects for longitudinal data. Biometrics, 57 (3), 795– 802. CrossRefGoogle ScholarPubMed
Zhao, H., Alexander, P. A., & Sun, Y. (2020). Relational reasoning’s contributions to mathematical thinking and performance in Chinese elementary and middle-school students. Journal of Educational Psychology, Google Scholar
Supplementary material: File

Liu and Wang Supplementary material

Liu and Wang Supplementary material
Download Liu and Wang Supplementary material(File)
File 128.9 KB