Joint Maximum Likelihood Estimation for High-Dimensional Exploratory Item Factor Analysis

Abstract

Joint maximum likelihood (JML) estimation is one of the earliest approaches to fitting item response theory (IRT) models. This procedure treats both the item and person parameters as unknown but fixed model parameters and estimates them simultaneously by solving an optimization problem. However, the JML estimator is known to be asymptotically inconsistent for many IRT models, when the sample size goes to infinity and the number of items keeps fixed. Consequently, in the psychometrics literature, this estimator is less preferred to the marginal maximum likelihood (MML) estimator. In this paper, we re-investigate the JML estimator for high-dimensional exploratory item factor analysis, from both statistical and computational perspectives. In particular, we establish a notion of statistical consistency for a constrained JML estimator, under an asymptotic setting that both the numbers of items and people grow to infinity and that many responses may be missing. A parallel computing algorithm is proposed for this estimator that can scale to very large datasets. Via simulation studies, we show that when the dimensionality is high, the proposed estimator yields similar or even better results than those from the MML estimator, but can be obtained computationally much more efficiently. An illustrative real data example is provided based on the revised version of Eysenck’s Personality Questionnaire (EPQ-R).

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Notes

  1. 1.

    Core(TM) i7CPU@5650U@2.2 GHz.

  2. 2.

    The small number of replications is due to the constraint that flexMIRT\(^\circledR \) needs to be run on a local \(\hbox {Windows}^\circledR \) machine.

References

  1. Andersen, E. B. (1973). Conditional inference and models for measuring. Copenhagen, Denmark: Mentalhygiejnisk Forlag.

    Google Scholar 

  2. Baker, F. B. (1987). Methodology review: Item parameter estimation under the one-, two-, and three-parameter logistic models. Applied Psychological Measurement, 11(2), 111–141.

    Article  Google Scholar 

  3. Bartholomew, D. J., Moustaki, I., Galbraith, J., & Steele, F. (2008). Analysis of multivariate social science data. Boca Raton, FL: CRC Press.

    Google Scholar 

  4. Béguin, A. A., & Glas, C. A. (2001). MCMC estimation and some model-fit analysis of multidimensional IRT models. Psychometrika, 66(4), 541–561.

    Article  Google Scholar 

  5. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F. M. Lord & M. R. Novick (Eds.), Statistical Theories of Mental Test Scores. Reading, MA: Addison-Wesley.

    Google Scholar 

  6. Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443–459.

    Article  Google Scholar 

  7. Bock, R. D., Gibbons, R., & Muraki, E. (1988). Full-information item factor analysis. Applied Psychological Measurement, 12(3), 261–280.

    Article  Google Scholar 

  8. Bolt, D. M., & Lall, V. F. (2003). Estimation of compensatory and noncompensatory multidimensional item response models using Markov chain Monte Carlo. Applied Psychological Measurement, 27(6), 395–414.

    Article  Google Scholar 

  9. Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111–150.

    Article  Google Scholar 

  10. Cai, L. (2010a). High-dimensional exploratory item factor analysis by a Metropolis–Hastings Robbins–Monro algorithm. Psychometrika, 75(1), 33–57.

    Article  Google Scholar 

  11. Cai, L. (2010b). Metropolis–Hastings Robbins–Monro algorithm for confirmatory item factor analysis. Journal of Educational and Behavioral Statistics, 35(3), 307–335.

    Article  Google Scholar 

  12. Cai, T., & Zhou, W.-X. (2013). A max-norm constrained minimization approach to 1-bit matrix completion. The Journal of Machine Learning Research, 14(1), 3619–3647.

    Google Scholar 

  13. Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1–29.

    Article  Google Scholar 

  14. Chiu, C.-Y., Köhn, H.-F., Zheng, Y., & Henson, R. (2016). Joint maximum likelihood estimation for diagnostic classification models. Psychometrika, 81(4), 1069–1092.

    Article  PubMed  Google Scholar 

  15. Dagum, L., & Menon, R. (1998). OpenMP: An industry standard API for shared-memory programming. Computational Science & Engineering, IEEE, 5(1), 46–55.

    Article  Google Scholar 

  16. Davenport, M. A., Plan, Y., van den Berg, E., & Wootters, M. (2014). 1-bit matrix completion. Information and Inference, 3(3), 189–223.

    Article  Google Scholar 

  17. Edelen, M. O., & Reeve, B. B. (2007). Applying item response theory (IRT) modeling to questionnaire development, evaluation, and refinement. Quality of Life Research, 16(1), 5–18.

    Article  PubMed  Google Scholar 

  18. Edwards, M. C. (2010). A Markov chain Monte Carlo approach to confirmatory item factor analysis. Psychometrika, 75(3), 474–497.

    Article  Google Scholar 

  19. Embretson, S. E., & Reise, S. P. (2000). Item response theory for psychologists. Mahwah, NJ: Lawrence Erlbaum Associates Publishers.

    Google Scholar 

  20. Eysenck, S. B., Eysenck, H. J., & Barrett, P. (1985). A revised version of the psychoticism scale. Personality and Individual Differences, 6(1), 21–29.

    Article  Google Scholar 

  21. Ghosh, M. (1995). Inconsistent maximum likelihood estimators for the Rasch model. Statistics & Probability Letters, 23(2), 165–170.

    Article  Google Scholar 

  22. Haberman, S. J. (1977). Maximum likelihood estimates in exponential response models. The Annals of Statistics, 5(5), 815–841.

    Article  Google Scholar 

  23. Haberman, S. J. (2004). Joint and conditional maximum likelihood estimation for the Rasch model for binary responses. ETS Research Report Series RR-04-20.

  24. Jöreskog, K. G., & Moustaki, I. (2001). Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behavioral Research, 36(3), 347–387.

    Article  PubMed  Google Scholar 

  25. Lee, K., & Ashton, M. C. (2009). Factor analysis in personality research. In R. W. Robins, R. C. Fraley, & R. F. Krueger (Eds.), Handbook of Research Methods in Personality Psychology. New York, NY: Guilford Press.

    Google Scholar 

  26. Lee, S.-Y., Poon, W.-Y., & Bentler, P. M. (1990). A three-stage estimation procedure for structural equation models with polytomous variables. Psychometrika, 55(1), 45–51.

    Article  Google Scholar 

  27. Lord, F. M. (1980). Applications of item response theory to practical testing problems. Mahwah, NJ: Routledge.

    Google Scholar 

  28. Meng, X.-L., & Schilling, S. (1996). Fitting full-information item factor models and an empirical investigation of bridge sampling. Journal of the American Statistical Association, 91(435), 1254–1267.

    Article  Google Scholar 

  29. Mislevy, R. J. & Stocking, M. L. (1987). A consumer’s guide to LOGIST and BILOG. ETS Research Report Series RR-87-43.

  30. Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16(1), 1–32.

    Article  Google Scholar 

  31. Parikh, N., & Boyd, S. (2014). Proximal algorithms. Foundations and Trends. Optimization, 1(3), 127–239.

  32. Reckase, M. (2009). Multidimensional item response theory. New York, NY: Springer.

    Google Scholar 

  33. Reckase, M. D. (1972). Development and application of a multivariate logistic latent trait model. Ph.D. thesis, Syracuse University, Syracuse NY.

  34. Reise, S. P., & Waller, N. G. (2009). Item response theory and clinical measurement. Annual Review of Clinical Psychology, 5, 27–48.

    Article  PubMed  Google Scholar 

  35. Schilling, S., & Bock, R. D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70(3), 533–555.

    Google Scholar 

  36. Sun, J., Chen, Y., Liu, J., Ying, Z., & Xin, T. (2016). Latent variable selection for multidimensional item response theory models via \(L_1\) regularization. Psychometrika, 81(4), 921–939.

    Article  PubMed  Google Scholar 

  37. von Davier, A. (2010). Statistical models for test equating, scaling, and linking. New York, NY: Springer.

    Google Scholar 

  38. Wirth, R., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 12(1), 58–79.

    Article  PubMed  PubMed Central  Google Scholar 

  39. Yao, L., & Schwarz, R. D. (2006). A multidimensional partial credit model with associated item and test statistics: An application to mixed-format tests. Applied Psychological Measurement, 30(6), 469–492.

    Article  Google Scholar 

  40. Yates, A. (1988). Multivariate exploratory data analysis: A perspective on exploratory factor analysis. Albany, NY: State University of New York Press.

    Google Scholar 

Download references

Acknowledgements

We would like to thank the Editor, the Associate Editor, and the reviewers for many helpful and constructive comments. We also would like to thank Dr. Barrett for sharing the EPQ-R dataset analyzed in Sect. 5. This work was partially supported by a NAEd/Spencer Postdoctoral Fellowship [to Yunxiao Chen] and NSF grant DMS 1712657 [to Xiaoou Li].

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yunxiao Chen.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 232 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Li, X. & Zhang, S. Joint Maximum Likelihood Estimation for High-Dimensional Exploratory Item Factor Analysis. Psychometrika 84, 124–146 (2019). https://doi.org/10.1007/s11336-018-9646-5

Download citation

Keywords

  • joint maximum likelihood estimator
  • item response theory
  • IRT
  • high-dimensional data
  • alternating minimization
  • projected gradient descent
  • personality assessment