Skip to main content
SpringerLink
Log in

Estimation Methods for Item Factor Analysis: An Overview

  • Chapter
  • First Online:
Modern Statistical Methods for Health Research

Part of the book series: Emerging Topics in Statistics and Biostatistics ((ETSB))

Abstract

Item factor analysis (IFA) refers to the factor models and statistical inference procedures for analyzing multivariate categorical data. IFA techniques are commonly used in social and behavioral sciences for analyzing item-level response data. Such models summarize and interpret the dependence structure among a set of categorical variables by a small number of latent factors. In this chapter, we review the IFA modeling technique and commonly used IFA models. Then we discuss the estimation methods for IFA models and their computation, with a focus on the situation where the sample size, the number of items, and the number of factors are all large. Existing statistical software for IFA are surveyed. This chapter is concluded with suggestions for practical applications of IFA methods and discussions of future directions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Orthogonal rotational methods (e.g., varimax rotation; 32) are available in factor analysis that requires the estimated factors to be orthogonal to each other. As the orthogonal requirement of the latent factors is somewhat artificial, we do not discuss them in this chapter.

References

  1. Andersen, E.B.: Conditional Inference and Models for Measuring. Mentalhygiejnisk Forlag, Copenhagen (1973)

    Google Scholar 

  2. Anderson, T.W., Rubin, H.: Statistical inference in factor analysis. In: Neyman J. (ed.) Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume V: Contributions to Econometrics, Industrial Research, and Psychometry, pp. 111–150. University of California Press, Berkerley, CA (1956)

    Google Scholar 

  3. Asparouhov, T., Muthén, B.: Comparison of computational methods for high dimensional item factor analysis. Unpublished manuscript retrieved from www.statmodel.com (2012)

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Birnbaum, A.: Some latent trait models and their use in inferring an examinee’s ability. In Lord F.M., Novick, M.R. (eds.) Statistical Theories of Mental Test Scores, pp. 397–479. Addison-Wesley, Reading, MA (1968)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Bolt, D.M., Lall, V.F.: Estimation of compensatory and noncompensatory multidimensional item response models using Markov chain Monte Carlo. Appl. Psychol. Meas. 27(6), 395–414 (2003)

    Article  MathSciNet  Google Scholar 

  8. Browne, M.W.: An overview of analytic rotation in exploratory factor analysis. Multivar. Behav. Res. 36(1), 111–150 (2001)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Cai, L.: Metropolis-Hastings Robbins-Monro algorithm for confirmatory item factor analysis. J. Educ. Behav. Stat. 35(3), 307–335 (2010)

    Article  Google Scholar 

  11. Cai, L., Wirth, R.: flexMIRT version 2: Flexible multilevel multidimensional item analysis and test scoring [Computer software]. Vector Psychometric Group, Chapel Hill, NC (2013)

    Google Scholar 

  12. Cai, L., Du Toit, S., Thissen, D.: IRTPRO: Flexible, Multidimensional, Multiple Categorical IRT Modeling [Computer software]. Scientific Software International, Chicago, IL (2011)

    Google Scholar 

  13. Celeux, G., Diebolt, J.: The SEM algorithm: A probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Comput. Stat. Q. 2, 73–82 (1985)

    Google Scholar 

  14. Chalmers, R.P.: mirt: A multidimensional item response theory package for the R environment. J. Stat. Softw. 48(6), 1–29 (2012)

    Google Scholar 

  15. Chatterjee, S.: Matrix estimation by universal singular value thresholding. Ann. Stat. 43(1), 177–214 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, Y., Li, X., Zhang, S.: Joint maximum likelihood estimation for high-dimensional exploratory item factor analysis. Psychometrika 84(1), 124–146 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, Y., Li, X., Zhang, S.: Structured latent factor analysis for large-scale data: Identifiability, estimability, and their implications. J. Am. Stat. Assoc 115(532), 1756–1770 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., Robins, J.: Double/debiased machine learning for treatment and structural parameters. Econom. J. 21(1), C1–C68 (2018)

    Article  MathSciNet  Google Scholar 

  19. Chung, K.L.: On a stochastic approximation method. Ann. Math. Stat. 25(3), 463–483 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dagum, L., Menon, R.: OpenMP: An industry standard API for shared-memory programming. Comput. Sci. Eng. IEEE 5(1), 46–55 (1998)

    Article  Google Scholar 

  21. Davis, C., Kahan, W.M.: The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal. 7(1), 1–46 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B (Methodol.) 39(1), 1–22 (1977)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Fabian, V.: On asymptotic normality in stochastic approximation. Ann. Math. Stat. 39(4), 1327–1332 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ghosh, M.: Inconsistent maximum likelihood estimators for the Rasch model. Stat. Probab. Lett. 23(2), 165–170 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gu, M.G., Kong, F.H.: A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems. Proc. Natl. Acad. Sci. 95(13), 7270–7274 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Haberman, S.J.: Maximum likelihood estimates in exponential response models. Ann. Stat. 5(5), 815–841 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  28. Holland, P.W.: On the sampling theory roundations of item response theory models. Psychometrika 55(4), 577–601 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ip, E.H.: On single versus multiple imputation for a class of stochastic algorithms estimating max. Comput. Stat. 17, 517–524 (2002)

    Article  MATH  Google Scholar 

  30. Jöreskog, K.G.: On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika 59(3), 381–389 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jöreskog, K.G., Moustaki, I.: Factor analysis of ordinal variables: A comparison of three approaches. Multivar. Behav. Res. 36(3), 347–387 (2001)

    Article  Google Scholar 

  32. Kaiser, H.F.: The varimax criterion for analytic rotation in factor analysis. Psychometrika 23(3), 187–200 (1958)

    Article  MATH  Google Scholar 

  33. Kotov, R., Krueger, R.F., Watson, D., Achenbach, T.M., Althoff, R.R., Bagby, R.M., Brown, T.A., Carpenter, W.T., Caspi, A., Clark, L.A., et al.: The hierarchical taxonomy of psychopathology (HiTOP): A dimensional alternative to traditional nosologies. J. Abnorm. Psychol. 126(4), 454–477 (2017)

    Article  Google Scholar 

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

    Article  Google Scholar 

  35. Mair, P., Hatzinger, R.: Extended Rasch modeling: The eRm package for the application of IRT models in R. J. Stat. Softw. 20(9), 1–20 (2007)

    Google Scholar 

  36. Martin, A.D., Quinn, K.M., Park, J.H.: MCMCpack: Markov chain Monte Carlo in R. J. Stat. Softw. 42(9), 1–21 (2011)

    Google Scholar 

  37. Meng, X.-L., Schilling, S.: Fitting full-information item factor models and an empirical investigation of bridge sampling. J. Am. Stat. Assoc. 91(435), 1254–1267 (1996)

    Article  MATH  Google Scholar 

  38. Muraki, E.: A generalized partial credit model: Application of an EM algorithm. Appl. Psychol. Meas. 16(2), 159–176 (1992)

    Article  Google Scholar 

  39. Muraki, E., Bock, D.: IRT Item Analysis and Test Scoring for Rating Scale Data [Computer software]. Scientific Software, Chicago, IL (1997)

    Google Scholar 

  40. Muraki, E., Carlson, J.E.: Full-information factor analysis for polytomous item responses. Appl. Psychol. Meas. 19(1), 73–90 (1995)

    Article  Google Scholar 

  41. Muthén, B.: A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika 49(1), 115–132 (1984)

    Article  Google Scholar 

  42. Muthén, L.K.: Mplus: The comprehensive modeling program for applied researchers: User’s guide. Muthén & Muthén, Los Angeles, CA (1998)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Nielsen, S.F.: The stochastic EM algorithm: Estimation and asymptotic results. Bernoulli 6(3), 457–489 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Patz, R.J., Junker, B.W.: Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. J. Educ. Behav. Stat. 24(4), 342–366 (1999)

    Article  Google Scholar 

  46. Rasch, G.: Probabilistic models for some intelligence and attainment tests. Danish Institute for Educational Research, Copenhagen (1960)

    Google Scholar 

  47. Reckase, M.: Multidimensional Item Response Theory. Springer, New York, NY (2009)

    Book  MATH  Google Scholar 

  48. Revelle, W., Condon, D.M., Wilt, J., French, J.A., Brown, A., Elleman, L.G.: Web and phone based data collection using planned missing designs. In: Fielding, N.G., Lee, R.M., Blank, G. (eds.) Handbook of Online Research Methods, pp. 578–595. Sage Publications, Thousand Oaks, CA (2016)

    Google Scholar 

  49. Rizopoulos, D.: ltm: An R package for latent variable modeling and item response theory analyses. J. Stat. Softw. 17(5), 1–25 (2006)

    Google Scholar 

  50. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22(3), 400–407 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  51. Robitzsch, A., Kiefer, T., Wu, M.: TAM: Test analysis modules. R package version 3.3-10 (2019)

    Google Scholar 

  52. Rupp, A.A.: Item response modeling with BILOG-MG and MULTILOG for Windows. Int. J. Test. 3(4), 365–384 (2003)

    Article  Google Scholar 

  53. Samejima, F.: Estimation of latent ability using a response pattern of graded scores. ETS Res. Bull. Ser. 1968(1), i–169 (1968)

    Google Scholar 

  54. Skitka, L.J., Sargis, E.G.: The internet as psychological laboratory. Annu. Rev. Psychol. 57(1), 529–555 (2006)

    Article  Google Scholar 

  55. Spiegelhalter, D., Thomas, A., Best, N., Lunn, D.: WinBUGS user manual: Version 1.4. MRC Biostatistics Unit, Cambridge (2003)

    Google Scholar 

  56. Stock, J.H., Watson, M.W.: Forecasting using principal components from a large number of predictors. J. Am. Stat. Assoc. 97(460), 1167–1179 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  57. Sun, J., Chen, Y., Liu, J., Ying, Z., Xin, T.: Latent variable selection for multidimensional item response theory models via L 1 regularization. Psychometrika 81(4), 921–939 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  58. van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  59. Varin, C., Reid, N., Firth, D.: An overview of composite likelihood methods. Stat. Sin. 21(1), 5–42 (2011)

    MathSciNet  MATH  Google Scholar 

  60. Wedin, P.-Å.: Perturbation bounds in connection with singular value decomposition. BIT Numer. Math. 12(1), 99–111 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  61. Wei, G.C.G., Tanner, M.A.: A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms. J. Am. Stat. Assoc. 85(411), 699–704 (1990)

    Article  Google Scholar 

  62. Yao, L., Schwarz, R.D.: A multidimensional partial credit model with associated item and test statistics: An application to mixed-format tests. Appl. Psychol. Meas. 30(6), 469–492 (2006)

    Article  MathSciNet  Google Scholar 

  63. Zhang, S., Chen, Y.: lvmcomp: Stochastic EM algorithms for latent variable models with a high-dimensional latent space. R package version 1.3.0 (2019)

    Google Scholar 

  64. Zhang, S., Chen, Y., Li, X.: mirtjml: Joint maximum likelihood estimation for high-dimensional item factor analysis. R package version 1.3.0 (2019)

    Google Scholar 

  65. Zhang, H., Chen, Y., Li, X.: A note on exploratory item factor analysis by singular value decomposition. Psychometrika 85, 358–372 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  66. Zhang, S., Chen, Y., Liu, Y.: An improved stochastic EM algorithm for large-scale full-information item factor analysis. Br. J. Math. Stat. Psychol. 73(1), 44–71 (2020)

    Article  MATH  Google Scholar 

  67. Zhao, Y., Joe, H.: Composite likelihood estimation in multivariate data analysis. Can. J. Stat. 33(3), 335–356 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. London School of Economics and Political Science, London, UK

    Yunxiao Chen & Siliang Zhang

Authors
  1. Yunxiao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siliang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunxiao Chen .

Editor information

Editors and Affiliations

  1. Department of Mathematics & Statistics, Georgia State University, Atlanta, GA, USA

    Yichuan Zhao

  2. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, KwaZulu-Natal, South Africa

    (Din) Ding-Geng Chen

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Chen, Y., Zhang, S. (2021). Estimation Methods for Item Factor Analysis: An Overview. In: Zhao, Y., Chen, (.DG. (eds) Modern Statistical Methods for Health Research. Emerging Topics in Statistics and Biostatistics . Springer, Cham. https://doi.org/10.1007/978-3-030-72437-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation