Skip to main content
SpringerLink
Log in
Book cover

Quantitative Psychology Research pp 43–58Cite as

Gauss–Hermite Quadrature in Marginal Maximum Likelihood Estimation of Item Parameters

  • Conference paper

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 140))

Abstract

Although many theoretical papers on the estimation method of marginal maximum likelihood of item parameters for various models under item response theory mentioned Gauss–Hermite quadrature formulas, almost all computer programs that implemented marginal maximum likelihood estimation employed other numerical integration methods (e.g., Newton–Cotes formulas). There are many tables that contain quadrature points and quadrature weights for the Gauss–Hermite quadrature formulas; but these tabled values cannot be directly used when quadrature points and quadrature weights are specified by the user of computer programs because the standard normal distribution is frequently employed in the marginalization of the likelihood. The two purposes of this paper are to present extensive tables of Gauss–Hermite quadrature for the standard normal distribution and to present examples that demonstrate the effects of using various numbers of quadrature points and quadrature weights as well as different quadrature formulas on item parameter estimates. Item parameter estimates obtained from more than 20 quadrature points and quadrature weights with either Gauss–Hermite quadrature or the Newton–Cote method were virtually identical.

This is a preview of subscription content, 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   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
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

Learn about institutional subscriptions

References

  • Abramowitz, M., & Stegun, I. A. (Eds.). (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York, NY: Wiley.

    MATH  Google Scholar 

  • Blocker, A. W. (2014). fastGHQuad: Fast Rcpp implementation of Gauss-Hermite quadrature. R package version 0.2.

    Google Scholar 

  • Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459; 47, 369 (Errata).

    Google Scholar 

  • Bock, R. D., & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.

    Article  Google Scholar 

  • Bock, R. D., & Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment. Applied Psychological Measurement, 6, 431–444.

    Article  Google Scholar 

  • Burden, R. L., & Faires, J. D. (1985). Numerical analysis (3rd ed.). Boston, MA: Prindle, Weber & Schmidt.

    Google Scholar 

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

    Google Scholar 

  • Cai, L., Thissen, D., & du Toit, S. (2010). IRTPRO: Item response theory for patient-reported outcomes [Computer software]. Skokie, IL: Scientific Software International.

    Google Scholar 

  • Cheney, W., & Kincaid, D. (1985). Numerical mathematics and computing (2nd ed.). Pacific Grove, CA: Brooks/Cole.

    MATH  Google Scholar 

  • Davis, P. J., & Rabinowitz, P. (1975). Methods of numerical integration. New York, NY: Academic.

    MATH  Google Scholar 

  • De Ayala, R. J. (2009). The theory and practice of item response theory. New York, NY: The Guilford Press.

    Google Scholar 

  • De Ayala, R. J., Schafer, W. D., & Sava-Bolesta, M. (1995). An investigation of the standard errors of expected a posteriori ability estimates. British Journal of Mathematical and Statistical Psychology, 47, 385–405.

    Article  Google Scholar 

  • de Toit, M. (Ed.). (2003). IRT from SSI. Lincolnwood, IL: Scientific Software International.

    Google Scholar 

  • Drasgow, F. (1989). An evaluation of marginal maximum likelihood estimation for the two-parameter logistic model. Applied Psychological Measurement, 13, 77–90.

    Article  Google Scholar 

  • Gauss, C. F. (1876). Methodus nova integralium valores per approximationem inveniendi [The new method of integral values by finding approximation]. In Carl Friedrich Gauss Werke (Vol. 3, pp. 163–196). Göttingen, Germany: Königlichen Gesellschaft der Wissenschaften.

    Google Scholar 

  • Golub, G. H., & Welsch, J. H. (1969). Calculation of Gauss quadrature rules. Mathematics of Computation, 23, 221–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Gradshteyn, I. S., & Ryzhik, I. M. (1994). Table of integrals, series, and products (5th ed.) (A. Jeffrey, Trans.). San Diego, CA: Academic.

    Google Scholar 

  • Hildebrand, F. B. (1974). Introduction to numerical analysis (2nd ed.). New York, NY: McGraw-Hill.

    MATH  Google Scholar 

  • Hochstrasser, W. (1972). Orthogonal polynomials. In M. Abramowitz & I. A. Stegun (Eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables (pp. 771–802). New York, NY: Wiley.

    Google Scholar 

  • Houts, C. R., & Cai, L. (2013). flexMIRT user’s manual version 2.0. Chapel Hill, NC: Vector Psychometric Group.

    Google Scholar 

  • Jeffrey, A. (2000). Handbook of mathematical formulas and integrals (2nd ed.). San Diego, CA: Academic.

    MATH  Google Scholar 

  • Johnson, M. S. (2007). Marginal maximum likelihood estimation of item response models in R. Journal of Statistical Software, 20(10), 1–24.

    Google Scholar 

  • Kennedy, W. J., Jr., & Gentle, J. E. (1980). Statistical computing. New York, NY: Marcel Dekker,

    MATH  Google Scholar 

  • Krylov, V. I. (1962). Approximate calculation of integrals (A. H. Stroud, Trans.). New York, NY: Macmillan.

    Google Scholar 

  • Linz, P., & Wang, R. L. C. (2003). Exploring numerical methods: An introduction to scientific computing using MATLAB. Boston, MA: Jones and Bartlett.

    Google Scholar 

  • Lord, F. M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49, 359–381.

    Article  MATH  Google Scholar 

  • Mislevy, R. J. (1986). Bayes modal estimation in item response models. Psychometrika, 51, 177–195.

    Article  MathSciNet  MATH  Google Scholar 

  • Mislevy, R. J., & Bock, R. D. (1984). BILOG II: Item analysis and test scoring with binary logistic models—User’s guide. Mooresville, IN: Scientific Software.

    Google Scholar 

  • Mislevy, R. J., & Bock, R. D. (1985). Implementation of the EM algorithm in the estimation of item parameters: The BILOG computer program. In D. J. Weiss (Ed.), Proceedings of the 1982 Item Response Theory and Computerized Adaptive Testing Conference (pp. 189–202). Minneapolis, MN: University of Minnesota, Department of Psychology, Computerized Adaptive Testing Laboratory.

    Google Scholar 

  • Mislevy, R. J., & Bock, R. D. (1986). PC-BILOG: Item analysis and test scoring with binary logistic models [Computer software]. Mooresville, IN: Scientific Software.

    Google Scholar 

  • Mislevy, R. J., & Bock, R. D. (1990). BILOG 3: Item analysis and test scoring with binary logistic models [Computer software]. Mooresville, IN: Scientific Software.

    Google Scholar 

  • Mislevy, R. J., & Stocking, M. L. (1989). A consumer’s guide to LOGIST and BILOG. Applied Psychological Measurement, 13, 57–75.

    Article  Google Scholar 

  • Muraki, E. (1984). Marginal maximum likelihood estimation for three-parameter polychotomous item response models: Application of an EM algorithm. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.

    Google Scholar 

  • Muraki, E., & Bock, R. D. (1993). PARSCALE: IRT based test scoring and item analysis for graded open-ended exercises and performance tasks [Computer software]. Chicago, IL: Scientific Software International.

    Google Scholar 

  • Muraki, E., & Bock, R. D. (2002). PARSCALE: Maximum likelihood item analysis and test scoring—polytomous model [Computer software]. Chicago, IL: Scientific Software International.

    Google Scholar 

  • Novomestky, F. (2013). gaussquad: Collection of functions for Gaussian quadrature. R package version 1.0-2.

    Google Scholar 

  • Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1996). Numerical recipes in Fortran 90: The art of parallel scientific computing (2nd ed.). New York, NY: Cambridge University Press.

    Google Scholar 

  • R Core Team. (2010). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from http://www.R-project.org.

    Google Scholar 

  • Rigdon, S. E., & Tsutakawa, R. K. (1983). Parameter estimation in latent trait models. Psycometrika, 48, 567–574.

    Article  MathSciNet  MATH  Google Scholar 

  • Salzer, H. E., Zucker, R., & Capuano, R. (1952). Table of the zeros and weight factors of the first twenty Hermite polynomials. Journal of Research of the National Bureau of Standards, 48, 111–116.

    Article  MathSciNet  Google Scholar 

  • Sanathanan, L., & Blumenthal, S. (1978). The logistic model and estimation of latent structure. Journal of the American Statistical Association, 73, 794–799.

    Article  MATH  Google Scholar 

  • Seong, T.-J. (1990). Validity of using two numerical analysis techniques to estimate item and ability parameters via MMLE: Gauss-Hermite quadrature formula and Mislevy’s histogram solution. Paper presented at the Annual Meeting of the National Council on Measurement in Education, Boston, MA.

    Google Scholar 

  • Shao, T. S., Chen, T. C., & Frank, R. M. (1964). Tables of zeros and Gaussian weights of certain associated Laguerre Polynomials and the related generalized Hermite polynomials. Mathematics of Computation, 26, 598–616.

    Article  MathSciNet  Google Scholar 

  • Smyth, G. K. (1998). Numerical integration. In P. Armitage & T. Colton (Eds.), Encyclopedia of biostatistics (pp. 3088–3095). Chichester: Wiley.

    Google Scholar 

  • Smyth, G. K. (2014). statmod: Statistical modeling. R package version 1.4.20.

    Google Scholar 

  • Stroud, A. H., & Secrest, D. (1966). Gaussian quadrature formulas. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.

    Article  MATH  Google Scholar 

  • Thissen, D. (1986). MULTILOG version 5 user’s guide. Mooresville, IN: Scientific Software.

    Google Scholar 

  • Thissen, D., Chen, W.-H., & Bock, R. D. (2002). MULTILOG [Computer software]. Lincolnwood, IL: Scientific Software International.

    Google Scholar 

  • Thisted, R. A. (1988). Elements of statistical computing: Numerical computation. New York, NY: Chapman and Hall.

    MATH  Google Scholar 

  • Tsutakawa, R. K. (1984). Estimation of two-parameter logistic item response curves. Journal of Educational Statistics, 9, 263–276.

    Article  Google Scholar 

  • Verhelst, N. D., Glas, C. A. W., & Verstralen, H. H. F. M. (1994). One parameter logistic model (OPLM) [Computer software]. Amhem: Cito.

    Google Scholar 

  • Wilson, D. T., Wood, R., & Gibbons, R. (1991). TESTFACT: Test scoring, item statistics, and item factor analysis (386/486 Version) [Computer software]. Chicago, IL: Scientific Software International.

    Google Scholar 

  • Wood, R., Wilson, D., Gibbons, R., Schilling, S., Muraki, E., & Bock, R. D. (2002). TESTFACT: Test scoring, item statistics, and item factor analysis (Version 4.0) [Computer software]. Lincolnwood, IL: Scientific Software International.

    Google Scholar 

  • Zwinderman, A. H., & van den Wollenberg, A. L. (1990). Robustness of marginal maximum likelihood estimation in the Rasch model. Applied Psychological Measurement, 14, 73–81.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Educational Psychology, The University of Georgia, 323 Aderhold Hall, Athens, GA, 30602-7143, USA

    Seock-Ho Kim, Yu Bao, Erin Horan, Meereem Kim & Allan S. Cohen

Authors
  1. Seock-Ho Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Erin Horan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Meereem Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Allan S. Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seock-Ho Kim .

Editor information

Editors and Affiliations

  1. University of Amsterdam, Amsterdam, The Netherlands

    L. Andries van der Ark

  2. University of Wisconsin, Madison, Wisconsin, USA

    Daniel M. Bolt

  3. The Hong Kong Institute of Education, Hong Kong, Hong Kong SAR

    Wen-Chung Wang

  4. University of Illinois, Champaign, Illinois, USA

    Jeffrey A. Douglas

  5. The Penn State University, University Park, Pennsylvania, USA

    Sy-Miin Chow

Appendices

Appendix 1

N = 10

XI

-3.436159118837738E+00 -2.532731674232789E+00 -1.756683649299880E+00 -1.036610829789513E+00 -3.429013272237046E-01

3.429013272237046E-01 1.036610829789513E+00 1.756683649299880E+00 2.532731674232789E+00 3.436159118837738E+00

AI

7.640432855232643E-06 1.343645746781229E-03 3.387439445548111E-02 2.401386110823148E-01 6.108626337353258E-01

6.108626337353258E-01 2.401386110823148E-01 3.387439445548111E-02 1.343645746781229E-03 7.640432855232643E-06

N = 20

XI

-5.387480890011237E+00 -4.603682449550741E+00 -3.944764040115622E+00 -3.347854567383215E+00 -2.788806058428129E+00

-2.254974002089274E+00 -1.738537712116586E+00 -1.234076215395323E+00 -7.374737285453945E-01 -2.453407083009013E-01

2.453407083009013E-01 7.374737285453945E-01 1.234076215395323E+00 1.738537712116586E+00 2.254974002089274E+00

2.788806058428129E+00 3.347854567383215E+00 3.944764040115622E+00 4.603682449550741E+00 5.387480890011237E+00

AI

2.229393645534086E-13 4.399340992273155E-10 1.086069370769280E-07 7.802556478532085E-06 2.283386360163550E-04

3.243773342237865E-03 2.481052088746362E-02 1.090172060200233E-01 2.866755053628341E-01 4.622436696006098E-01

4.622436696006098E-01 2.866755053628341E-01 1.090172060200233E-01 2.481052088746362E-02 3.243773342237865E-03

2.283386360163550E-04 7.802556478532085E-06 1.086069370769280E-07 4.399340992273155E-10 2.229393645534086E-13

Appendix 2

N = 10

XK

-4.859462828332313E+00 -3.581823483551926E+00 -2.484325841638953E+00 -1.465989094391158E+00 -4.849357075154976E-01

4.849357075154976E-01 1.465989094391158E+00 2.484325841638953E+00 3.581823483551926E+00 4.859462828332313E+00

A(XK)

4.310652630718300E-06 7.580709343122154E-04 1.911158050077032E-02 1.354837029802678E-01 3.446423349320191E-01

3.446423349320191E-01 1.354837029802678E-01 1.911158050077032E-02 7.580709343122154E-04 4.310652630718300E-06

N = 20

XK

-7.619048541679765E+00 -6.510590157013650E+00 -5.578738805893197E+00 -4.734581334046053E+00 -3.943967350657314E+00

-3.189014816553388E+00 -2.458663611172368E+00 -1.745247320814127E+00 -1.042945348802751E+00 -3.469641570813560E-01

3.469641570813560E-01 1.042945348802751E+00 1.745247320814127E+00 2.458663611172368E+00 3.189014816553388E+00

3.943967350657314E+00 4.734581334046053E+00 5.578738805893197E+00 6.510590157013650E+00 7.619048541679765E+00

A(XK)

1.257800672437891E-13 2.482062362315165E-10 6.127490259982936E-08 4.402121090230865E-06 1.288262799619300E-04

1.830103131080495E-03 1.399783744710101E-02 6.150637206397688E-02 1.617393339840000E-01 2.607930634495547E-01

2.607930634495547E-01 1.617393339840000E-01 6.150637206397688E-02 1.399783744710101E-02 1.830103131080495E-03

1.288262799619300E-04 4.402121090230865E-06 6.127490259982936E-08 2.482062362315165E-10 1.257800672437891E-13

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Kim, SH., Bao, Y., Horan, E., Kim, M., Cohen, A.S. (2015). Gauss–Hermite Quadrature in Marginal Maximum Likelihood Estimation of Item Parameters. In: van der Ark, L., Bolt, D., Wang, WC., Douglas, J., Chow, SM. (eds) Quantitative Psychology Research. Springer Proceedings in Mathematics & Statistics, vol 140. Springer, Cham. https://doi.org/10.1007/978-3-319-19977-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation