Skip to main content
SpringerLink
Log in

A Nonparametric Estimator of a Monotone Item Characteristic Curve

  • Conference paper
Quantitative Psychology Research

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

Abstract

This paper presents a nonparametric approach to estimating item characteristic curves (ICCs) when they should be monotonic. First, it addresses the uni-dimensional case; before generalizing it to the multidimensional case.

This is a two-stage process. The first stage uses a nonparametric estimator of the ICC by means of nonparametric kernel regression; the second uses the above result to estimate the density function of the inverse ICC.

By integrating this density function, we obtain an isotonic estimator of the inverse ICC: symmetrized with respect to the bisector of the unit square, to obtain the ICC estimator. We also present the multidimensional case, in which we proceed on a coordinate-by-coordinate basis.

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 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

Institutional subscriptions

References

  • Baker, F. B., & Kim, S.-H. (2004). Item response theory: Parameter estimation techniques (2nd ed.). New York: Marcel Dekker.

    Google Scholar 

  • Barlow, R. E., Bartholomew, D. J., Bremner, J. M., & Brunk, H. D. (1972). Statistical inference under order restrictions: the theory and application of isotonic regression. New York: Wiley.

    MATH  Google Scholar 

  • Boomsma, A., Van Duijn, J., & Snijders, A. B. (Eds.). (2001). Essays on item response theory. New York: Springer.

    MATH  Google Scholar 

  • Brunk, H. D. (1955). Maximum likelihood estimates of monotone parameters. Annals of Mathematical Statistics, 26, 607–616.

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng, K. F., & Linn, P. E. (1981). Nonparametric estimation of a regression function. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 57, 223–233.

    Article  MathSciNet  MATH  Google Scholar 

  • De Koning, E., Sijstma, K., & Hamers, J. H. M. (2002). Comparison of four IRT models when analyzing two tests for inductive reasoning. Applied Psychological Measurement, 26(3), 302–320.

    Article  MathSciNet  Google Scholar 

  • Delecroix, M., & Thomas-Agnan, C. (2000). Spline and kernel regression under shape restrictions. In M. G. Schimek (Ed.), Smoothing and regression: Approaches, computation and application. New York: Wiley.

    Google Scholar 

  • Dette, H., Neumeyer, N., & Pilz, K. (2006). A simple nonparametric estimator of a strictly monotone regression function. Bernoulli, 12(3), 469–490.

    Article  MathSciNet  MATH  Google Scholar 

  • Douglas, J. (1997). Joint consistency of nonparametric item characteristic curve and ability estimation. Psychometrika, 62, 7–28.

    Article  MathSciNet  MATH  Google Scholar 

  • Douglas, J., & Cohen, A (2001). Nonparametric item response function estimation for assessing parametric model fit. Applied Psychological Measurement, 25, 234–243.

    Article  MathSciNet  Google Scholar 

  • Douglas, J., Kim, H. R., Habing, B., & Gao, F. (1998). Investigating local dependence with conditional covariance functions. Journal of Educational and Behavioral Statistics, 23, 129–151.

    Article  Google Scholar 

  • Ellis, J. L., & Junker, B. W. (1997). Tail-measurability in monotone latent variable models. Psychometrika, 62, 495–523.

    Article  MathSciNet  MATH  Google Scholar 

  • Fischer, G. H., & Molenaar, I. (Eds.). (1995). Rasch models: Foundations, recent developments and applications. New York: Springer.

    MATH  Google Scholar 

  • Friedman, J., & Tibshirani, R. (1984). The monotone smoothing of scatterplots. Technometrics, 26, 243–250.

    Article  Google Scholar 

  • Giampaglia, G. (1990). Lo scaling unidimensionale nella ricerca sociale. Napoli: Liguori Editore.

    Google Scholar 

  • Gijbels, I. (2005). Monotone regression. In N. Balakrishnan, S. Kotz, C. B. Read, & B. Vadakovic (Eds.), The encyclopedia of statistical sciences (2nd ed.). Hoboken/New York: Wiley.

    Google Scholar 

  • Grayson, D. A. (1988). Two-group classification in latent trait theory: Scores with monotone likehood ratio. Psychometrika, 53, 383–392.

    Article  MathSciNet  MATH  Google Scholar 

  • Hall, P., & Huang, L. S. (2001). Nonparametric kernel regression subject to monotonicity constraints. The Annals of Statistics, 29, 624–647.

    Article  MathSciNet  MATH  Google Scholar 

  • Hambleton, R., Swaminathan, H., & Rogers, H. (1991). Fundamentals of item response theory. Thousand Oaks: Sage Publications.

    Google Scholar 

  • Härdle, W. (1990) Applied nonparametric regression. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Holland, P. W. (1990). On the sampling theory foundations of item response theory models. Psychometrika, 55, 577–601.

    Article  MathSciNet  MATH  Google Scholar 

  • Holland, P. W., & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523–1543.

    Article  MathSciNet  MATH  Google Scholar 

  • Junker, B. W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 21, 1359–1378.

    Article  MathSciNet  MATH  Google Scholar 

  • Junker, B. W. (2001). On the interplay between nonparametric and parametric IRT, with some thoughts about the future. In A. Boomsma, M. A. J. Van Duijn, & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 247–276). New York: Springer.

    Chapter  Google Scholar 

  • Junker, B. W. & Sijtsma, K. (2001). Nonparametric item response theory in action: An overview of the special issue. Applied Psychological Measurement, 25, 211–220.

    Article  MathSciNet  Google Scholar 

  • Kelly, C., & Rice, J. (1990). Monotone smoothing with application to dose response curves and the assessment of synergism. Biometrics, 46, 1071–1085.

    Article  Google Scholar 

  • Lee, Y. S. (2007). A comparison of methods for nonparametric estimation of item characteristic curves for binary items. Applied Psychological Measurement, 31(2), 121–134.

    Article  MathSciNet  Google Scholar 

  • Lee, Y. S., Douglas, J., & Chewning, B. (2007). Techniques for developing health quality of life scales for point of service use. Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, 83(2), 331–350.

    Article  Google Scholar 

  • Lewis, C. (1983). Bayesian inference for latent abilities. In S. B. Anderson & J. S. Helmick (Eds.), On educational testing (pp. 224–251). San Francisco: Jossey-Bass.

    Google Scholar 

  • Loevinger, J. (1947). A systematic approach to the construction and evaluation of tests of ability. Psychological Monographs, 61(4), 1–49.

    Article  Google Scholar 

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

    Google Scholar 

  • Luzardo, M., & Forteza, D. (2014). Modelo no paramétrico multidimensional para la estimación de los rasgos y de las curvas características del ítem mediante regresión no paramétrica con núcleos. Montevideo: CSIC.

    Google Scholar 

  • Mammen, E. (1991). Estimating a smooth monotone regression function. The Annals of Statistics, 19, 724–740.

    Article  MathSciNet  MATH  Google Scholar 

  • Meijer, R. R., Sijtsma, K., & Smid, N. G. (1990). Theoretical and empirical comparison of the Mokken and the Rasch approach to IRT. Applied Psychological Measurement, 14, 283–298.

    Article  Google Scholar 

  • Mokken, R. J. (1971). A theory and procedure of scale analysis. New York/Berlin: De Gruyter.

    Book  Google Scholar 

  • Mokken, R.J. (1997). Nonparametric models for dichotomous responses. In Handbook of Modern Item Response Theory, ed. W. J. van der Linden and R. K. Hambleton, 351–368. New York: Springer.

    Chapter  Google Scholar 

  • Mokken, R. J., & Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses. Applied Psychological Measurement, 6, 417–430.

    Article  Google Scholar 

  • Molenaar, I. W., & Sijtsma, K. (2000). User’s manual MSP5 for Windows. Groningen: iecProGAMMA.

    Google Scholar 

  • Mukerjee, H. (1988). Monotone nonparametric regression. The Annals of Statistics, 16, 741–750.

    Article  MathSciNet  MATH  Google Scholar 

  • Niemoller, K., & van Schuur, W. (1983). Stochastic models for unidimensional scaling: Mokken and Rasch. In D. McKay & N. Schofield (Eds.), Data analysis and the social sciences. London: Frances Printer Ltd.

    Google Scholar 

  • Ramsay, J. O. (1988). Monotone regression splines in action. Statistical Science, 3(4), 425–461.

    Article  Google Scholar 

  • Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.

    Article  MathSciNet  MATH  Google Scholar 

  • Ramsay, J. O. (1998). Estimating smooth monotone functions. Journal of the Royal Statistical Society. Series B. Statistical Methodology, 60, 365–375.

    Article  MathSciNet  MATH  Google Scholar 

  • Ramsay, J. O. (2000). TestGraf: A computer program for nonparametric analysis of testing data. Unpublished manuscript, McGill University.

    Google Scholar 

  • Robertson, T., Wright, F. T. & Dykstra, R. L. (1988). Order restricted statistical inference. New York: Wiley.

    MATH  Google Scholar 

  • Rosenbaum, P. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–435.

    Article  MathSciNet  MATH  Google Scholar 

  • Schriever, B. F. (1985). Order dependence. Unpublished Ph.D. thesis, Free University, Amsterdam.

    Google Scholar 

  • Sijtsma, K. (1988). Contributions to Mokken’s nonparametric item response theory. Amsterdam: Free University Press.

    Google Scholar 

  • Sijtsma, K. (1998). Methodology review: Nonparametric IRT approaches to the analysis of dichotomous item scores. Applied Psychological Measurement, 22, 3–32.

    Article  MathSciNet  Google Scholar 

  • Sijtsma, K. (2001). Developments in measurement of persons and items by means of item response models. Behaviormetrika, 28, 65–94.

    Article  MATH  Google Scholar 

  • Sijtsma, K., & Molenaar, I. W. (2002). Introduction to nonparametric item response theory. Thousand Oaks: Sage Publications.

    MATH  Google Scholar 

  • Stout, W. F. (1987). A nonparametric approach for assessing latent trait unidimensionality. Psychometrika, 52, 589–617.

    Article  MathSciNet  MATH  Google Scholar 

  • Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325.

    Article  MathSciNet  MATH  Google Scholar 

  • Van der Linden, W., & Hambleton, R. (1997). Handbook of modern item response theory. New York: Springer.

    Book  MATH  Google Scholar 

  • Wahba, G. (1990). Spline Models for observational data. Philadelphia: Society for Industrial and Applied Mathematics.

    Book  MATH  Google Scholar 

  • Wright, E. T. (1981). The asymptotic behavior of monotone regression estimates. The Annals of Statistics, 9, 443–448.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Psychology, University of the Republic, Montevideo, Uruguay

    Mario Luzardo

  2. University Center East Regional, University of the Republic, Maldonado, Uruguay

    Pilar Rodríguez

Authors
  1. Mario Luzardo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pilar Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Luzardo .

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

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Luzardo, M., Rodríguez, P. (2015). A Nonparametric Estimator of a Monotone Item Characteristic Curve. 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_8

Download citation

Publish with us

Policies and ethics

Search

Navigation