Psychometrika

, Volume 56, Issue 4, pp 611–630 | Cite as

Kernel smoothing approaches to nonparametric item characteristic curve estimation

  • J. O. Ramsay
Article

Abstract

The option characteristic curve, the relation between ability and probability of choosing a particular option for a test item, can be estimated by nonparametric smoothing techniques. What is smoothed is the relation between some function of estimated examinee ability rankings and the binary variable indicating whether or not the option was chosen. This paper explores the use of kernel smoothing, which is particularly well suited to this application. Examples show that, with some help from the fast Fourier transform, estimates can be computed about 500 times as rapidly as when using commonly used parametric approaches such as maximum marginal likelihood estimation using the three-parameter logistic distribution. Simulations suggest that there is no loss of efficiency even when the population curves are three-parameter logistic. The approach lends itself to several interesting extensions.

Key words

fast Fourier transform polytomous response bootstrapping option characteristic curve trace line 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Azzalini, A., Bowman, A. W., & Härdle, W. (1989) On the use of nonparametric regression for model checking.Biometrika, 76, 1–11.Google Scholar
  2. Barry, D. (1986). Nonparametric Bayesian regression.Annals of Statistics, 14, 934–953.Google Scholar
  3. Becker, R. A., Chambers, J. M. & Wilks, A. R. (1988).The new S language. Pacific Grove, CA.: Wadsworth & Brooks/Cole.Google Scholar
  4. Bock, R. D., & Aitkin M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.Psychometrika, 46, 443–459.Google Scholar
  5. Bock, R. D., & Lieberman, M. (1970). Fitting a response model forn dichotomously scored items.Psychometrika, 35, 179–197.Google Scholar
  6. Buja, A., Hastie, T., & Tibshirani, R. (1989). Linear smoothers and additive models (with discussion).Annals of Statistics, 17, 453–555.Google Scholar
  7. Copas, J. B. (1983). Plottingp againstx.Applied Statistics, 32, 25–31.Google Scholar
  8. Craven, P., & Wahba, G. (1979). Smoothing noisy data with spline functions.Numerische Mathematik, 31, 377–403.Google Scholar
  9. Cressie, N., & Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models.Psychometrika, 48, 129–142.Google Scholar
  10. Drasgow, F., Levine, M. V., Williams, B., McLaughlin, M. E., & Candell, G. L. (1989). Modeling incorrect responses to multiple-choice items with multilinear formula score theory.Applied Psychological Measurement, 13, 285–299.Google Scholar
  11. Eubank, R. L. (1988).Spline smoothing and nonparametric regression. New York: Marcel Dekker.Google Scholar
  12. Gasser, T., Müller, H. G., & Mammitzsch, V. (1985). Kernels for nonparametric curve estimation.Journal of the Royal Statistical Society, Series B, 47, 238–252.Google Scholar
  13. Härdle, W. (1987). Resistant smoothing using the fast Fourier transform.Applied Statistics, 36, 104–111.Google Scholar
  14. Härdle, W. (1991).Smoothing techniques with implementation in S. New York: Springer-Verlag.Google Scholar
  15. Hastie, T., & Tibshirani, R. (1986). Generalized additive models.Statistical Science, 1, 297–318.Google Scholar
  16. Hastie, T., & Tibshirani, R. (1987). Generalized additive models: Some applications.Journal of the American Statistical Association, 82, 371–386.Google Scholar
  17. Levine, M. V. (1984).An introduction to multilinear formula score theory (Measurement Series 84-5). Champaign, IL: University of Illinois, Department of Educational Psychology, Model-Based Measurement Laboratory.Google Scholar
  18. Levine, M. V. (1985). The trait in latent trait theory. In D. J. Weiss (Ed.),Proceedings of the 1982 item response theory/computerized adaptive testing conference. Minneapolis, MN: University of Minnesota, Department of Psychology, Computerized Adaptive Testing Laboratory.Google Scholar
  19. Lewis, C. (1986). Test theory andPsychometrika: The past twenty-five years.Psychometrika, 51, 11–22.Google Scholar
  20. Lord, F. M. (1980).Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  21. Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.Google Scholar
  22. Lord, F. M., & Pashley, P. J. (1988).Confidence bands for the three-parameter logistic item response curve (Research Report No. 88-67). Princeton, NJ: Educational Testing Service.Google Scholar
  23. Mislevy, R. J., & Bock, R. D. (1982).BILOG: Item analysis and test scoring with binary logistic models [Computer program]. Mooresville, IN: Scientific Software.Google Scholar
  24. Mokken, R. J. (1971).A theory and procedure of scale analysis. The Hague: Mouton.Google Scholar
  25. O'Sullivan, F., Yandell, B. S., & Raynor, W. J. Jr. (1986). Automatic smoothing of regression functions in generalized linear models.Journal of the American Statistical Association, 81, 96–103.Google Scholar
  26. Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. (1986).Numerical recipes: The art of scientific computing. Cambridge: Cambridge University Press.Google Scholar
  27. Ramsay, J. O. (1988). Monotone regression splines in action (with discussion).Statistical Science, 3, 425–461.Google Scholar
  28. Ramsay, J. O. (1989). A comparison of three simple test theory models.Psychometrika, 54, 487–499.Google Scholar
  29. Ramsay, J. O., & Abrahamowicz, M. (1989). Binomial regression with monotone splines: A psychometric application.Journal of the American Statistical Association, 84, 906–915.Google Scholar
  30. Ramsay, J. O., & Winsberg, S. (1991). Maximum marginal likelihood estimation for semiparametric item analysis.Psychometrika, 56, 365–379.Google Scholar
  31. Samejima, F. (1979).A new family of models for the multiple choice item (Research Report No. 79-4). Knoxville, TN: University of Tennessee, Department of Psychology.Google Scholar
  32. Samejima, F. (1981).Efficient methods of estimating the operating characteristics of item response categories and challenge to a new model for the multiple-choice item. Unpublished manuscript, University of Tennessee, Department of Psychology.Google Scholar
  33. Samejima, F. (1984).Plausibility functions of Iowa Vocabulary Test items estimated by the simple sum procedure of the conditional P.D.F. approach (Research Report No 84-1). Knoxville, TN: University of Tennessee, Department of Psychology.Google Scholar
  34. Samejima, F. (1988).Advancement of latent trait theory, Unpublished manuscript, University of Tennessee, Department of Psychology.Google Scholar
  35. Silverman, B. (1986).Density estimation for statistics and data analysis. London: Chapman and Hall.Google Scholar
  36. Thissen, D., & Wainer, H. (1982). Some standard errors in item response theory.Psychometrika, 47, 397–412.Google Scholar
  37. Wahba, G. (1990).Spline models for observational data. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
  38. Wand, M. P., & Schucany, W. R. (1990). Gaussian-based kernels.Canadian Journal of Statistics, 18, 197–204.Google Scholar
  39. Wingersky, M. S., Patrick, R., & Lord, F. M. (1988). LOGIST users guide. Princeton, NJ: Educational Testing Service.Google Scholar
  40. Winsberg, S., & Ramsay, J. O. (1980). Monotonic transformations to addivity using splines.Biometrika, 67, 669–674.Google Scholar
  41. Winsberg, S., & Ramsay, J. O. (1983). Monotonic spline transformations for dimension reduction.Psychometrika, 48, 403–423.Google Scholar

Copyright information

© The Psychometric Society 1991

Authors and Affiliations

  • J. O. Ramsay
    • 1
  1. 1.McGill UniversityCanada

Personalised recommendations