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Kernel Smoothing Approaches to Nonparametric Item Characteristic Curve Estimation

Published online by Cambridge University Press:  01 January 2025

J. O. Ramsay
J. O. Ramsay*
Affiliation:
McGill University
*
Requests for reprints should be sent to J. O. Ramsay, Department of Psychology, 1205 Dr. Penfield Ave., Montreal, Quebec, CANADA H3A 1B1.
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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.

Keywords

fast Fourier transformpolytomous responsebootstrappingoption characteristic curvetrace line

Information

Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 4 , December 1991 , pp. 611 - 630
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

The author wishes to acknowledge the support of the National Sciences and Engineering Research Council of Canada through grant A320 and the support of Educational Testing Service during his leave there.

References

Azzalini, A., Bowman, A. W., Härdle, W. (1989). On the use of nonparametric regression for model checking. Biometrika, 76, 1–11.CrossRefGoogle Scholar
Barry, D. (1986). Nonparametric Bayesian regression. Annals of Statistics, 14, 934–953.CrossRefGoogle Scholar
Becker, R. A., Chambers, J. M., Wilks, A. R. (1988). The new S language, Pacific Grove, CA.: Wadsworth & Brooks/Cole.Google Scholar
Bock, R. D., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R. D., Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.CrossRefGoogle Scholar
Buja, A., Hastie, T., Tibshirani, R. (1989). Linear smoothers and additive models (with discussion). Annals of Statistics, 17, 453–555.Google Scholar
Copas, J. B. (1983). Plotting p against x. Applied Statistics, 32, 25–31.CrossRefGoogle Scholar
Craven, P., Wahba, G. (1979). Smoothing noisy data with spline functions. Numerische Mathematik, 31, 377–403.CrossRefGoogle Scholar
Cressie, N., Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika, 48, 129–142.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Eubank, R. L. (1988). Spline smoothing and nonparametric regression, New York: Marcel Dekker.Google Scholar
Gasser, T., Müller, H. G., Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. Journal of the Royal Statistical Society, Series B, 47, 238–252.CrossRefGoogle Scholar
Härdle, W. (1987). Resistant smoothing using the fast Fourier transform. Applied Statistics, 36, 104–111.CrossRefGoogle Scholar
Härdle, W. (1991). Smoothing techniques with implementation in S, New York: Springer-Verlag.CrossRefGoogle Scholar
Hastie, T., Tibshirani, R. (1986). Generalized additive models. Statistical Science, 1, 297–318.Google Scholar
Hastie, T., Tibshirani, R. (1987). Generalized additive models: Some applications. Journal of the American Statistical Association, 82, 371–386.CrossRefGoogle Scholar
Levine, M. V. (1984). An introduction to multilinear formula score theory, Champaign, IL: University of Illinois, Department of Educational Psychology, Model-Based Measurement Laboratory.Google Scholar
Levine, M. V. (1985). The trait in latent trait theory. In Weiss, D. J. (Eds.), 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
Lewis, C. (1986). Test theory and Psychometrika: The past twenty-five years. Psychometrika, 51, 11–22.CrossRefGoogle Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Lord, F. M., Pashley, P. J. (1988). Confidence bands for the three-parameter logistic item response curve, Princeton, NJ: Educational Testing Service.CrossRefGoogle Scholar
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
Mokken, R. J. (1971). A theory and procedure of scale analysis, The Hague: Mouton.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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
Ramsay, J. O. (1988). Monotone regression splines in action (with discussion). Statistical Science, 3, 425–461.Google Scholar
Ramsay, J. O. (1989). A comparison of three simple test theory models. Psychometrika, 54, 487–499.CrossRefGoogle Scholar
Ramsay, J. O., Abrahamowicz, M. (1989). Binomial regression with monotone splines: A psychometric application. Journal of the American Statistical Association, 84, 906–915.CrossRefGoogle Scholar
Ramsay, J. O., Winsberg, S. (1991). Maximum marginal likelihood estimation for semiparametric item analysis. Psychometrika, 56, 365–379.CrossRefGoogle Scholar
Samejima, F. (1979). A new family of models for the multiple choice item, Knoxville, TN: University of Tennessee, Department of Psychology.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Samejima, F. (1984). Plausibility functions of Iowa Vocabulary Test items estimated by the simple sum procedure of the conditional P.D.F. approach, Knoxville, TN: University of Tennessee, Department of Psychology.Google Scholar
Samejima, F. (1988). Advancement of latent trait theory, Unpublished manuscript, University of Tennessee, Department of Psychology.CrossRefGoogle Scholar
Silverman, B. (1986). Density estimation for statistics and data analysis, London: Chapman and Hall.Google Scholar
Thissen, D., Wainer, H. (1982). Some standard errors in item response theory. Psychometrika, 47, 397–412.CrossRefGoogle Scholar
Wahba, G. (1990). Spline models for observational data, Philadelphia: Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Wand, M. P., Schucany, W. R. (1990). Gaussian-based kernels. Canadian Journal of Statistics, 18, 197–204.CrossRefGoogle Scholar
Wingersky, M. S., Patrick, R., Lord, F. M. (1988). LOGIST users guide, Princeton, NJ: Educational Testing Service.Google Scholar
Winsberg, S., Ramsay, J. O. (1980). Monotonic transformations to addivity using splines. Biometrika, 67, 669–674.CrossRefGoogle Scholar
Winsberg, S., Ramsay, J. O. (1983). Monotonic spline transformations for dimension reduction. Psychometrika, 48, 403–423.CrossRefGoogle Scholar