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The Person Response Function as a Tool in Person-Fit Research

Published online by Cambridge University Press:  01 January 2025

Klaas Sijtsma  and
Rob R. Meijer
Klaas Sijtsma*
Affiliation:
Tilburg University
Rob R. Meijer
Affiliation:
University of Twente
*
Requests for reprints should be sent to Klaas Sijtsma, Department of Research Methodology, FSW, Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands. E-Mail: k.sijtsma@kub.nl
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Abstract

Item responses that do not fit an item response theory (IRT) model may cause the latent trait value to be inaccurately estimated. In the past two decades several statistics have been proposed that can be used to identify nonfitting item score patterns. These statistics all yield scalar values. Here, the use of the person response function (PRF) for identifying nonfitting item score patterns was investigated. The PRF is a function and can be used for diagnostic purposes. First, the PRF is defined in a class of IRT models that imply an invariant item ordering. Second, a person-fit method proposed by Trabin & Weiss (1983) is reformulated in a nonparametric IRT context assuming invariant item ordering, and statistical theory proposed by Rosenbaum (1987a) is adapted to test locally whether a PRF is nonincreasing. Third, a simulation study was conducted to compare the use of the PRF with the person-fit statistic ZU3. It is concluded that the PRF can be used as a diagnostic tool in person-fit research.

Keywords

appropriateness measurementinvariant item orderingnonparametric item response theoryperson-fit methodperson response function
Type
Articles
Information
Psychometrika , Volume 66 , Issue 2 , June 2001 , pp. 191 - 207
Copyright
Copyright © 2001 The Psychometric Society

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Footnotes

The authors are grateful to Coen A. Bernaards for preparing the figures used in this article, and to Wilco H.M. Emons for checking the calculations.

References

Agresti, A. (1990). Categorical data analysis. New York: Wiley.Google Scholar
Drasgow, F., Levine, M.V., McLaughlin, M.E. (1987). Detecting inappropriate test scores with optimal and practical appropriateness indices. Applied Psychological Measurement, 11, 59–79.CrossRefGoogle Scholar
Drasgow, F., Levine, M.V., Zickar, M.J. (1996). Optimal identification of mismeasured individuals. Applied Measurement in Education, 9, 47–64.CrossRefGoogle Scholar
Ellis, J.L., van den Wollenberg, A.L. (1993). Local homogeneity in latent trait models. A characterization of the homogeneous monotone IRT model. Psychometrika, 58, 417–429.CrossRefGoogle Scholar
Grayson, D.A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 383–392.CrossRefGoogle Scholar
Hambleton, R.K., Swaminathan, H. (1985). Item response theory. Principles and applications. Boston, MA: Kluwer-Nijhoff Publishing.CrossRefGoogle Scholar
Hemker, B.T., Sijtsma, K., Molenaar, I.W., Junker, B.W. (1997). Stochastic ordering using the latent trait and the sum score in polytomous IRT models. Psychometrika, 62, 331–347.CrossRefGoogle Scholar
Holland, P.W., Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Huynh, H. (1994). A new proof for monotone likelihood ratio for the sum of independent bernoulli random variables. Psychometrika, 59, 77–79.CrossRefGoogle Scholar
Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 21, 1359–1378.CrossRefGoogle Scholar
Klauer, K.C. (1991). An exact and optimal standardized person test for assessing consistency with the Rasch model. Psychometrika, 56, 535–547.CrossRefGoogle Scholar
Klauer, K.C., Rettig, K. (1990). An approximately standardized person test for assessing consistency with a latent trait model. British Journal of Mathematical and Statistical Psychology, 43, 193–206.CrossRefGoogle Scholar
Levine, M.V., Drasgow, F. (1982). Appropriateness measurement: Review, critique and validating studies. British Journal of Mathematical and Statistical Psychology, 35, 42–56.CrossRefGoogle Scholar
Levine, M.V., Rubin, D.B. (1979). Measuring the appropriateness of multiple-choice test scores. Journal of Educational Statistics, 4, 269–290.CrossRefGoogle Scholar
Lindgren, B.W. (1993). Statistical theory. New York: Chapman & Hall.Google Scholar
Lord, F.M. (1952). A theory of test scores. Psychometrika Monograph No. 7.Google Scholar
Lord, F.M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Erlbaum.Google Scholar
Lumsden, J. (1978). Tests are perfectly reliable. British Journal of Mathematical and Statistical Psychology, 31, 19–26.CrossRefGoogle Scholar
Meijer, R.R. (Ed.). (1996). Person-fit research: Theory and applications [Special issue]. Applied Measurement in Education, 9(1).Google Scholar
Meijer, R.R. (1998). Consistency of test behaviour and individual difference in precision of prediction. Journal of Occupational and Organizational Psychology, 71, 147–160.CrossRefGoogle Scholar
Meijer, R.R., Molenaar, I.W., Sijtsma, K. (1994). Influence of test and person characteristics on nonparametric appropriateness measurement. Applied Psychological Measurement, 18, 111–120.CrossRefGoogle Scholar
Meijer, R.R., Sijtsma, K. (1995). Detection of aberrant item score patterns: A review of recent developments. Applied Measurement in Education, 8, 261–272.CrossRefGoogle Scholar
Mokken, R.J., Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses. Applied Psychological Measurement, 6, 417–430.CrossRefGoogle Scholar
Molenaar, I.W., Hoijtink, H. (1990). The many null distributions of person fit statistics. Psychometrika, 55, 75–106.CrossRefGoogle Scholar
Nering, M.L., Meijer, R.R. (1998). A comparison of the person response function and thel z person-fit statistic. Applied Psychological Measurement, 22, 53–69.CrossRefGoogle Scholar
Ramsay, J.O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.CrossRefGoogle Scholar
Ramsay, J.O. (1995). A similarity-based smoothing approach to nondimensional item analysis. Psychometrika, 60, 323–339.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen, Denmark: Nielsen & Lydiche.Google Scholar
Rosenbaum, P.R. (1987). Probability inequalities for latent scales. British Journal of Mathematical and Statistical Psychology, 40, 157–168.CrossRefGoogle Scholar
Rosenbaum, P.R. (1987). Comparing item characteristic curves. Psychometrika, 52, 217–233.CrossRefGoogle Scholar
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281–304.CrossRefGoogle Scholar
Sijtsma, K. (1998). Methodology Review: Nonparametric IRT approaches to the analysis of dichotomous item scores. Applied Psychological Measurement, 22, 3–31.CrossRefGoogle Scholar
Sijtsma, K., Junker, B.W. (1996). A survey of theory and methods of invariant item ordering. British Journal of Mathematical and Statistical Psychology, 49, 79–105.CrossRefGoogle ScholarPubMed
Sijtsma, K., Molenaar, I.W. (1987). Reliability of test scores in nonparametric item response theory. Psychometrika, 52, 79–97.CrossRefGoogle Scholar
Stout, W.F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325.CrossRefGoogle Scholar
Suppes, P., Zanotti, M. (1981). When are probabilistic explanations possible?. Synthese, 48, 191–199.CrossRefGoogle Scholar
Trabin, T.E., Weiss, D.J. (1983). The person response curve: Fit of individuals to item response theory models. In Weiss, D.J. (Eds.), New horizons in testing (pp. 83–108). New York: Academic Press.Google Scholar
Van der Flier, H. (1982). Deviant response patterns and comparability of test scores. Journal of Cross-Cultural Psychology, 13, 267–298.CrossRefGoogle Scholar
Weiss, D.J. (1973). The stratified adaptive computerized ability test. Minneapolis: University of Minnesota, Department of Psychology, Psychometric Methods Program.Google Scholar