Hostname: page-component-5f745c7db-q8b2h Total loading time: 0 Render date: 2025-01-06T06:21:25.927Z Has data issue: true hasContentIssue false
  • English
  • Français

Departure from Normal Assumptions: A Promise for Future Psychometrics with Substantive Mathematical Modeling

Published online by Cambridge University Press:  01 January 2025

Fumiko Samejima
Fumiko Samejima*
Affiliation:
University of Tennessee
*
Requests for reprints should be sent to Fumiko Samejima, Psychology Department, 405 Austin Peay Building, University of Tennessee, Knoxville, TN 37996-0900.
Get access Rights & Permissions [Opens in a new window]

Abstract

Normal assumptions have been used in many psychometric methods, to the extent that most researchers do not even question their adequacy. With the rapid advancement of computer technologies in recent years, psychometrics has extended its territory to include intensive cognitive diagnosis, etcetera, and substantive mathematical modeling has become essential. As a natural consequence, it is time to consider departure from normal assumptions seriously. As examples of models which are not based on normality or its approximation, the logistic positive exponent family of models is discussed. These models include the item task complexity as the third parameter, which determines the single principle of ordering individuals on the ability scale.

Keywords

latent trait modelsitem response theorysubstantive mathematical modelinglogistic positive exponent family of modelstask complexitydichotomous responses
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 4 , December 1997 , pp. 471 - 493
Copyright
Copyright © 1997 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (Eds.), Statistical theories of mental test scores (pp. 395–479). Reading, MA: Addison Wesley.Google Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29–51.CrossRefGoogle Scholar
Hemker, B. T., Sijtsma, K., Molenaar, I. W., Junker, B. W. (1996). Polytomous IRT models and monotone likelihood ratio of the total test score. Psychometrika, 61, 679–693.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
Levine, M. (1984). An introduction to multilinear formula scoring theory, Champaign, IL: University of Chicago.Google Scholar
Lord, F. M. (1952). A Theory of Test Scores. Psychometric Monograph, No. 7.Google Scholar
Lord, F. M. (1970). Item characteristic curves estimated without knowledge of their mathematical form: A confrontation of Birnbaum's logistical model. Psychometrika, 35, 43–50.CrossRefGoogle Scholar
Lord, F. M. (1983). Unbiased estimators of ability parmaeters, of their variance, and of their parallel-forms reliability. Psychometrika, 48, 233–245.CrossRefGoogle Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison Wesley.Google Scholar
Luce, R. D. (1959). Individual choice behavior, New York: Wiley.Google Scholar
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.CrossRefGoogle Scholar
Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, No. 17.Google Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph, No. 18.Google Scholar
Samejima, F. (1973). A comment on Birnbaum's three-parameter logistic model in the latent trait theory. Psychometrika, 38, 221–233.CrossRefGoogle Scholar
Samejima, F. (1993). An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete. Psychometrika, 58, 119–138.CrossRefGoogle Scholar
Samejima, F. (1993). The bias function of the maximum likelihood estimate of ability for the dichotomous response level. Psychometrika, 58, 195–209.CrossRefGoogle Scholar
Samejima, F. (1995). Acceleration model in the heterogeneous case of the general graded response model. Psychometrika, 60, 549–572.CrossRefGoogle Scholar
Samejima, F. (1996). Evaluation of mathematical models for ordered polychotomous responses. Behaviormetrika, 23, 17–35.CrossRefGoogle Scholar
Samejima, F. (in press). Rationale and actual procedures of efficient nonparametric approaches for estimating the operating characteristics of discrete item responses. Psychometrika.Google Scholar
Samejima, F. (accepted). Virtues of asymmetric item characteristic curves with the logistic model as a transition. Psychometrika.Google Scholar
Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427–450.CrossRefGoogle Scholar
Yen, W. M., Burket, G. R., Sykes, R. C. (1991). Nonunique solutions to the likelihood equation for the three-parameter logistic model. Psychometrika, 56, 39–54.CrossRefGoogle Scholar