# Item Discrimination, Guessing and Carelessness Asymptotes: Estimating IRT Parameters with Rasch

Fred Lord's three-parameter-logistic Item Response Theory (3-PL IRT) model (Birnbaum, 1968) incorporates an item discrimination parameter, modeling the slope of the item characteristic curve, and a lower asymptote parameter modeling "guessing" or, better, "item guessability". Here is a 3-PL model, written in log-odds format, with ci as the lower asymptote, ai as the item discrimination, θn as the person ability and bi as the item difficulty:
 (1)

Lord's 4-PL model (Barton & Lord, 1981) incorporates an upper asymptote parameter for item-specific "carelessness". Here is a "carelessness" model, written in log-odds format, with di as the upper asymptote:
 (2)

 4-PL IRT Item Characteristic Curve

Upper and lower asymptotes are notoriously difficult to estimate, so it appears that Lord abandoned his 4-PL model, and the value of ci in the 3-PL model is, on occasion, imputed from the number of options in a multiple-choice item, instead of being estimated directly from the data. Even the estimation of item discrimination usually requires constraints, such as "ai cannot be negative or too big."

The dichotomous Rasch model, however, provides an opportunity to estimate a first approximation to these parameters. These estimates can be useful in diagnosing whether the behavior they reflect could be distorting the Rasch measures. In the dichotomous Rasch model, ci=0, di=1 and ai=1. We can, however, treat the Rasch values as starting values in a Newton-Raphson iterative processed apparently intended to find the maximum-likelihood values of each of these parameters, in a context in which all other parameter values are known.

Following Wright & Masters (1982, 72-77), and using the standard approach of first and second derivatives of the log-likelihood of the data with respect to the parameter of interest, we obtain the following Newton-Raphson estimation equations for the first approximations:

Item discrimination (ICC slope):
 (3)

 with S.E. = 1/ sqrt ( ) (4)

The Rasch expectation of ai is 1.

A corollary is that, when data fit the dichotomous Rasch model, there is zero correlation between the observation residuals and their generating measure differences.

There is a similar result for polytomous items. The Generalized Partial Credit can be written:
 (5)

The "generalized" item discrimination (ICC slope), equivalent to a Rasch item discrimination index, is:
 (6)

 where and Mni0 = 0 (7)

And for the discrimination of polytomous inter-category "generalized" thresholds:
 (8)

the "generalized" threshold discrimination is:

 (9)

Returning to the dichotomous model, the lower asymptote (guessability) is:
 (10)

where 0 <= ci <= 1

 with S.E. = 1/ sqrt (11)

The Rasch expectation of ci is 0.

The upper asymptote (carelessness) is:
 (12)

where 0 <= di <= 1

 with S.E. = 1/ sqrt (13)

The Rasch expectation of di is 1.

In practice, it is convenient to use only observations in the lower tail for estimating the lower asymptote, in the center for estimating discrimination, and in the upper tail for estimating the upper asymptote.

John Michael Linacre

Birnbaum A. (1968) Some latent trait models and their uses in inferring an examinee's ability. In F.M. Lord & M.R. Novick, Statistical theories of mental test scores (pp. 395-479). Reading, MA: Addison-Wesley.

Barton M.A. & Lord F.M. (1981) An upper asymptote for the three-parameter logistic item-response model. Princeton, N.J.: Educational Testing Service.

1. The Item Discrimination Index: Does it Work? Tristan Lopez A. … Rasch Measurement Transactions, 1998, 12:1 p. 626
2. BICAL item discrimination index. Wright BD, Mead RJ, Bell SR. … Rasch Measurement Transactions, 2002, 16:1 p.869
3. Item Discrimination Indices. Kelley T., Ebel R., Linacre, JM. … Rasch Measurement Transactions, 2002, 16:3 p.883-4
4. Discrimination, Guessing and Carelessness Asymptotes: Estimating IRT Parameters with Rasch. Linacre J.M. … Rasch Measurement Transactions, 2004, 18:1 p.959-960

Discrimination, Guessing and Carelessness Asymptotes: Estimating IRT Parameters with Rasch, Linacre J.M. … Rasch Measurement Transactions, 2004, 18:1 p.959-960

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

 Forum Rasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
May 17 - June 21, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024
June 21 - July 19, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com