Linking Constants with Common Items and Judges

Best Test Design (Wright & Stone, 1979 p. 96) gives formulae for evaluating the statistical quality of a linking constant between dichotomous tests with common items.

When Test A is given to a sample of NA persons, and Test B to a different sample of NB persons, then each item i of the K equally trustworthy common items is estimated to have two difficulties, DiA in Test A and DiB in Test B. DiA and DiB have standard errors of approximately 2.5/NA½ and 2.5/NB½ respectively.

If a cross-plot of the two sets of common items indicates that there is a single constant that adds to all difficulties and abilities in Test B to translate them onto the scale of Test A, then that constant is


with standard error


If NANBN, then


A test of the hypothesis that this linking constant explains the difference between difficulties of common items is


Links with Polytomous Items and Judges

These formulae can be generalized for tests containing polytomous items with both common items and common judges. Tests A and B have K common items and J common judges. Common items have difficulties DiA and DiB. Common judges have severities SjA and SjB. The standard errors of the item and judge measures are obtained from the reported results of analyses of Test A and Test B separately.

If cross-plots of the two sets of common item difficulties and of the two sets of judge severities exhibit approximately 45 trends, then the constants to add to difficulties (the Item Link), severities (the Judge Link) and abilities (GAB) in Test B to translate them onto the scale of Test A are




with standard errors of




A test of the hypothesis that each piece of the linking constant explains the difference between common measures in its facet has the form


Information-weighted Linking

When the standard errors of measures within the sets of linking items or judges differ noticeably and their precision is deemed to reflect the influence they should have on the linking constant, then the construction of information-weighted linking constants for the items or judges may be preferred. For instance, the Fisher information in each item difficulty shift is:


Then, the linking constant for the items is


with standard error of


A test of the hypothesis that this linking constant explains the difference between common items is


These item and judge links substitute directly in the earlier formulae for GAB and its standard error.

John Michael Linacre

Linacre J. M. (1998) Linking Constants with Common Items and Judges. Rasch Measurement Transactions 12:1 p. 621.

Linking Constants with Common Items and Judges. Linacre J. M. … Rasch Measurement Transactions, 1998, 12:1 p. 621.




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
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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
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Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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