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 Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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