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 N_A persons, and Test B to a different sample of N_B persons, then each item i of the K equally trustworthy common items is estimated to have two difficulties, D_iA in Test A and D_iB in Test B. D_iA and D_iB have standard errors of approximately 2.5/N_A^½ and 2.5/N_B^½ 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 N_AN_BN, then

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

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 D_iA and D_iB. Common judges have severities S_jA and S_jB. 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 (G_AB) 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

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 G_AB 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
in Spanish:	Análisis de Rasch para todos, Agustín Tristán		Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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.

The URL of this page is www.rasch.org/rmt/rmt121m.htm

Website: www.rasch.org/rmt/contents.htm