Disattenuating Correlation Coefficients

When two sets of measures, {x} and {y}, are correlated, measurement error lowers the correlation coefficient below the level it would have reached had the measures been precise. The reliability (Cronbach Alpha, KR-20, Rasch, etc.) of a set of measures is the proportion of observed variance not due to measurement error, rxx for set {x} and ryy for set {y}. Measurement error can be removed from a correlation coefficient, rxy, to estimate the correlation coefficient disattenuated of measurement error, Rxy, by the formula (Spearman 1904, 1910):

Rxy = rxy / sqrt (rxx ryy)

For two sets of person scores or measures, use the person "test" reliabilities.
For two sets of item p-values or measures, use the item (not the "test") reliabilities.

If you have the standard error of each score or measure, then the reliability of the set of scores or measures is:
rxx = [(observed variance of the measures) - sum(SE² of each measure)/(count of measures)] / [(observed variance of the measures)]

Disattenuated values greater than 1.00 indicate that measurement error is not randomly distributed. Report the disattenuated correlation as 1.0.

Muchinsky (1996) summarizes features of the disattenuated correlation coefficient:

1.Disattenuation does not change the quality of the measures or their predictive power.

2.Disattenuated correlations are not directly comparable with uncorrected correlations.

3.Disattenuated correlations are not suited to statistical hypothesis testing.

4.Disattenuation is not a substitute for precise measurement.

5.But, disattenuation tells us whether the correlation between two sets of measures is low because of measurement error or because the two sets are really uncorrelated.

Randall E. Schumacker

Muchinsky P.M. (1996) The correction for attenuation. Educational & Psychological Measurement 56:1, 63-75.

Spearman C. (1904) The proof and measurement of association between two things. American Journal of Psychology, 15, 72-101.

Spearman C. (1910) Correlation calculated from faulty data. British Journal of Psychology, 3, 271-295

Zimmerman, D. W., & Williams, R. H. (1997). Properties of the Spearman correction for attenuation for normal and realistic non-normal distributions. Applied Psychological Measurement, 21, 253-270.

The reliabilities, rxx and ryy can be computed from tables of measures with standard errors:
rxx = ( S.D.(measures for set(x))**2 - RMSE(set(x))**2 ) / S.D.(measures for set(x))**2
ryy = ( S.D.(measures for set(y))**2 - RMSE(set(y))**2 ) / S.D.(measures for set(y))**2

Table of Disattenuated of Correlation Coefficients
Reliability (Test 1)
multiplied by
Reliability (Test 2)
Reported Test 1 x Test 2 Correlation Coefficient

"The correlation coefficient corrected for attenuation between two tests x and y is the correlation between their true scores [or true measures]. If, on the basis of a sample of examinees, the corrected coefficient is near unity, the experimenter concludes that the two tests are measuring the same trait." (p. 117) in Joreskog, K.G.(1971) Statistical analysis of sets of congeneric tests, Psychometrica 36, 109-133

Disattenuating correlation coefficients. Schumacker RE, Muchinsky PM. … Rasch Measurement Transactions, 1996, 10:1 p.479

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

To be emailed about new material on www.rasch.org
please enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Rasch.org

www.rasch.org welcomes your comments:

Your email address (if you want us to reply):


ForumRasch 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


The URL of this page is www.rasch.org/rmt/rmt101g.htm

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