We are so accustomed to thinking in terms of raw scores, with their apparent pin-point precision, that we find the concept of measurement error at first abhorrent and later perplexing. We understand, finally, that measurement error always exists, but we don't know how to handle it when reporting our results.
Measurement error always affects our findings. When reporting our results we need to decide whether the result is supposed to describe the local details of this data collection, or to infer some more general "reality". Local results, such as an individual's performance on a test and the test's reliability, must contain their measurement error. General inferences, however, such as the math level of third graders, are intended to be estimated without bias from measurement error. To ascertain what, if anything, of global import we have discovered, we need to deduct the effect of measurement error from our final estimates.
Consider person n with measure bn of standard error sn. The more observations we make, the more we know about this person, and the smaller sn will be. What about the dispersion of a sample of N people? When measurement error is uncorrelated with what is measured, the variance of the sample of observed measures contains the sample variance and also the measures' error variance. But it is only the sample variance that estimates the population sampled.
The estimation for this corrected sample variance is:
s^2 = sum( bn^2 - b.^2 - sn^2 ) / (N-1)
where b. = sum(bn)/N
Correlation coefficients are also affected. Measurement error reduces the maximum observable correlation. Adjusting observed correlation coefficients for measurement error isolates how much sample variance is left unexplained.
Consider the typical regression analysis, in which dependent measures are mistaken for points without error. Let R^2 be the observed multiple squared correlation (proportion of explained variance), E be the observed RMS residual from the regression, and S be the observed standard deviation of the measures. Then E contains both unexplained variance and measurement error, and R^2 has been reduced accordingly. R^2 is defined as
R^2 = (S^2 -E^2) / S^2
If r^2 is the squared multiple correlation corrected for measurement error in the dependent variable and e is the RMS measurement error, then
r^2 = (S^2 - E^2) / (S^2 - e^2)
= R^2 / [ 1 - (e/E)^2 (1 - R^2)]
e<=E; r>=R; when e=0, then r=R; when e=E, then r=1.
It is this larger correlation r, and not R, which tells us how successfully our independent variables have explained the variance of the dependent measure.
Errors, Variances and Correlations, B Wright Rasch Measurement Transactions, 1991, 5:2 p. 147
Please help with Standard Dataset 4: Andrich Rating Scale Model
|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.
|Coming Rasch-related Events|
|April 26-30, 2017, Wed.-Sun.||NCME, San Antonio, TX, www.ncme.org - April 29: Ben Wright book|
|April 27 - May 1, 2017, Thur.-Mon.||AERA, San Antonio, TX, www.aera.net|
|April 29, 2017, Sat., 16:35 to 18:05.||NCME Presidents Invitational Symposium: a new book commemorating Ben Wright's life and career, 16:35 to 18:05, San Antonio, TX, www.ncme.org|
|May 26 - June 23, 2017, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 30 - July 29, 2017, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|July 31 - Aug. 3, 2017, Mon.-Thurs.||Joint IMEKO TC1-TC7-TC13 Symposium 2017: Measurement Science challenges in Natural and Social Sciences, Rio de Janeiro, Brazil, imeko-tc7-rio.org.br|
|Aug. 7-9, 2017, Mon-Wed.||In-person workshop and research coloquium: Effect size of family and school indexes in writing competence using TERCE data (C. Pardo, A. Atorressi, Winsteps), Bariloche Argentina. Carlos Pardo, Universidad Catòlica de Colombia|
|Aug. 7-9, 2017, Mon-Wed.||PROMS 2017: Pacific Rim Objective Measurement Symposium, Sabah, Borneo, Malaysia, proms.promsociety.org/2017/|
|Aug. 10, 2017, Thurs.||In-person Winsteps Training Workshop (M. Linacre, Winsteps), Sydney, Australia. www.winsteps.com/sydneyws.htm|
|Aug. 11 - Sept. 8, 2017, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|Aug. 18-21, 2017, Fri.-Mon.||IACAT 2017: International Association for Computerized Adaptive Testing, Niigata, Japan, iacat.org|
|Sept. 15-16, 2017, Fri.-Sat.||IOMC 2017: International Outcome Measurement Conference, Chicago, jampress.org/iomc2017.htm|
|Oct. 13 - Nov. 10, 2017, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|Jan. 5 - Feb. 2, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|Jan. 10-16, 2018, Wed.-Tues.||In-person workshop: Advanced Course in Rasch Measurement Theory and the application of RUMM2030, Perth, Australia (D. Andrich), Announcement|
|Jan. 17-19, 2018, Wed.-Fri.||Rasch Conference: Seventh International Conference on Probabilistic Models for Measurement, Matilda Bay Club, Perth, Australia, Website|
|May 25 - June 22, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|June 29 - July 27, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com|
|Aug. 10 - Sept. 7, 2018, Fri.-Fri.||On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com|
|Oct. 12 - Nov. 9, 2018, Fri.-Fri.||On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com|
|The HTML to add "Coming Rasch-related Events" to your webpage is:|
The URL of this page is www.rasch.org/rmt/rmt52i.htm