Immediate Raw Score to Logit Conversion

A supposed flaw in the Rasch model can be used to great advantage. Bruce Thompson informs us that Fan (1998) and MacDonald and Paunonen (2002) support his perception that the correlation between Rasch measures and raw scores is always .97 ±.02, i.e., is effectively linear. Malec et al. (2000) report a correlation of .98 for their clinical data. If this also holds true for your data then you can immediately convert raw scores to logits!

What conditions must hold for this hold true?
(a) The raw scores must all be on the same set of items.
(b) The proportion of very high and very low scores is low.

Then we have these convenient relationships. For each person n and item i of a test of length L, there is an observation Xni. Its Rasch model expectation is Eni, and the modeled variance of the observation around its expectation is Qni (see Wright and Masters, 1982, p. 100). Thus, person n's raw score, Rn, and raw score "error" variance, Vn , are given by:

variance equations


An approximate conversion factor between raw scores and logits for person n of ability Bn, at the center of the test characteristic curve is the slope of the curve: 1/Vn.

Suppose we know the observed standard deviation, S, of the raw scores of a sample on a test and the reliability estimate (KR-20, Cronbach Alpha) of the test for the same sample, R. Then, from the definition of Reliability as "True Variance" / "Observed Variance", raw score error variance = S2(1-R). So that the raw-score-to-Rasch-measure conversion factor is 1/(S2(1-R)) .

It is conventional to set the origin of the logit scale in the center of the test, i.e., where the raw score is about 50%. This gives the convenient raw score-to-measure conversion:

Bn = (Rn - (Maximum score + Minimum score)/2 ) / S2(1-R)

And the standard error of Bn is 1/sqrt(Vn) = 1/(S sqrt(1-R)) logits.

Applying this to the Wright & Masters (1982) "Liking for Science" data: Raw score S.D. = 8.6, Reliability = .87, minimum score = 0, maximum score = 50. Measure for raw score of 20 = -0.52, for 30 = 0.52, with S.E. ±.32. Winsteps says -0.55, 0.61 with S.E. ±.34. So that the results are statistically equivalent.

John M. Linacre

Fan, X. (1998) Item Response Theory and classical test theory (CTT): An empirical comparison of their item/person statistics. Educational and Psychological Measurement, 58, 357-381.

MacDonald, P., & Paunonen, S.V. (2002) A Monte Carlo comparison of item and person statistics based on item response theory versus classical test theory. Educational and Psychological Measurement, 62.

Malec J. F., Moessner, A. M., Kragness, M., and Lezak, M.D. (2000) Refining a measure of brain injury sequelae to predict postacute rehabilitation outcome: rating scale analysis of the Mayo-Portland Adaptability Inventory (MPAI). Journal of Head Trauma Rehabilitation, 15 (1), 670-682.

Immediate raw score to logit conversion. Linacre, JM. … 16:2 p.877


Immediate raw score to logit conversion. Linacre, JM. … Rasch Measurement Transactions, 2002, 16:2 p.877



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

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
June 29 - July 27, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 25 - July 27, 2018, Wed.-Fri. Pacific-Rim Objective Measurement Symposium (PROMS), (Preconference workshops July 23-24, 2018) Fudan University, Shanghai, China "Applying Rasch Measurement in Language Assessment and across the Human Sciences", www.promsociety.org
July 29 - August 4, 2018 Vth International Summer School `Applied Psychometrics in Psychology and Education`, Institute of Education at the Higher School of Economics, St. Petersburg, Russia, https://ioe.hse.ru/en/announcements/215681182.html
July 30 - Nov., 2018Online Introduction to Classical and Rasch Measurement Theories (D.Andrich), University of Western Australia, Perth, Australia, http://www.education.uwa.edu.au/ppl/courses
Aug. 10 - Sept. 7, 2018, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
August 25 - 28, 2018, Sat.-Tue.Análisis de Rasch introductorio (en español). (Agustín Tristán), Instituto de Evaluación e Ingeniería Avanzada. San Luis Potosí, México. www.ieia.com.mx
Sept. 3 - 6, 2018, Mon.-Thurs. IMEKO World Congress, Belfast, Northern Ireland, www.imeko2018.org
Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com

 

The URL of this page is www.rasch.org/rmt/rmt162e.htm

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