# 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:

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

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