Exact Match: OBS% and EXP%


|ENTRY    RAW                   MODEL|   INFIT  |  OUTFIT  |PTMEA|EXACT MATCH|                      |



|    72     14     25   -1.32     .37|2.02   2.9|5.16   5.7|A .04| 60.0  65.8| JACKSON, SOLOMON     |


Suppose your dataset consists of observations, {Xni}, of person n on item i. Based on the Rasch parameters (measures), there is an expected value Eni corresponding to each observation Xni. Eni is obtained by a calculation from the Rasch model.


When the absolute value of (Xni-Eni) is less than 0.5 then the observed data point is within 0.5 score points of its expected value, so the match is the closest possible. Thus, across all observations of item i,

Count ( |Xni-Eni| <0.5 ) = A - these observations are of the closest categories to their expectations

Count ( |Xni-Eni| =0.5 ) = B - these observations are on the borderline of matching their expectations

Count ( |Xni-Eni| >0.5 ) = C - these observations are at least one category away from their expectations


So that A+B+C = Count (Xni)


OBS% = Observed % = 100 * ( A + B/2 ) / ( A+B+C )


B/2: consider the situation when person ability = item difficulty with dichotomous observations. Then every observation is 1 or 0, and every expectation is 0.5, so B = 100%. Under these conditions, it would be reasonable to say that 50% (B/2) of the observations matched their expectations.


Each possible value of Xni has a probability according to the Rasch model. Based on these, the expected value of OBS% can be computed, this is the EXP%. So, if the possible values of Xni are j=0,1,2,...,m, with probabilities Pnij, then

A = sum ( (|j-Eni|<0.5 )*Pnij )

B = sum ( (|j-Eni|=0.5 )*Pnij )

C = sum ( (|j-Eni|>0.5 )*Pnij )


So that A+B+C = Count (Xni)


EXP% = Expected % = 100 * ( A + B/2 ) / ( A+B+C )


If OBS%<EXP% then the local data are more random than the model predicts.

If OBS%>EXP% then the local data are more predictable than the model predicts.

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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 with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind 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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April 10-12, 2018, Tues.-Thurs. Rasch Conference: IOMW, New York, NY, www.iomw.org
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May 22 - 24, 2018, Tues.-Thur. EALTA 2018 pre-conference workshop (Introduction to Rasch measurement using WINSTEPS and FACETS, Thomas Eckes & Frank Weiss-Motz), https://ealta2018.testdaf.de
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Oct. 12 - Nov. 9, 2018, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com



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