# Item Discrimination Indices

Item discrimination indicates the extent to which success on an item corresponds to success on the whole test. Since all items in a test are intended to cooperate to generate an overall test score, any item with negative or zero discrimination undermines the test. Positive item discrimination is generally productive, unless it is so high that the item merely repeats the information provided by other items on the test. This is the "attenuation paradox."

The Discrimination Index (D) is computed from equal-sized high and low scoring groups on the test. Subtract the number of successes by the low group on the item from the number of successes by the high group, and divide this difference by the size of a group. The range of this index is +1 to -1. Using Truman Kelley's "27% of sample" group size, values of 0.4 and above are regarded as high and less than 0.2 as low by R.L. Ebel (1954, Procedures for the Analysis of Classroom Tests, Educational and Psychological Measurement, 14, 352-364).

The Point-biserial Correlation is the Pearson correlation between responses to a particular item and scores on the total test (with or without that item). The Biserial Correlation models the responses to the item to represent stratification of a normal distribution and computes the correlation accordingly. Again the ranges are +1 to -1. The biserial is always more extreme than the point-biserial. Jm Nunnally (Psychometric Theory, 1967, p. 123) states that "to use the biserial is to paint a faulty picture of the actual size of the correlations obtainable from existing data." A convenient substitute for these correlations, particularly when data are missing, is the correlation between the Rasch person measures and their responses to the item, the point-measure correlation.

The 2-PL model parameterizes item discrimination in the model and uses it to estimate person ability. A 2-PL model can be written:

 (1)

At its core, the estimation process is:

 (2) (3)

Unconstrained, this produces a feed-back loop. In (2), success on highly discriminating items (Xni=1) raises the person measure, failure (Xni=0) lowers the person measure. In (3), success on an item by those with high measures, coupled with failure by those with low measures, raises the item discrimination. This raised discrimination then feeds back into (2) to increase the measure difference between the successful and unsuccessful, which then, in (3), increases the item discrimination, ad infinitum. To avoid this, 2-PL estimation programs introduce constraints such as a maximum limit on item discrimination estimates, and a pre-set person distribution. Rasch models have pre-set item discriminations in the model, so feedback does not occur. Person and item measures can be estimated in (2) because ai is set to 1 at this step. Then those measures can be used in (3) to estimate Rasch" item discriminations. There is no return to (2).

The plot shows the relationship between these indices. It reports item discrimination indices for dichotomous data reported in W J. Micheels and M. R. Karnes (1950, Measuring Educational Achievement, p. 478-9). For these data, the biserial correlations sometimes exceed 1, so that index is contra-indicated. The Discrimination Index (D) has been computed with the top 27% of the person sample in the high group and the bottom 27% in the low group. The trendlines (.... point-biserial, - - - discrimination index) show that all indices give similar information. The item ringed on left side of the plot has a low correlation but high Rasch item discrimination. It is an easy item with a few misfitting incorrect responses. The item ringed in the bottom left of the plot has negative Rasch discrimination. Its model and empirical ICCs are shown here. These results suggest that the point-measure or, for complete data, the point-biserial correlation capture the useful item discrimination information.

John M. Linacre

Note: the biserial correlation originated in Karl Pearson, "On a New Method of Determining Correlation ....", Biometrika, Vol. VII, pp. 96-105, 1909, and the point-biserial correlation originated in Richardson, M.W. & Stalnaker, J.M. (1933). "A note on the use of bi-serial r in test research". Journal of General Psychology, 8, 463-465.

 Item 31 (Micheels & Karnes)

1. The Item Discrimination Index: Does it Work? Tristan Lopez A. … Rasch Measurement Transactions, 1998, 12:1 p. 626
2. BICAL item discrimination index. Wright BD, Mead RJ, Bell SR. … Rasch Measurement Transactions, 2002, 16:1 p.869
3. Item Discrimination Indices. Kelley T., Ebel R., Linacre, JM. … Rasch Measurement Transactions, 2002, 16:3 p.883-4
4. Discrimination, Guessing and Carelessness Asymptotes: Estimating IRT Parameters with Rasch. Linacre J.M. … Rasch Measurement Transactions, 2004, 18:1 p.959-960

Item Discrimination Indices. Kelley T., Ebel R., Linacre, JM. … Rasch Measurement Transactions, 2002, 16:3 p.883-4

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

 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
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
Oct. 25-27, 2017, Wed.-Fri. In-person workshop: Applying the Rasch Model hands-on introductory workshop, Melbourne, Australia (T. Bond, B&FSteps), Announcement
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
April 13-17, 2018, Fri.-Tues. AERA, New York, NY, www.aera.net
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