## DIF in Polytomous Items

Zwick & Thayer (Z&T, 1996) present elaborations of the Mantel-Haenszel dichotomous DIF method in order to estimate DIF (differential item functioning) in polytomous items. Since the Mantel-Haenszel statistic is a log-odds estimator, it produces similar DIF findings to Rasch techniques. How do the polytomous versions compare?

Extract from
Zwick & Thayer's Table 2
Rating Category
on Target Item
Subject Group 1 2 3
Low Performers: Reference
Focal
13
5
5
14
7
4
High Performers: Reference
Focal
28
1
54
2
98
10

Z&T present a small data set which can be easily analyzed with Rasch programs. Examining their Table, one can see that, though there are many more subjects in the Reference than the Focal groups (as expected), the average rating for both the Low and High Focal groups looks higher than for the corresponding Reference group. Could this be accidental?

```+------------------------------------------------------------+
|Obsvd    Exp.  Obsvd  Obs-Exp| Bias  Model        |         |
|Score   Score  Count  Average|Measure S.E. Z-Score|Group    |
+-----------------------------+--------------------+---------+
|  474    480.0   205     -.03|  -.05   .09    .57 |Reference|
|   80     74.0    36      .17|   .29   .22  -1.30 |Focal    |
+-----------------------------+--------------------+---------+
|  277.0  277.0   120.5    .07|  -.12   .16   -.36 |Mean     |
|  197.0  203.0    84.5    .10|   .17   .06    .93 |S.D.     |
+------------------------------------------------------------+
|Fixed (all = 0) chi-square: 2.0  d.f.: 2  significance: .36 |
+------------------------------------------------------------+
```

The Facets Rasch analysis program incorporates a post-hoc bias/interaction measurement routine. For this analysis, all low performers (regardless of group) are asserted to have the same measure, and similarly all high performers. All performers share the same three category rating scale. The analysis finds the rating scale to be not very discriminating with only .14 logits between the step difficulties for categories 12 and 23. The high performers measure .90 logits higher than the low performers.

The Facets Bias/Interaction Table (shown here) reports that the 205 ratings of the Reference group sum to 474 and the 36 for the Focal group total 80. On average the Reference group was rated .03 points lower and the Focal group .17 points higher than expected after allowing for the relative performance of the high and low strata and the structure of the rating scale. This points difference gives a measured advantage (DIF) to the Focal group on this item of .29 - -.05 = .34 logits with a joint standard error of sqrt(.09^2+.22^2) = .24 logits. The Z statistic for the DIF is .34/.24 = 1.42, slightly more conservative than Z&T's two Z statistics of 1.45 and 1.55, but equivalent in meaning as "not significantly improbable". Rasch and Z&T's methods produce similar results.

Facets also tests the hypothesis that the two reported Biases represent the same zero bias value. This fixed chi-square test yields a significance of .36, suggesting that, though the group values are somewhat far apart, it is reasonable to consider them as reflecting the same underlying common value.

Z&T are also concerned about how differences in item discrimination across groups affect DIF. In Rasch methodology this is easily investigated. Merely allow each group to define its own rating scale structure and compare results.

John Michael Linacre

Zwick R, Thayer DT (1996) Evaluating the magnitude of Differential Item Functioning in polytomous items. Journal of Educational Statistics 21:3 187-201.

DIF in polytomous items. Linacre J.M. … Rasch Measurement Transactions, 1996, 10:3 p. 520.

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
March 21, 2019, Thur. 13th annual meeting of the UK Rasch user group, Cambridge, UK, http://www.cambridgeassessment.org.uk/events/uk-rasch-user-group-2019
April 4 - 8, 2019, Thur.-Mon. NCME annual meeting, Toronto, Canada,https://ncme.connectedcommunity.org/meetings/annual
April 5 - 9, 2019, Fri.-Tue. AERA annual meeting, Toronto, Canada,www.aera.net/Events-Meetings/Annual-Meeting
April 12, 2019, Fri. On-line course: Understanding Rasch Measurement Theory - Master's Level (G. Masters), https://www.acer.org/au/professional-learning/postgraduate/rasch
May 24 - June 21, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - 30, 2019, Wed.-Thu. Measuring and scale construction (with the Rasch Model), University of Manchester, England, https://www.cmist.manchester.ac.uk/study/short/intermediate/measurement-with-the-rasch-model/
June 4 - 7, 2019, Tue.-Fri.In-Person Italian Rasch Analysis Workshop based on RUMM (Fabio La Porta and Serena Caselli; entirely in Italian). Prof David Andrich from Western Australia University will be hosted by the workshop. For enquiries and registration email to workshop.rasch@gmail.com
June 17-19, 2019, Mon.-Wed. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Introduction to Rasch measurement (Trevor Bond, Winsteps), Announcement
June 20-21, 2019, Thurs.-Fri. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Advanced Rasch measurement with Facets (Trevor Bond, Facets), Announcement
June 28 - July 26, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 2-5, 2019, Tue.-Fri. 2019 International Measurement Confederation (IMEKO) Joint Symposium, St. Petersburg, Russia,https://imeko19-spb.org
July 11-12 & 15-19, 2019, Thu.-Fri. A Course in Rasch Measurement Theory (D.Andrich), University of Western Australia, Perth, Australia, flyer - http://www.education.uwa.edu.au/ppl/courses
Aug 5 - 10, 2019, Mon.-Sat. 6th International Summer School "Applied Psychometrics in Psychology and Education", Institute of Education at HSE University Moscow, Russia.https://ioe.hse.ru/en/announcements/248134963.html
Aug. 9 - Sept. 6, 2019, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
August 25-30, 2019, Sun.-Fri. Pacific Rim Objective Measurement Society (PROMS) 2019, Surabaya, Indonesia https://proms.promsociety.org/2019/
Oct. 11 - Nov. 8, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Nov. 3 - Nov. 4, 2019, Sun.-Mon. International Outcome Measurement Conference, Chicago, IL,http://jampress.org/iomc2019.htm
Jan. 24 - Feb. 21, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - June 19, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 26 - July 24, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 7 - Sept. 4, 2020, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 9 - Nov. 6, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com