Convergence, Collapsed Categories and Construct Validity

While analyzing a dataset of 10 polytomous partial-credit items, I found that the estimates of item difficulties and their ordering varied depending on the convergence limits set for estimation. The ordering is important because it is used as evidence for the construct validity of the instrument. In my investigation, the item locations were calibrated twice. Once with the item convergence limits set at 0.01 and again at 0.0005. The sample size was 6,520 and the person ability distribution was roughly normal.

Figure 1 is a graphical presentation of the differences between item locations at the two convergence limits. As expected, the tighter (smaller) convergence limit resulted in more dispersed item difficulty estimates. The location differences between convergence at 0.01 and convergence at 0.0005 are surprisingly large. The absolute values of the differences vary from 0.38 to 0.92 logits. What could be the reason?

When category frequencies were examined, it was found that items 6-10 had no observations in their extreme highest categories (see Table 1). These had been automatically accommodated by my RUMM2030 analysis. To examine the impact of the unobserved categories on the item locations, the unobserved extreme category 5s were combined with their adjacent category 4s. After collapsing those extreme categories, the item locations were again estimated twice with convergence limits at 0.01 and more tightly at 0.0001.

Figure 2 shows the resulting item estimates. Compared with the estimates in Figure 1, the location differences for each item at the two convergence limits are much smaller. This time, the absolute differences varied from 0.10 to 0.42 logits. Although, there were no changes made to items 1-5, the location differences for most of these items also reduced between the two convergence limits. The difference for item 4 remained somewhat large, perhaps because item 4 has only 1 observation in category 4, its top category, making estimation of its difficulty location less stable.

As expected, the items with collapsed categories, items 6-10, have become relatively easier than in the first, uncollapsed analysis. This is because the definition of item difficulty is "the location on the latent variable at which the top and bottom categories are equally probable." Collapsing the two highest categories for each item has moved the combined top category toward the middle of the original rating scale, and so moved the item location down the latent variable.

In conclusion, these analyses indicate that convergence limits should be set tightly enough to be substantively stable. These analyses also show that collapsing categories can make conspicuous changes to the item difficulty hierarchy. If categories are collapsed and the item hierarchy must be maintained for measure interpretation and construct validity, then pivot-anchoring (RMT 11:3 p. 576-7) may be required.

Edward Li


Figure 1. Item locations with original categories, including unobserved categories.


Figure 2. Item locations with unobserved extreme categories collapsed with neighboring categories

ItemCat 1Cat 2Cat 3Cat 4Cat 5
164039281952  
27685539213  
346130550501181
43155819071 
5450388619971852
650019403982980#
78347191685330#
891833132222670#
9400528982470#
01654442543740#
Table 1. Original category frequencies of the data
Note: # this unobserved category collapsed with adjacent category in the second analysis.


Convergence, Collapsed Categories and Construct Validity. Edward Li ... Rasch Measurement Transactions, 2012, 25:4, 1339




Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com

 

The URL of this page is www.rasch.org/rmt/rmt254a.htm

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