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


Please help with Standard Dataset 4: Andrich Rating Scale Model



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

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
Jan. 25-26, 2017, Wed.-Thurs. In-person workshop: Measurement with the Rasch Model (M. Pampaka, J. Williams, Winsteps), Manchester, UK, website
Feb. 27 - June 24, 2017, Mon.-Sat. On-line: Advanced course in Rasch Measurement Theory (EDUC5606), Website
March 31, 2017, Fri. Conference: 11th UK Rasch Day, Warwick, UK, www.rasch.org.uk
April 2-3, 2017, Sun.-Mon. Conference: Validity Evidence for Measurement in Mathematics Education (V-M2Ed), San Antonio, TX, Information
April 26-30, 2017, Wed.-Sun. NCME, San Antonio, TX, www.ncme.org
April 27 - May 1, 2017, Thur.-Mon. AERA, San Antonio, TX, www.aera.net
May 26 - June 23, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 30 - July 29, 2017, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
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. PROMS 2017: Pacific Rim Objective Measurement Symposium, Sabah, Borneo, Malaysia, proms.promsociety.org/2017/
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
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
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
The HTML to add "Coming Rasch-related Events" to your webpage is:
<script type="text/javascript" src="http://www.rasch.org/events.txt"></script>

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

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