## Speedy JMLE (UCON) Estimation of Rasch Measures

Rasch estimation programs address the incompatible goals of statistical exactness, flexibility, data set capacity and speed. Compromises must be made. Gerhard Fischer's (1974) Rasch computer program features exactness. BIGSTEPS features flexibility and capacity. In real-time instructional use, speed may be the most valuable attribute. Demonstrating Rasch methodology by enabling students to perform many analyses quickly can be more valuable than exactness or flexibility.

I am writing an item analysis program which will go with a French psychometry textbook and probably also an English one. This program implements both classical and Rasch methodology. Flexibility is not important. The data are restricted to dichotomies with no missing responses (a "classical" constraint). Speed is important with capacity the next consideration.

Rasch estimation has the convenient feature that the same raw scores across the same items yield the same measure estimates. When there are L items (with no missing data), there are L+1 possible raw scores: 0,1,..,L. Of these, L-1 measures are estimated directly from the data. The measures for 0 and L can be estimated indirectly by some Bayesian rule, e.g., as the measures for 0.5 and L-0.5. Thus, regardless of how many subjects there are, only L+1 person estimates are required. Similarly, when there are N persons (with no missing data), there are at most N+1 item measures. This leads to computational convenience and the possibility of fast, memory-efficient programs.

PROX (BTD, p. 61-62) is the obvious estimation method, because there is a non-iterative form for complete data. But PROX is based on symmetric, unimodal distributions. My application requires demonstrating the impact of skewed and multi-modal distributions, as well as guessing, copying and other unmodelled activity. So JMLE (UCON) (BTD, p. 63-65) becomes the method of choice. Here is a speedy version of that method, building on the similar approach presented in Wright & Panchapakesan (1969):

Step 1) Cycle through data counting the raw score for every element (person or item). Count up the number of non-extreme items, IACTIVE, and non-extreme persons, PACTIVE. If there are extreme scores for any elements (0 or >=PACTIVE for items, 0 or >=IACTIVE for persons), flag the extreme elements. Cycle through again counting the raw score for every element (person or item) obtained only on non-extreme elements (items or persons). Repeat this until no new extreme elements are identified. Extreme elements will all now have scores of 0 or PACTIVE (for items) and IACTIVE (for persons).

Step 2) Make an entry in table PSCORE for each of the different person raw scores observed on the final set of non-extreme items. In the matching table PFREQ record the number of persons who made each score. These tables have PTABMAX entries with a maximum of IACTIVE+1. Similarly store in table ISCORE each of the observed item raw scores and in matching table IFREQ their frequencies of occurrence. These tables have ITABMAX entries with a maximum of PACTIVE+1.

Step 3) Define tables PEXPECTED, PVAR and PLOGIT to match PSCORE, and tables IEXPECTED, IVAR and ILOGIT to match ISCORE. PEXPECTED and IEXPECTED contain expected score fields. PVAR and IVAR contain variance fields. For each person score, define an initial measure,
PLOGIT(P) = loge(PSCORE(P) / (IACTIVE - PSCORE(P))).
For each item score, define an initial calibration
ILOGIT(I)=loge((PACTIVE-ISCORE(I))/ISCORE(I)).
For extreme scores, adjust PSCORE(P) and ISCORE(I) up by 0.5 for scores of 0, and down by 0.5 for perfect scores.

Step 4) Center the non-extreme item calibrations:
ITOTAL=0
FOR I = 1 to ITABMAX
'not extreme score
IF ISCORE(I)>0 and ISCORE(I)<PACTIVE THEN
'sum non-extreme items
ITOTAL = ITOTAL + ILOGIT(I)*IFREQ(I)
ENDIF
NEXT I
IMEAN = ITOTAL / IACTIVE
FOR I = 1 to ITABMAX: ILOGIT(I) = ILOGIT(I) - IMEAN: NEXT I

Step 5) Initialize the expected score and variance counters:
FOR I = 1 to ITABMAX: IEXPECTED(I) = 0: IVAR(I) = 0: NEXT I
FOR P = 1 to PTABMAX: PEXPECTED(P) = 0: PVAR(P) = 0: NEXT P

Step 6) Compute expected marginal scores and variances corresponding to each raw score:
FOR P = 1 to PTABMAX
FOR I = 1 TO ITABMAX
'expected
PROB = 1 / (1 + EXP(ILOGIT(I) - PLOGIT (P)))
' variance
V = PROB*(1-PROB)
IF PSCORE(P)>0 and PSCORE(P)<IACTIVE THEN
' bypass extremes
IEXPECTED(I)=IEXPECTED(I) + PROB*PFREQ(P)
IVAR(I) = IVAR(I) + V*PFREQ(P)
ENDIF
IF ISCORE(I)>0 and ISCORE(I)<PACTIVE THEN
' bypass extremes
PEXPECTED(P)=PEXPECTED(P) + PROB*IFREQ(I)
PVAR(P) = PVAR(P) + V*IFREQ(I)
ENDIF
NEXT I
NEXT P

Step 7) Compute revised unconditional (joint) maximum-likelihood estimates:
' to save largest change
MAXCHANGE=0
' to save largest raw score residual
MAXRESIDUAL=0
FOR P = 1 TO PTABMAX
' marginal score discrepancy
RESIDUAL = PSCORE(P) - PEXPECTED(P)
IF ABS(RESIDUAL)>MAXRESIDUAL THEN
MAXRESIDUAL = ABS(RESIDUAL)
ENDIF
' Newton-Raphson method
CHANGE = RESIDUAL / PVAR(P)
IF ABS(CHANGE)>MAXCHANGE THEN
MAXCHANGE = ABS(CHANGE)
ENDIF
PLOGIT(P) = PLOGIT(P) + CHANGE
NEXT P
FOR I = 1 TO ITABMAX
RESIDUAL = ISCORE(I) - PEXPECTED(I)
IF ABS(RESIDUAL)>MAXRESIDUAL THEN
MAXRESIDUAL = ABS(RESIDUAL)
ENDIF
CHANGE = RESIDUAL / IVAR(I)
IF ABS(CHANGE)>MAXCHANGE THEN
MAXCHANGE = ABS(CHANGE)
ENDIF
ILOGIT(I) = ILOGIT(I) - CHANGE
NEXT I

Center the ILOGIT(I) item calibrations, as in Step 4 above.

Step 8) Go back and repeat Step 5 onwards until MAXRESIDUAL is less than 0.5 score points, i.e., all residuals are unobservably small, and MAXCHANGE is less than .01 logits, i.e., all changes are unprintably small.

Step 9) The final person measures are PLOGIT(P) with model S.E. of 1/sqrt(PVAR(P), and the final item calibrations are ILOGIT(I) with model S.E. of 1/sqrt(IVAR(I).

Step 10) To compute fit statistics, the original non-extreme response strings must be accessed. For an OUTFIT mean-square, consider non-extreme person Q, with observed score recorded at PSCORE(P). This person has measure PLOGIT(P). Q's response to non-extreme item J is XQJ. Item J has observed score ISCORE(I) and calibration ILOGIT(I). Q's expected response to item J is PQJ. Then,

Programmers will discover ways to optimize this code for particular languages and applications. Comparing this speed- oriented approach with the flexible approach used by BIGSTEPS (no slouch itself) reduced the time required for an analysis of 30 items and 400 persons from 19.2 seconds to 1.0 seconds on my target hardware.

BTD: Wright BD & Stone MH (1979) Best Test Design. Chicago: MESA Press.

Fischer G (1974) Introduction to Mental Test Theory. Bern: Huber.

Wright, B. D., and Panchapakesan, N. A. 1969. A procedure for sample-free item analysis. Educational and Psychological Measurement 29: 23-48

Speedy UCON estimation of Rasch measures. Peladeau N. … Rasch Measurement Transactions, 1995, 9:1 p.418

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

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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 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