Paired Comparisons with Standard Rasch Software

Rasch computer programs that can handle incomplete data (such as responses made to computer-adaptive tests) can also analyze paired comparisons, such as consumer preferences and sports competitions.

Set up your data matrix so that each row corresponds to a player (team, competitor, brand, procedure, etc.) and each column an occasion (match, game, aspect of quality, consumer, etc.). The rows can be used for occasions and the columns for players, if more convenient.

There are as many columns as there are occasions (matches, etc.). In each column make only two entries, the results: enter a "1" in the winner's row and a "0" in the loser's. The column total is always 1. Each row total is that player's total number of wins.

The test length is the number of occasions. The resulting measures for the columns, the occasions, can be ignored, but large column misfit flags unexpected outcomes. The measures for the rows, the players, are in logits, but doubled by this estimation method. The reported standard errors are 1.4 times too large.

Here is confirmation that the reported logits need to be divided by two. Player n of ability B_n wins R_n times against Player m of ability B_m, who wins R_m times. By direct application of the Rasch model (or the Bradley-Terry-Luce paired comparison model):

Whenever persons n and m meet, there results an occasion column, o, with two observations in it, one "1" and one "0", summing to 1. Thus all occasions on which n and m meet produce columns with the same raw score of 1 across the same two rows, n and m, and so have the same occasion difficulty, D_o. Because any wins for player m against player n appear in the data matrix as losses for player n on occasions with difficulty D_o, an ability estimate for player n, B'_n, is given by

Similarly, because each win for player n against player m appears in the data matrix as a loss for player m on an occasion with difficulty D_o, an ability estimate for player m, B'_m, is given by

Yielding,

and

confirming that the reported estimates, B'_n and B'_m are twice as large as the direct Rasch estimates, B_n and B_m.

If your Rasch software supports measure rescaling, an adjustment of 0.5 can be made, but this adjustment produces an S.E. that is 71% of its correct value.

Figure 1 shows the match results from a chess tournament. The columns represent chess matches in the order they were reported in the magazine, "Chess". Drawn matches have been ignored. A halving adjustment produces the measures in Table 1. Browne has the highest ability measure with 7 wins in 8 non-drawn matches. Hort's and Zichichi's results show misfit because Hort unexpectedly lost to Zichichi.

Paired comparison with ties (draws, "no preference", etc.) are equally simple. Merely enter "2" for win, "1" for tie, "0" for loss. Each column (occasion) now has a score of 2. Rescoring the chess data matrix from "1D0" to "210" yields the adjusted measures in Table 2. A reasonable adjustment to the initially reported logits is: logit measures halved, standard errors divided by 1.4. Comparing Table 2 with Table 1, the 5 times that Hort drew with weaker players have dragged him down below Mariotti. His loss to Zichichi is no longer as surprising, so the OUTFIT mean-squares are more reasonable.

John M. Linacre

Available Rasch software is listed at www.rasch.org/software.htm.

Paired Comparisons with Standard Rasch Software. Linacre J.M. … Rasch Measurement Transactions, 1997, 11:3 p. 584-5.

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

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