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.
|
If you are doing a paired comparison analysis that models the comparisons directly, then no adjustment to the standard errors or measures is necessary. For instance, using a Facets one-facet model for paired comparisons: The Facets estimates are exactly correct without adjustments. |
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.
| Players | Chess Matches at the Venice Tournament, 1971 |
| Browne Mariotti Tatai Hort Kavalek Damjanovic Gligoric Radulov Bobotsov Cosulich Westerinen Zichichi |
1D.0..1...1....1.....1......D.......D........1.........1.......... 0.1.D..0...1....1.....1......D.......1........D.........1......... .D0..0..1...D....D.....1......1.......1........1.........D........ ...1D1...D...D....1.....D......D.......D........1.........0....... ......010D....D....D.....1......D.......1........1.........D...... ..........00DDD.....D.....D......D.......1........D.........1..... ...............00D0DD......D......1.......1........1.........0.... .....................000D0DD.......D.......1........D.........1... ............................DD0DDD0D........0........0.........1.. ....................................D00D00001.........1.........1. .............................................0D000D0D10..........1 .......................................................00D1D010000 |
Here is confirmation that the reported logits need to be divided by two. Player n of ability Bn wins Rn times against Player m of ability Bm, who wins Rm times. By direct application of the Rasch 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, Do. Because any wins for player m against player n appear in the data matrix as losses for player n on occasions with difficulty Do, 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 Do, an ability estimate for player m, B'm, is given by
Yielding,
B'n - B'm 2*loge(Rn/Rm) 2*(Bn - Bm)
confirming that the reported estimates, B'n and B'm are twice as large as the direct Rasch estimates, Bn and Bm.
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.
| Wins | Non-Draw Matches |
Adj. Measure |
Adj. S.E. |
OUTFIT Mn-Sq |
Players |
| 7 4 6 4 5 2 2 3 3 2 2 1 |
8 5 8 6 7 4 6 7 9 8 9 5 |
2.78 2.25 1.76 1.55 1.47 .25 -.36 -.98 -1.40 -1.85 -2.02 -2.70 |
.83 1.00 .70 .77 .74 1.08 .86 .76 .57 .71 .66 .83 |
.32 9.90 .35 .47 .37 .15 .17 .60 .31 .30 8.67 .57 |
Browne Hort Mariotti Kavalek Tatai Damjanovic Radulov Gligoric Cosulich Westerinen Zichichi Bobotsov |
| Score from 11 matches | Adj. Measure |
Adj. S.E. |
OUTFIT Mn-Sq |
Players |
| 17 15 14 14 13 11 10 9 8 8 7 6 |
1.09 .68 .56 .56 .33 -.01 -.17 -.34 -.51 -.51 -.69 -.88 |
.36 .33 .32 .32 .31 .31 .31 .31 .31 .31 .33 .34 |
1.02 .96 1.54 .83 .80 .37 .91 .52 1.00 1.15 .89 1.90 |
Browne Mariotti Hort Tatai Kavalek Damjanovic Gligoric Radulov Bobotsov Cosulich Westerinen 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
Paired Comparisons with Standard Rasch Software. Linacre J.M. … Rasch Measurement Transactions, 1997, 11:3 p. 584-5.
| 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 2nd. 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 |
|
|
|
| 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 | |
|---|---|
| May 15-17, 2013, Wed.-Fri. | In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| May 20-22, 2013, Mon.-Wed. | In-person workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| May 31 - June 28, 2013, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
| June 7, 2013, Fri. | SPHERE workshop: Response-Shift and subjective measures in health science, Nantes, France, www.sphere-nantes.fr/ |
| June 19-21, 2013, Wed.-Fri. | SIS 2013 Conference on Advances in Latent Variables: Methods, Models and Applications, Brescia, Italy, meetings.sis-statistica.org/index.php/sis2013/ALV |
| July 1 - Nov. 30, 2013, Mon.-Sun. | Online Course: Introduction to Rasch Measurement Theory (D. Andrich, RUMM), uwa.edu.au |
| July 5 - Aug. 2, 2013, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
| Aug.1-5, 2013, Thur.-Mon. | TERA-PROMS Annual Meeting, Kaohsiung, Taiwan, tera.education.nsysu.edu.tw |
| Aug. 9 - Sept. 6, 2013, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
| Aug. 22, 2013, Thursday. | Symposium in honor of Svend Kreiner, Copenhagen, Denmark, biostat.ku.dk/kreinersymposium |
| Sept. 4-6, 2013, Wed.-Fri. | IMEKO TC1-TC7-TC13 Symposium: Measurement Across Physical and Behavioural Sciences, Genoa, Italy, www.imeko-genoa-2013.it |
| Sept. 13 - Oct. 11, 2013, Fri.-Fri. | On-line workshop: Rasch Applications in Clinical Assessment, Survey Research, and Educational Measurement (W.P. Fisher), www.statistics.com |
| Sept. 18-20, 2013, Wed.-Fri. | In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| Sept. 23-25, 2013, Mon.-Wed. | In-person workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| Sept. 26-27, 2013, Thurs.-Fri. | In-person workshop: Advanced Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| Oct. 18 - Nov. 15, 2013, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
| Oct. 20 - Oct. 25, 2013, Sun.-Fri. | International Association for Educational Assessment (IAEA) 39th Annual Conference, Tel Aviv, Israel, www.iaea-2013.com |
| Dec. 11-13, 2013, Wed.-Fri. | In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| March 12-14, 2014, Wed.-Fri. | In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| May 14-16, 2014, Wed.-Fri. | In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| May 19-21, 2013, Mon.-Wed. | In-person workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| July 4 - Aug. 1, 2014, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
| Aug. 8 - Sept. 5, 2014, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
| Sept. 10-12, 2014, Wed.-Fri. | In-person workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| Sept. 12 - Oct. 10, 2014, Fri.-Fri. | On-line workshop: Rasch Applications in Clinical Assessment, Survey Research, and Educational Measurement (W.P. Fisher), www.statistics.com |
| Sept. 15-17, 2014, Mon.-Wed. | In-person workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| Sept. 18-19, 2014, Thurs.-Fri. | In-person workshop: Advanced Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric |
| The javascript 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/rmt113o.htm
Website: www.rasch.org/rmt/contents.htm