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. |
More robust, especially if you are doing a paired comparison analysis with ties (draws, equalities)
and using a Facets one-facet model for paired comparisons: |
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.
Chess Matches at the Venice Tournament, 1971. Each column is one match. |
1D.0..1...1....1.....1......D.......D........1.........1.......... Browne 0.1.D..0...1....1.....1......D.......1........D.........1......... Mariotti .D0..0..1...D....D.....1......1.......1........1.........D........ Tatai ...1D1...D...D....1.....D......D.......D........1.........0....... Hort ......010D....D....D.....1......D.......1........1.........D...... Kavalek ..........00DDD.....D.....D......D.......1........D.........1..... Damjanovic ...............00D0DD......D......1.......1........1.........0.... Gligoric .....................000D0DD.......D.......1........D.........1... Radulov ............................DD0DDD0D........0........0.........1.. Bobotsov ....................................D00D00001.........1.........1. Cosulich .............................................0D000D0D10..........1 Westerinen .......................................................00D1D010000 Zichichi |
Figure 1. Match results of a chess tournament |
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):
B_{n} - B_{m} = log_{e}(R_{n}/R_{m}) | (1) |
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
B'_{n} - D_{o} = log_{e}(R_{n}/R_{m}) | (2) |
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
B'_{m} - D_{o} = log_{e}(R_{m}/R_{n}) | (3) |
Yielding,
D_{o} = (B'_{n} + B'_{m})/2 | (4) |
and
B'_{n} - B'_{m} = 2*log_{e}(R_{n}/R_{m}) = 2*(B_{n} - B_{m}) | (5) |
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.
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
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.
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