Baseball.txt: 1 facet paired comparison (BTL): League Baseball

Each baseball team in the American league in 1948 played every other team about 22 times. This is modeled with the Bradley-Terry-Luce (BTL) model for paired comparisons:

Team A strength (in logits) - Team B strength (in logits) = log (Probability Team A wins / Probability Team B wins)

 

For each pair of teams, the number of the victories of the first team against the second team is modeled as a count of successes on Bernoulli trials. There is only one facet to be measured: the teams. The probability of one team's success is obtained from the difference between its ability and the ability of the team it played.

This is modeled by specifying that the outcome is the result of the first team's measure, indicated by "?", from which is subtracted the second team's measure, indicated by "-?". Both teams are coded as elements in the same facet, 1.

To balance the data, enter each paired comparison twice: A vs. B, and B vs. A.

Adjust for this by giving the Models= a weighting of 0.5.

Teams did not all meet the same number of times. To allow for this, an extra facet, 2, is introduced: the number of games teams played against each other. This extra facet acts as a model selector, permitting pairs of teams to be matched with model statements specifying the correct number of Bernoulli trials. The elements of this extra facet serve no measurement purpose, so their measures are anchored at 0 (the umean= value), preventing them from affecting the teams' measures.

 

Facets specifications and data (in file Baseball.txt):

 

Title = American League Baseball 1948 (Fred Mosteller)

facets = 3 ; three facets in data: game count and two teams

arrange = m,N ; arrange tables by measure-descending, element number-ascending

entered in data = 2,1,1 ; number of encounters from facet 2, then two teams from facet 1

positive = 1 ; more wins - higher measure

non-centered = 0 ; the only active facet, teams, is centered

unexpected = 2 ; report ratings if standardized residual >=|2|

models = ; each model is weighted by 0.5 because each datapoint is entered twice.

;#,?, X,R23,1.0          ; do this analysis first: to check the raw scores

22,?,-?,B22,0.5 ; first team opposes second team 22 times

21,?,-?,B21,0.5 ; Washington-Chicago played 21 times

23,?,-?,B23,0.5 ; Cleveland-Boston played 23 times

*

labels =

1,Teams ; name of first facet

1=Cleveland ; names of elements, cities

 |

8=Chicago

*

2,Games played,A ; dummy facet - all elements anchored at 0

21,21 games,0 ; used for model selection only - anchored at 0

22,22 games,0

23,23 games,0

*

data =

23,1,2,12 ; Cleveland beat Boston 12 out of 23 games

|  ; all the other 26 paired comparisons

21,7,8,12 ; Washington beat Chicago 12 out of 21.

 ; same data again with teams reversed

23,2,1,11 ; Boston beat Cleveland 11 out of 23 games

|  ; all the other 26 paired comparisons

21,8,7,9 ; Chicago beat Washington 9 out of 21. Last line of data, and end of file

 

 

The reported standard error (.17 in this example) is relative to the latent variable. The standard error of the measure of each team for comparison with other teams is S.E./2. So that the S.E. of the measure difference between Cleveland and Boston is ((.17/2)²+(.17/2)²) = .17 in this case.

 

Bradley, R.A. and Terry, M.E. (1952). Rank analysis of incomplete block designs, I. the method of paired comparisons. Biometrika, 39, 324–345.

Luce, R.D. (1959). Individual Choice Behaviours: A Theoretical Analysis. New York: J. Wiley.

Mosteller, F., 1951, Remarks on the method of paired comparisons: III, Psychometrika, 16/2, 207-218, published this dataset.


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