I have reached the correlation section in a course that I teach and I hit upon the idea of using data from the weekly Bowl Championship Series (BCS) rankings to illustrate different techniques for assessing correlation.
For those not familiar with college football in the United States (where "football" refers to American football, not what is called soccer here and football in most other countries) I should explain that many, many universities and colleges have football teams but each team only plays 10-15 games per season, so not every team will play every other team. The game is so rough that it is not feasible to play more than one match per week and a national playoff after the regular season is impractical. It would take too long and the players are, in theory, students first and athletes second. In place of a national playoff there are various polls of coaches or sports writers that purport to rank teams nationally. Several analysts also publish computer-based rankings that use complicated formulas based on scores in individual games, strength of the opponent, etc. to rank teams. Rankings from two of the "human polls" (the Harris poll of sports writers and the USA Today poll of the coaches) and from six of the computer polls are combined to produce the official BCS ranking. The Wikipedia entry for "Bowl Championship Series" gives the history and evolution of the actual formula that is currently used. This season has been notable for the volatility of those rankings. One is reminded of the biblical prophesy that "The first shall be last and the last shall be first". Another notable feature this year is the extent to which the computer-based rankings and the rankings in the human polls disagree. I enclose a listing of the top 25 teams and the components of the rankings as of last Sunday (2007-10-21). (Almost all college football games are played on Saturdays and the rankings are published on Sundays). The columns are Rec - won-loss record Hvot - total number of Harris poll votes Hp - proportion of maximum Harris poll votes HR - rank in the Harris poll (smaller is better) Uvot, Up, UR - same for the USA Today poll Cavg - Average score (it's actually a trimmed mean) on computer-based rankings (larger is better) BCS - BCS score - the average of Hp, Up and Cavg Pre - BCS rank in the previous week As I understand it, the votes in the Harris and USA Today polls are calculated by asking each voter to list their top 25 teams then awarding 25 points for a team ranked 1, 24 points for a team ranked 2, etc. on each ballot and calculating the total. Apparently there are now 114 Harris poll participants and 60 USA Today poll participants giving maximum possible scores of 2850 and 1500, respectively. The Cavg column is calculated from 6 scores of 0 to 25 (larger is better) dropping the largest and smallest scores. The raw score is out of 100 and the proportion is reported as Cavg. The data frame is available (for a little while) as http://www.stat.wisc.edu/~bates/BCS.rda The raw scores and the rankings from the Harris and USA Today polls are in fairly good agreement but the Cavg scores are very different. Although scatterplots will show this I feel that correlation measures may be thrown off by the large number of zeros in the Cavg scores. What would be the preferred of measuring correlation in such a case? What would be a good graphical presentation showing the lack of agreement of the various components of the BCS score?
Rec Hvot Uvot Cavg Pre Hp HR Up UR BCS Ohio St. (8-0) 2847 1498 0.93 2 0.99895 1 0.9987 1 0.9759 Boston College (7-0) 2676 1412 0.97 23 0.93895 2 0.9413 2 0.9501 LSU (7-1) 2550 1319 0.96 31 0.89474 3 0.8793 3 0.9114 Arizona St. (7-0) 2003 1089 0.86 35 0.70281 8 0.7260 7 0.7629 Oregon (6-1) 2281 1225 0.67 3 0.80035 5 0.8167 5 0.7623 Oklahoma (7-1) 2521 1306 0.51 32 0.88456 4 0.8707 4 0.7551 West Virginia (6-1) 2157 1134 0.61 36 0.75684 6 0.7560 6 0.7076 Virginia Tech (6-1) 1831 1052 0.69 4 0.64246 10 0.7013 9 0.6779 Kansas (7-0) 1671 911 0.75 6 0.58632 11 0.6073 10 0.6479 South Florida (6-1) 1627 813 0.81 13 0.57088 12 0.5420 12 0.6410 Florida (5-2) 1867 906 0.61 8 0.65509 9 0.6040 11 0.6230 USC (6-1) 2100 1060 0.17 7 0.73684 7 0.7067 8 0.5378 Missouri (6-1) 1568 790 0.53 9 0.55018 13 0.5267 13 0.5356 Kentucky (6-2) 1156 604 0.55 34 0.40561 15 0.4027 15 0.4528 Virginia (7-1) 650 466 0.76 12 0.22807 20 0.3107 18 0.4329 South Carolina (6-2) 1031 474 0.39 33 0.36175 17 0.3160 17 0.3559 Hawaii (7-0) 1265 617 0.00 11 0.44386 14 0.4113 14 0.2851 Georgia (5-2) 711 402 0.23 14 0.24947 19 0.2680 19 0.2492 Texas (6-2) 1054 527 0.00 15 0.36982 16 0.3513 16 0.2404 Michigan (6-2) 643 325 0.26 18 0.22561 21 0.2167 21 0.2341 California (5-2) 873 397 0.02 5 0.30632 18 0.2647 20 0.1970 Auburn (5-3) 333 179 0.33 10 0.11684 23 0.1193 23 0.1887 Connecticut (6-1) 80 75 0.33 29 0.02807 29 0.0500 28 0.1360 Alabama (6-2) 322 177 0.15 27 0.11298 24 0.1180 24 0.1270 Penn St. (6-2) 404 294 0.01 20 0.14175 22 0.1960 22 0.1159
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