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Tuesday, December 05, 2006
More college football rankings than you knew existed, 23 to be exact, are available here. How many independent sources of variation are there across the different rankings? One might naturally expect there are relatively few: after all, how many different ways can there be to rank teams? One way to determine the number of independent sources of variation across a number of different variables is principal components analysis. It is common practice to consider only those components with an eigenvalue greater than one as a "significant" independent source of variation. Here are the first ten eigenvalues from the principal components analysis of the 23 different rankings:
There appears to be seven independent sources of variation across the different rankings; a bit more than I expected. However, the first component accounts for 75% of the variation across the different ranking systems  this suggests that there is considerable agreement amongst the various ranking systems (which is not surprising). The second component accounts for an additional 8% and the seventh component (the last with an Eigenvalue greater than one) contributes only 1.3% of the total variation. Overall, the first seven components account for 94% of the variation across the different ranking systems. This suggests some disagreement across the ranking systems but does the disagreement occur more so at top or at the bottom of the rankings? If we look at a scatter plot of the standard deviation of a team's ranking versus its average rank we find a humpshaped curve: This implies a general consensus of the best and worst teams but that the controversy occurs the middle (again, not surprising). And now in order of lowest average rank to highest average rank:
I was barely wrong. In the 2006 season the Pacific Ten had a .01667 better average ranking than the SEC. The Big East came in third, but has only eight teams and the Sun Belt is a bad conference  but hey they are playing football. Noting that the SEC has two more teams than the PAC10 and the SEC's worst team was better than the Pac Ten's worst team, perhaps a better comparison is to take the ten best teams in the SEC and create a virtual conference of those teams (perhaps this virtual conference would not have the same rankings, I 'll admit) and pit them against the Pacific 10. We get the following: In this comparison the SEC is much stronger than the Pacific 10. Indeed, the top ten teams in the SEC would be the strongest (morphed) conference in the nation and if we limited our measure to only eight teams, the SEC would have an average rank of 18.375 (sd = 12.8). This would blow away the Big East and the Sun Belt conferences. The average rank of OSU opponents was 55, the average rank of UM opponents was 46, and the average rank of UF opponents was 39. While the rematch between OSU and Michigan would have been interesting, UF seems deserving given the strength of their opponents. Does it not seem odd that UF has a much stronger schedule, has won one more game than OSU and their consensus ranking is #3 not #2? STATA data file
Comments:
Craig  I hope you keep posting, even though you don't get much comment traffic. I enjoy them very much. I have a folder in Firefox that I read a few times per week and your blog is in that. Much appreciated.
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