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Tuesday, December 05, 2006

On college football rankings

More college football rankings than you knew existed, 23 to be exact, are available here.

  • Here's the consensus Top 25 based on the average rank. The columns of the table include team name (team), team wins (w), team losses (l), average rank across the 23 different polls (mean), the ordinal ranking of the team (rank), and the team's position in teh USA-Today Coaches poll (usa) :

    . list team w l mean rank usa

    +------------------------------------------------+
    | team w l mean rank usa |
    |------------------------------------------------|
    1. | Ohio St 12 0 1.16 1 1 |
    2. | Michigan 11 1 3.12 2 3 |
    3. | Florida 12 1 4.17 3 2 |
    4. | USC 10 2 4.37 4 7 |
    5. | Louisville 11 1 5.42 5 6 |
    |------------------------------------------------|
    6. | LSU 10 2 7.03 6 4 |
    7. | Boise St 12 0 8.77 7 9 |
    8. | Wisconsin 11 1 11.39 8 5 |
    9. | Oklahoma 11 2 11.44 9 8 |
    10. | Notre Dame 10 2 12.41 10 11 |
    |------------------------------------------------|
    11. | West Virginia 10 2 12.7 11 12 |
    12. | Auburn 10 2 13.38 12 10 |
    13. | California 9 3 14.34 13 19 |
    14. | Tennessee 9 3 14.41 14 18 |
    15. | Arkansas 10 3 14.56 15 13 |
    |------------------------------------------------|
    16. | Virginia Tech 10 2 14.96 16 14 |
    17. | Rutgers 10 2 15.88 17 17 |
    18. | Texas 9 3 17.81 18 16 |
    19. | BYU 10 2 19.14 19 20 |
    20. | Wake Forest 11 2 21.22 20 15 |
    |------------------------------------------------|
    21. | Oregon St 9 4 22.96 21 25 |
    22. | Boston College 9 3 24.19 22 23 |
    23. | UCLA 7 5 25.54 23 . |
    24. | Nebraska 9 4 26.26 24 22 |
    25. | Texas A&M 9 3 27 25 21 |
    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:

    (principal components; 22 components retained)
    Component Eigenvalue Difference Proportion Cumulative
    ------------------------------------------------------------------
    1 64.73326 57.46783 0.7527 0.7527
    2 7.26543 3.92295 0.0845 0.8372
    3 3.34249 1.40829 0.0389 0.8761
    4 1.93420 0.48150 0.0225 0.8986
    5 1.45270 0.18988 0.0169 0.9154
    6 1.26282 0.08367 0.0147 0.9301
    7 1.17915 0.33907 0.0137 0.9438
    8 0.84009 0.12055 0.0098 0.9536
    9 0.71953 0.05399 0.0084 0.9620
    10 0.66554 0.10714 0.0077 0.9697

    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 hump-shaped curve:



    This implies a general consensus of the best and worst teams but that the controversy occurs the middle (again, not surprising).

  • There is always debate about which is the best conference. Personally, I lean towards the SEC. But why take my word for it? I calculated the the average rank within each conference. First, the conferences in alphabetical order:

    Conference | Obs Mean Std. Dev. Min Max
    -------------+--------------------------------------------------------
    ACC | 12 52.33333 31.16914 16 113
    BIG10 | 11 45.45455 32.75474 1 97
    BIG12 | 12 48.91667 28.09225 9 93
    BIGEAST | 8 38.5 27.0238 5 74
    C-USA | 12 76.83333 23.26462 39 109
    INDIE | 4 69.25 51.33144 10 117
    MAC | 12 87.5 23.48113 46 116
    MTNWEST | 9 71.33333 32.17919 19 111
    PAC10 | 10 34.9 24.05757 4 90
    SEC | 12 34.91667 27.26456 3 80
    SUNBELT | 8 97.5 15.84749 71 119
    WAC | 9 75.33333 39.47468 7 118
    And now in order of lowest average rank to highest average rank:

    Conference | Obs Mean Std. Dev. Min Max
    -------------+--------------------------------------------------------
    PAC10 | 10 34.9 24.05757 4 90
    SEC | 12 34.91667 27.26456 3 80
    BIGEAST | 8 38.5 27.0238 5 74
    BIG10 | 11 45.45455 32.75474 1 97
    BIG12 | 12 48.91667 28.09225 9 93
    ACC | 12 52.33333 31.16914 16 113
    INDIE | 4 69.25 51.33144 10 117
    MTNWEST | 9 71.33333 32.17919 19 111
    WAC | 9 75.33333 39.47468 7 118
    C-USA | 12 76.83333 23.26462 39 109
    MAC | 12 87.5 23.48113 46 116
    SUNBELT | 8 97.5 15.84749 71 119

    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:

    Conference| Obs Mean Std. Dev. Min Max
    -------------+--------------------------------------------------------
    PAC10| 10 34.9 24.05757 4 90
    SEC | 10 26.7 21.32838 3 69
    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.

  • Given the annual controversy over who plays in the BCS Championship game, it seems only fitting to provide a final comparison between the average ranking of the DI-A opponents of Ohio State, Michigan and Florida:

    . sum rank if osu

    Variable | Obs Mean Std. Dev. Min Max
    -------------+--------------------------------------------------------
    rank | 12 55.91667 31.95297 2 108

    . sum rank if uf

    Variable | Obs Mean Std. Dev. Min Max
    -------------+--------------------------------------------------------
    rank | 12 39 27.2797 6 98

    . sum rank if um

    Variable | Obs Mean Std. Dev. Min Max
    -------------+--------------------------------------------------------
    rank | 12 46.83333 29.20098 1 83
    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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