# [R] Appropriate measure of correlation with 'zero-inflated' data?

Douglas Bates bates at stat.wisc.edu
Thu Oct 25 19:58:24 CEST 2007

```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?
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