JohnW wrote, in part..
>I assume that Wayne is calculating this as a way to correct for the
>different in the number of games played by ECAC teams this season, and
 
Yes.  There are any number of ways one *might* order teams.  I have
posted ECAC HEAL as much to start discussion as to suggest it might be
better than total points or winning percentage or whatever.
 
>the unbalanced nature of the schedule.  However, HEAL seems like a
>strange choice of rating system for this job, for the following
>reason:
>
>If a group of teams play a balanced schedule, each team playing each
>other team an equal number of times, I think we'd all agree that the
>correct way to rank the teams based on those results is by their
>overall won-lost-tied record.  (Whether you refer to total points or
 
No.  Maybe sane people of letters would agree, but not all.  It is not
obvious to me that won-lost-tied (win% or total points) is "the correct
way to rank".  Maybe it is, but maybe it isn't.
 
>winning percentage, the results are the same, since winning percentage
>is just points divided by twice the number of games played.)  KRACH
>(as one of the defining properties of the system) and RPI (because all
>teams' "Opponents' Winning Percentage" are equal after a balanced
>schedule) are both guaranteed to give the same ranking as
>won-lost-tied record if the schedule is balanced.  HEAL lacks this
>very desirable property, since wins over good and bad teams are rated
>differently, while losses to any team are rated the same.  So if HEAL
>ratings were calculated for last season's ECAC race, or this season's
>Hockey East race, which both used a balanced schedule, there's no
>guarantee that the results would reproduce the final standings.
 
That is correct.  Though HEAL was not developed to measure a balanced
schedule differently than win%, it may.  For the same reason that
win% is best (it, obviously, just is), treating a loss just as a loss
is reasonable IMHO (it just is ... winning is everything :-)).
 
HEAL was invented because when winning percentage was the rule, teams
would schedule weaker teams and, due to any reason they could dream up,
sometimes avoid playing games with stronger opponents.  But I digress.
(I must, since I'm unable to properly defend HEAL (or KRACH or anything)
in statistical terms!)
 
So, HEAL may yield results different from win% with a balanced schedule.
IMHO, this attribute of HEAL causes better selection, because higher
ranked teams generally have had better success against better teams.
They may have fallen against poor teams, but who cares about that?  We
aren't crowning a league or national champion with the ranking, but
ordering teams for the best tournament, using results that matter.
 
It is an anomaly of HEAL that sometimes win% is important (PI) and
sometimes the sum of your wins is important (TI).
 
Perhaps a statistician (I'm on thin ice here, folks) would say that a
loss to a good team is more probable than to a poor team, so the ranking
should take this into account.  I recall explaining this effect on a
local radio station a couple of years ago.  Team A and Team B were close
in the standings.  Team A played a top team and lost.  Team B played a
poor team and won.  Team A charged into the (RPI) ranking lead, just
because they scheduled a top team.  The RPI result was poor, IMHO;  the
win% result was poor, IMHO;  HEAL more correctly dealt with it.
 
>To give an example of how this can happen, imagine a league with four
>teams, A, B, C and D, who each play each other once each, with the
>following results:
>
>head2head               ____HEAL____
>  A B C D   W-L-T  Pts  P.I.    T.I.   Pct   RPI  KRACH
>A   W W T   2-0-1   5  33.33  133.33  .833  .617  350.2
>B L   W T   1-1-1   3  20.00   66.67  .500  .500  100.0
>C L L   W   1-2-0   2  13.33   44.44  .333  .442  56.51
>D T T L     0-1-2   2  13.33   88.89  .333  .442  56.51
>
>The teams have played a balanced schedule, so there is nothing to
>correct for, but while team D's won-lost-tied record of 0-1-2 is worse
>than team B's record of 1-1-1, they have the higher HEAL rating
>(Tournament Index).  As shown above, both RPI and KRACH give the same
>rankings as winning percentage for this balanced schedule.
 
Good example.  That's about as perverse as it gets!  (HEAL and RPI
normally have more in common than each of them have in common with KRACH
or other methods).
 
HEAL has measured "how successful were you over teams with good win%",
more or less.  This doesn't maximize the ability to predict any game
outcome or exactly measure the outcome of all games, but may give you
the best teams for the best tournament games?  If you are Team A, which
team do you want to play in the semi-final?  I'd pick Team C. That
leaves B vs D for the other semi.  Which should be higher ranked?  They
tied head-to-head.  HEAL considers the Team D ties with powerhouse Team
A as more important than the Team B win over lowly Team C. Is that
correct?  Guess it's in the eye of the beholder.
 
If all games are to be treated equally and the results are final, I'll
take Bradley-Terry/KRACH.  But KRACH was not offered as a solution to
the ECAC "standings problem" (until JohnW's article).  If my computer
was smart enough to calculate KRACH I might have used it :-)
 
Cheers and best regards to John Whelan and all his great contributions
to the college hockey lists.
 
Wayne T. Smith
[log in to unmask]                          Old Town Landing
Co-owner of the College Hockey lists - Hockey-L and Hockey3
 
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