Since Erik Biever first posted the RPICH ratings to HOCKEY-L several
weeks ago, there has been much discussion about the relative weights
assigned to each factor in the rating.  It seems that giving 20%
weight to a teams own winning percentage while giving 40% each to
thier opponents winning percentage and their opponents opponents
winning percentage is heavily biased toward strength of schedule.
Noone has yet been able to explain this but I think I may have found
the answer.  Read on....
 
A team's own winning percentage is given a weight of 20%.  If you
look at the numbers from the latest RPICH (2/15/92), you'll note that
RPICH leader Maine sports the best winning percentage at 88.24% while
lowly Union bottoms out at 10.42%.  There is a difference here of
77.82 points between the best and the worst.
 
Before taking a look at the opponents winning percentage statistics,
it is important to remember that most of the Division I teams play in
leagues, and league games comprise the majority of their schedule.
Some leagues play as many as 32 games per team, while others are
lower at 21, but in all cases, they make up the majority of league
games.  By necessity, the average winning percenatge of all league
teams in league games is .5000.  What does this mean?
 
What it means is that only two things are important in a teams
opponents winning percentage:
 
1) How strong a team's non-league opponents are.
2) How well the league's other members did out of conference.
 
Since these games are only a small percentage of games played in the
league, they will both have a relatively small impact of the
opponents winning percentage.  A team's own non-league opponents is
much more important in this factor than how well the entire league
has done.  This is of primary importance in the opponents opponents
winning percentage.
 
Let's look at the opp win pct. factors for the CCHA, which plays the
most league games at 32.  We see that the highest factor for a CCHA
is 53.23% for Bowling Green, and the lowest is 50.99% for Western
Michigan.  We note that all CCHA teams have factors above .5000, but
we also note that there is a difference of only 2.24 points between
the best and the worst.  The factors for the WCHA, whose teams also
play alot of league games, go from 53.54% to 48.75% for a difference
of 4.79 points.  For Hockey East, with the fewest league games, the
factors range from 57.76% to 47.07% for a difference of 10.69 points.
 
What these numbers indicate is that opponents winning percentages for
teams in a league will be centered around .5000, since the league
games are the majority of games played by those teams.  The
cenetering point will be higher if league teams sport good out of
conference records, or lower otherwise.  But, in any event, only a
small portion of this factor is relevant.  It may seem odd to give
twice as much weight to this as to a teams own win percentage, but
when you account for the fact that most of this factor is the same
for everybody, it actually affects the RPICH value much less than it
appears.
 
One more thing - what about the independents?  Since they do not play
in a league it is not true that most of their opponents win
percenatges will average out to .5000.  When we look at the numbers,
we see that high-rated Alaska-Anchorage has opponents win percentage
at 50.59% while Air Force is way down there at 38.26%.  The lowest
factor for a team in a league belongs to Clarkson at 45.67%.
 
The opponents opponents win percentage factor is affected even more
by league play.  This factor is largely a measure of how strong a
league is, and is determined almost entirely by how the league teams
performed in non-conference games.  For all the non-independents, the
highest value here is 52.29% while the lowest is 47.66%.  Note how
just these two average to 49.98%, almost exactly .5000 !!!  Again the
independents come in lower, with Army as low as 45.86%.  What we see
from these numbers is that they are highly centered around .5000 and
that only a very small fraction of them is relevant.
 
The summary of it all is this:
A team's own win percentage is given a weight of 20% which directly
affects the RPICH value.  The team's opponents win percentage is
given a weight of 40%, but actually has much less impact because most
of these numbers are the same for everybody.  Likewise, the teams
opponents opponents win percentage is assigned a weight of 40%, but
has even less impact because so much of it is the same for everybody.
 
I don't attempt to give any quantitative estimate of exactly how much
impact each factor ends up having, but I'm sure there are some
mathematicians out there who will be anxious to figure it out.
 
One may also conclude that the weights in this system were designed
to rate teams in leagues and to compare how one league performs
against another.  Because the independents generally do not play the
best teams from each league, they end up getting shortchanged in this
system.  Something else for the mathematicians to ponder.
 
There's my theory.  Fire Away !!!
 
Timothy J. Danehy
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Let's Go SLU !!!!                          1992 ECAC Champions !!!!!