The **W**omen's **C**ollege **H**ockey **O**ffensive/**D**efensive
**R**atings
are based on a multiplicative maximum likelihood model..

**What's a multiplicative model?**

We assume that a team's scoring rate in a particular game depends on
its offensive ability, the quality of the defense it faces, and a home
ice factor. Under the multiplicative WCHODR model, these factors will interact
as products. So the predicted scoring rate,l
_{AB},
for Team A playing Team B on neutral ice would be

where *AVG* represents the average rating (scoring rate) for all
teams. If Team A was playing this game at home, we would multiply the predicted
scoring rate by a home ice advantage *H*, assumed to be constant for
all teams. If Team B was the home team, we would divide Team A's predicted
scoring rate by the same amount *H*.

**How do we compute the ratings under the new system?**

We assume that the scoring in any game follows a *Poisson distribution*,
with the scoring rates determined by the formula above. Thus if Team A
has a scoring rate of l , the probability that
they score exactly *k* goals would be given by

Offensive ratings, Defensive ratings, and the Home Ice Adjustment are
then chosen to *maximize* the probabilities of all previous game scores
(hence the ratings are maximum likelihood estimates of a team's ability).
.

**How does this affect how we interpret the ratings?**

We can interpret an offensive rating as the expected scoring rate against
the hypothetical "average" team and a defensive rating as the expected
goals allowed. The Home Ice Advantage can now be viewed as a percentage,
i.e. *H=1.04* would mean about a 4% increase in scoring rate for the
home team and corresponding decrease for the visitor.

**What about the Overall Rating?**

**New in 2004-5:** If we know the expected scoring rates for any
game, we can use the Poisson probability function to compute an expected probability
that Team A beats Team B (just sum up the joint probabilities for all game scores
with that outcome) and find the expected probability that Team A ties Team B.
To get a measure of overall ability, we compute an **Expected Winning
Pct** by finding the average P(win)+1/2P(Tie) based on the current ratings
assuming a team were to play a balanced schedule (on neutral ice) against all
Division I teams.