BELOW – Bringing Elo to the WCHA

BELOW:  Bringing Elo to the WCHA

BELOW is an Elo-style rating system designed to help determine the relative strength of the ten members of the men’s complement of the Western Collegiate Hockey Association (WCHA).  It’s the project of Geof Morris, a 2002 graduate of The University of Alabama in Huntsville.

How to pronounce it

BEE-low.

What’s going on here

BELOW is a rating system that defines an average team as 1500.  Matchups between two teams use a formula to determine an expected value of a positive result for Team A and Team B.  The results of those matchups are used to confirm a new rating for each team.

The principle

BELOW is the result of pairwise comparisons (note: not the Pairwise Ranking).  If we have an estimate of the relative strength of Team A and Team B (e.g., BELOW), we can determine which team should win a matchup.  If Team A is favored, a win for them adds a small amount to their rating; that amount is subtracted from Team B’s rating; a loss by Team A subtracts a larger amount from their rating, which is given to Team B.

The math

Each team gets an expected value in a matchup.  That value is calculated as follows:

Elo Expected Value

, where E(A) is the expected value of the matchup for Team A, R(A) is the rating of Team A, and R(B) is the rating of Team B.

A new rating for Team A is calculated like so:

Elo Re-Rating

, where R'(A) is the rating after the matchup, K is a factor that we’ll discuss, and S(A) is the actual result.

How it’s calculated

This can be a chicken-or-egg solution.  I generally started back in 2014 with the winning percentage of each WCHA team and used that to calculate BELOW.  If an average team has a 1500 BELOW rating, then we can use a team’s actual winning percentage to back that out.  It really doesn’t matter too much where you start because BELOW will work itself out over time.  I could have put every team at 1500 and get to a good estimate by the end of the year; I just jump-started things a bit.

From there, every BELOW for every team is calculated for every WCHA game (it has to be a league game, Smokey; this is BELOW, there are rules).

An example

Team A: BELOW 1571, Expected Value of 60%.

Team B: BELOW 1500, Expected Value of 40%.

Constant: Variable based on game outcome; a discussion of this will be saved for discussion of the ABOVE model.

If Team A wins the comparison, they will gain points in their BELOW rating.  Consider getting a 1-goal win:

New BELOW = 1571 + 40 * (1-0.6) = 1587 for a net gain of 16 points.

But if Team B blows Team A out by three goals or more:

New BELOW = 1571 + 60 * (0-0.6) = 1529 for a net loss of 42 points.

What the rating means to you

An average WCHA team will have a BELOW rating hovering near 1500.  If your team’s rating is 1570 or above, you can figure that you have a pretty good chance at home ice in the playoffs; if it’s below 1430, you can simply hope that you’ll squeak into the playoffs.

Last season, the highest BELOW went to Michigan Tech at 1744, an expected winning percentage of 80%.  The lowest BELOW of last season was by Alabama-Huntsville at 1319 (26%).  The 2-35-1 Huntsville team in 2012-13 would’ve finished around 1050; the 21-4-3 Minnesota State team in 2014-15 would’ve finished around 1745.

Why this all matters

There are many reasons:

  1. The WCHA plays an un-balanced schedule, so we don’t have a 1:1 comparison in all matchups.  BELOW helps to provide a little context for that by using recent events to help adjust our estimates of how good individual teams are.
  2. WCHA schedules also are often out-of-whack with games-in-hand; BELOW gives you a sense of relative strength coming down the stretch.
  3. BELOW is predictive as a random variable, so you can run a Monte Carlo simulation (think lots of weighted coin flips) to determine how each matchup will play out.  Then you can extend each of those opportunities all the way to the end of the season and through the playoffs.
  4. Numbers are fun, and BELOW is a nice four-digit number that’s easy to calculate.

A handy table of numbers for a reference

BELOWEWP v. 1500BELOWEWP v. 1500
200095%10005%
195093%10507%
188290%111810%
185088%115012%
182587%117513%
180085%120015%
177583%122517%
175081%125019%
174180%125920%
172579%127521%
169175%130925%
167573%132527%
164770%135330%
162567%137533%
160064%140036%
157160%142940%
155057%145043%
152554%147546%
150050%150050%

I broke this into four columns so it’s a little easier to read.