Usage Adjusted Rating, as I discussed previously, has Alternate Win Score (AWS) as its base. Alternate Win Score is a simple per minute measure of performance, which has proven to be the best linear weights metric for prediction across high continuity and low continuity contexts. High continuity contexts are situations where a team is the largely the same as it had been when the players compiled the statistics being used to make predictions. Low continuity contexts are the opposite. AWS, as Neil Paine has demonstrated, is the best linear weights metric for prediction when dealing with both of those situations. So how is Alternate Win Score defined?
AWS equals Points+0.7*(Offensive Rebounds)+0.3*(Defensive Rebounds)+Steals+0.5*(Blocks)+0.5*(Assists)-0.7*(FG missed)-(FG made)-0.35*(Free Throws Missed)-0.5*(Free Throws Made)-Turnovers -0.5*(Fouls Committed) all divided by Minutes Played.
I wanted to make some tweaks to this basic formula. Namely, I wanted to include a usage-efficiency tradeoff. As I mentioned in the previous post, APBRmetrics forum poster v-zero provided a way to do that. I incorporated his math into the formula for AWS and after some tweaking, I arrived at UAR.
About that tweaking. Some people have expressed interest in knowing exactly how I arrived at the numbers I came up with. So here goes. I broke AWS into two separate figures. The scoring (and offensive turnover) portion and the Non-Scoring aspect. The Non-Scoring portion of UAR simply is equal to .7*OREB+.3*DREB+Steals+.5*Blocks+.5*Assists-.5*Fouls Committed per pace adjusted 48 minutes.
Then I moved on to the Scoring portion of UAR, which includes turnovers because turnovers use a possession just the same as a shot attempt or free throw attempts, except turnovers obviously always result in 0 points. I calculated the league average for points per possession (PPP), using the simple formula for possessions (FGA+.44*FTA+TOV), and similarly calculated the league average for possessions per 48 minutes (USGper48), again using the simple possession definition. I then used the coefficients v-zero provided to create what I call average ScoreRating, which is simply 5*(PPP)+.076*(USGper48). For this season, thus far, the league average for that number has been roughly 6.2. Next I calculated the Score Rating for every player in the league and subtracted out the league average rate, so that if you’re an average scorer you break-even in Score Rating, if you’re above average you contribute a positive value through your combined scoring volume and efficiency whereas if you’re below average, you detract value from your team through your inability to score. I also had to multiply Score Rating by a coefficient in order to properly value scoring in UAR relative to the NonScoring parts of UAR. The Scoring Rating needed to be worth roughly 2.7 times the Non-Scoring Rating, based on some math resulting from the Four Factor weights discovered by Evan Zamir here. In order to get the scale right, the coefficient turned out to be roughly 2.4. This owed to the league average for Score Rating being 6.2 and the league average for Non-Score Rating being about 5.5. Then I set total league average UAR to 0.
These numbers change year over year but they are pretty consistently in this range. I then added the Scoring and Non-Scoring parts together to get UAR. The equation for this year basically looks like this:
UAR = (2.4*(5*(PPP)+.076(USGper48))+ (7*OREB+.3*DREB+Steals+.5*Blocks+.5*Assists-.5*Fouls Committed per pace adjusted 48 minutes)-((lg avg Score Rating)+(lg avg non-score Rating))
The numbers, as I said, vary year over year depending on what the average numbers league wide are.