Finding a Newer, Better NBA Pythagorean Exponent

Possession-based analysis in the NBA is often some of the best. Dean Oliver, noted as one of the first to use possessions in his stats, created Offensive and Defensive Ratings, which measure the amount of points scored or allowed by a team per 100 possessions.

Similarly, pace factor is an estimate of a team's possessions per 48 minutes. It is also used greatly in normalizing a player's stats, so players on Golden State won't have an advantage over those on the Clippers because they get more possessions per game.
In baseball, the Pythagenpat and Pythagenport try to find an exponent based on a team's run environment (RPG). For basketball's Pyth. formula, I tried to find a floating exponent based on a team's pace factor and total possessions.

Normal exponents for basketball range from 13-17, with 14 usually creating the lowest RMSE. I used databasebasketball.com's downloadable database to look at all teams since the ABA-NBA merger (1976-'77), with the 1998 strike-shortened season excluded, a total of 811 team seasons, to find the exponent that yielded the lowest RMSE.

I came up with two equations to find that exponent, based on total possessions and pace factor. (The formulae for possessions and pace factor are rather lengthy, and can be found here.)

Using possessions, the formula for the floating exponent is sqrt(Poss.) / log(Poss.) x 0.75 - 2.8 . The formula using pace factor is 1.25 x (sqrt(pace) + 1.5) . I basically guessed-and-checked to find the best exponent for both, so I'm sure the formulae can be improved to lower the RMSE's of each.

Both formulae produced a 3.074 RMSE (with the possession formula just a tick better), an upgrade of the 3.102 RMSE generated from a concrete 14 exponent, the best in the 13-17 range. The new formulae form one more win over the 14 exponent per league year (30 teams), if you use average absolute error instead of RMSE. The possession formula is off by 73.4 total wins per 30 teams, while the 14 exponent is off by 74.4 total wins per 30 teams.

(I myself prefer the Pace Factor exponent, since it is easier and shorter than the Possession exponent formula.)

Here's how each team performed in comparison of their real wins and their expected wins based on the Pyth. formula.

Expected Wins vs. Real Wins

Team	Exp.Wins	Wins	Diff
TOR	49.2	41	8.2
UTA	58.9	54	4.9
ORL	55.8	52	3.8
MIA	18.2	15	3.2
DAL	53.7	51	2.7
LAL	59.7	57	2.7
MEM	24.6	22	2.6
PHI	42.2	40	2.2
DET	61.2	59	2.2
IND	37.1	36	1.1
BOS	67.0	66	1.0
DEN	50.9	50	0.9
MIN	22.8	22	0.8
NYK	23.3	23	0.3
NOH	55.6	56	-0.4
HOU	54.6	55	-0.4
PHO	54.2	55	-0.8
CHI	32.1	33	-0.9
SAS	54.9	56	-1.1
GSW	46.9	48	-1.1
ATL	35.8	37	-1.2
SEA	18.6	20	-1.4
LAC	21.4	23	-1.6
POR	38.1	41	-2.9
WAS	40.1	43	-2.9
CHR	28.8	32	-3.2
MIL	22.7	26	-3.3
SAC	34.7	38	-3.3
CLE	40.0	45	-5.0
NJN	26.9	34	-7.1

Toronto was the fifth-most underachieving team in all of NBA history last year, going eight games under their expected record. The Nets, on the other end, were the 10th-most overachieving team last year in NBA history.

In a few days I'll try to see how teams' real win-loss record compared with their pace-factor Pyth. win-loss record from the year before (i.e. how a team's 2007 record compared with their 2006 record using the Pyth. formula and the pace factor exponent).