Rating 3pt Statistics with the Colley Matrix Method

Taking into account opponent strength is one area I continually question when studying all forms of NBA statistics, whether it be at the team, 5-player unit, or individual player level. Having the ability to quantify this is something I want to get a handle on, so I’ve spent time studying methods the BCS uses to rank college football teams. More specifically, I have studied the Colley Matrix Method created by Wes Colley.

I feel I have a strong understanding of how the method works, so my first application will be to a team’s 3pt shooting statistics. I’m still studying how to quantify the uncertainty in the method, so while I have a belief as to how to measure this uncertainty I will leave that for a future time until I’ve got a firmer grip on what I believe to be true.

The Method for Rating 3pt Shooting Statistics

To setup the Colley matrix, I create 60 “teams”: one “team” for each team’s offensive 3pt shots attempted, and one team for each team’s defensive 3pt shots faced.

So an offensive team’s “wins” are the number of 3pt shots made, and the number of offensive team’s “losses” are the number of 3pt shots missed. The reverse is true for defensive team’s “wins” and “losses”.

In addition to the Colley matrix, the b vector is created using the win-loss information as outlined above.

Solving for the Ratings

To solve for the ratings (the r vector), one must solve:

r = C^-1 x b

Solving this equation gives you the ratings for each team.

The Results

Below are the offensive and defensive ratings using data from all games of the 2008-2009 season played on or before December 23rd:

Offensive Ratings

Rank	Team	Rating	Rank	Team	Rating	Rank	Team	Rating
1	SAS	0.4681	11	ATL	0.4437	21	MEM	0.4167
2	PHX	0.4664	12	OKC	0.4430	22	MIA	0.4100
3	NOH	0.4627	13	TOR	0.4425	23	UTA	0.4081
4	POR	0.4586	14	ORL	0.4423	24	DAL	0.4014
5	BOS	0.4544	15	CHA	0.4386	25	WAS	0.3994
6	DET	0.4522	16	DEN	0.4349	26	GSW	0.3941
7	LAL	0.4504	17	NYK	0.4334	27	SAC	0.3824
8	HOU	0.4495	18	IND	0.4235	28	LAC	0.3813
9	CHI	0.4456	19	CLE	0.4229	29	MIN	0.3783
10	NJN	0.4450	20	MIL	0.4191	30	PHI	0.3642

Defensive Ratings

Rank	Team	Rating	Rank	Team	Rating	Rank	Team	Rating
1	ATL	0.6070	11	DET	0.5883	21	TOR	0.5611
2	BOS	0.6070	12	PHI	0.5847	22	CHA	0.5603
3	NYK	0.6066	13	UTA	0.5820	23	OKC	0.5588
4	HOU	0.6028	14	LAC	0.5800	24	MIA	0.5538
5	DAL	0.6016	15	LAL	0.5794	25	MEM	0.5462
6	MIL	0.6002	16	SAS	0.5715	26	POR	0.5405
7	CHI	0.6001	17	IND	0.5690	27	MIN	0.5370
8	DEN	0.5939	18	WAS	0.5683	28	GSW	0.5323
9	ORL	0.5935	19	PHX	0.5643	29	SAC	0.5119
10	CLE	0.5891	20	NOH	0.5641	30	NJN	0.5117

Using These Ratings

One noteworthy aspect of the Colley Matrix Method is that the mean rating is 0.5. Thus you can interpret these ratings in terms of “against 0.500 level competition”. This means the log5 method can be used to calculate expectations for any given matchup.

For example, suppose the Boston Celtics play the Golden State Warriors. What % of 3pt shots should we expect the Celtics to make? The Warriors?

Applying the log5 method:

The Celtics Expectation Is

0.4544 x (1-0.5323) / ( 0.4544 x (1-0.5323) + (1-0.4544) x 0.5323) = 0.423 = 42.3%

The Warriors Expectation Is

0.3941 x (1-0.6070) / (0.3941 x (1-0.6070) + (1-0.3941) x 0.6070) = 0.296 = 29.6%

Future Work with the Colley Matrix Method

Applying the Colley Matrix Method to team level 3pt shooting statistics is mainly just to help show how this might be applied to other areas of basketball statistics. I am most interested in applying this to 5-player unit level statistics, with the ideal goal of using those to extract each player’s impact.

Where would you like to see the Colley Matrix Method applied?