Factors of Efficiency

[CamdenHu]

The Four Factors have long been established as the factors behind both offensive and defensive efficiency, but I have yet to see any study on their relative importance. (maybe it's just me) So I ran some regression of several factors with efficiency for teams from 1996-1997 to 2012-2013 (both offensive and defensive efficiencies) and try to determine their significance. There are a total of 502 teams over this period so n = 1004.

My efficiency will differ slightly from the traditional ones seen as I use (FGA+0.44FTA+TO-OREB%*Misses) as possessions to try to encompass team offensive rebounds.
Misses = FGA-FGM +0.56 (FTA-FTM). 0.56 might be slight overestimation due to technical free throws, but I imagine the actual OREB% (with team rebounds considered) would be higher than the OREB% calculated from OREB/(OREB+oppDREB).
By the way, shouldn't the official box score start to include team rebounds? Now, the team total row is just the sum of individual player stats. It would make a lot of sense to include team rebounds there (minus the team offensive rebounds following a missed free throw on the front end. Those should not even be team offensive rebounds). The same goes for team turnovers like 24 second shot clock violation.

Going back to the results of regression, I use TS%, OREB% and TO% as the factors because the traditional four factors ignore FT% which make the results less accurate. While adding FT% to the four factors solves the issue, I was hoping for one that is simple to understand. The result of the regression: Efficiency = 51*OREB%+174*TS%-138*TO%+17. The coefficients are rounded to the nearest integer. The R square is 0.9982. The standard error is 0.158. I also ran another regression with constant set as 0. The result:
Efficiency = 59*OREB%+198*TS%-121TO%. The R square is even higher at 0.9999 but the standard error goes up to 0.519. (I am not sure which regression makes more sense in the context, but I will be sticking with the traditional regression where the constant is 17).

Although all three factors are on the same scale, they don't have the same range. In the 2012-2013 season, TS% ranges from 50.7% to 58.8% on offense (similar on defense), TO% ranges from 11.7% to 14.9% on offense (similar on defense), OREB% ranges from 20.1% to 31.4% on offense (24.5% to 29.0% on defense). From here, we can deduce that TS% is about three times as important as OREB% on offense, given their coefficients and similar range. However, on defense, TS% is about six times as important as OREB%, because the best defensive rebounding teams are not that much better than the worst defensive rebounding teams. TS% is also about 3 times as important as TO%, given that the range of TS% is about three times that of TO% and that their coefficients are not that far apart.

This has confirmed my belief that shooting the ball effectively is much more important than have more opportunities to shoot the ball. Now we also know how much more important they are relative to one another. One particular finding I want to point out is how insignificant defensive rebounding is to the team's defense. From what I gather, only slightly more than 10% of a team's defense depends on its defensive rebounding, while the majority rests in its ability to create turnovers and shot defense.

[Crow]

Haven't really read and digested your post yet but you might be interested in this old post:

viewtopic.php?f=2&t=100&hilit=4+factors

[Crow]

"I use TS%, OREB% and TO% as the factors because the traditional four factors ignore FT% which make the results less accurate."

Actually the original factor FT/FGA does incorporate FT% because "FT" in this formula = FT made = FGA * FT%.

Sometimes folks see FGA in FT/FGA and think that the FT part means FTA but it doesn't. It is literally FT made.

[CamdenHu]

Free Throw Rate itself means FTA/FGA. I think that is the one generally accpected in the Four Factors. (Hoopdata for example)
There are some who use FT/FGA, but I think that would be combining two skills into one stat, the ability to get to foul line and the ability to knock them down.