New Player Metric: 2.5-Year Adjusted Four Factor +/- (A4PM)

Blake meet Ekpe.

There are four factors of an offense or defense that define its efficiency: shooting percentage, turnover rate, offensive rebounding percentage, and getting to the foul line. Striving to control those factors leads to a more successful team. (Dean Oliver, “Basketball on Paper”)

A while back I did some work regressing the four factors (FF or 4F) on point differential at the *team* level (Part I and Part II). The result was the following equation:

$$ p.d. = 10.41 + 1.49eFG(own) – 1.63eFG(opp) + 0.187FTR(own) – 0.213 FTR(opp) -1.51TOR(own)+ 1.37TOR(opp) + 0.327ORR(own) -0.365ORR(opp) $$


  • effective FG% (eFG): $$ eFG=(FG+0.5 *3PT)/FGA$$
  • foul rate (FTR): $$ FTR = FTA/FGA$$
  • turnover rate (TOR): $$ TOR=TOV / (FGA + 0.44 * FTA + TOV)$$
  • offensive rebounding rate (ORR): $$ ORR=ORB / (ORB + Opp DRB)$$

Since the time I wrote that post, I’ve thought it would be useful to translate the team level FF relationship to point differential (and winning) down to the player level. The way to do this (or, at least, one way) is to calculate adjusted versions of each of the four factors (i.e. APM-style), and then regress those adjusted factors onto player-level APM or RAPM. (It should be noted that Joe Sill calculated adjusted FF a few years ago, but those data and the articles have been taken down since he started working in the NBA.) Here, I’m using data from 2009-2010 through last Thursday’s games to calculate my own version of RAPM and adjusted four factors for each player.

To cut to the chase (the FF will be dumped at the end of the post), here is the player-level equation for adjusted FF +/- (which I call “A4PM”):

$$ A4PM = 1.77eFG(own) – 1.60eFG(opp) + 0.14FTR(own) – 0.11FTR(opp)-1.37TOR(own)+ 0.80TOR(opp) + 0.31ORR(own) -0.41ORR(opp) $$

$$ R^2=0.88$$

Undoubtedly, the most interesting finding here is the difference in the coefficient on the offensive (-1.37) and defensive (+0.80) turnover rates (OTOR and DTOR). (DTOR is really better called a take-away rate, of course.) According to this, an (adjusted) offensive turnover lowers player value more than a defensive take-away increases it. This result is not immediately intuitive, but it suggests that there is additional offensive value represented by the ability of a player to handle the ball well.


Go to Google Spreadsheet
PID is the unique basketball-value player ID. A4PM, O4PM, and D4PM are the total, offensive, and defensive adjusted four factor +/- ratings. RAPM is my 2.5-year version of ridge-regressed APM.

Adjusted Four Factors

On offense, positive values are better in each case, except for OTOR. On defense, negative values are better, except for DTOR.
Go to Google Spreadsheet

FF Contributors

O1..D4 are the adjusted four factor components multiplied by their respective coefficients from the A4PM regression equation. This enables one to see how much each factor actually contributes to the A4PM ratings for each player. (In each case better ratings are more positive.)
Go to Google Spreadsheet

There’s much more to do with this metric. One thing, of course, is to figure out whether it’s more or less predictive than straight-up RAPM. I haven’t done that yet, so until then, I certainly wouldn’t recommend using this as a replacement for other forms of APM or RAPM. Notably, until proven otherwise, I believe Jeremias Engelmann’s RAPM ratings are the gold standard right now. For now, what A4PM really gives is a neat way to see (at least, roughly) how each player actually gets their rating from each of the four factors. Also, in the same way that you can add up APM or RAPM for 5 players to predict the +/- for a unit, you can use the adjusted four factors to predict how a particular unit will rebound or shoot or turnover the ball, and so on. Obviously, another use for these data is to sort by each of the factors and figure out which players do what best (or worst) in the NBA right now. Enjoy!


Daniel (@DSMok1) made a cool Tableau visualization for A4PM:

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