# A Model for Offensive Rebounding Rates

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A few months ago I took my first look at trying to neutralize rebounding rates. Since that time I’ve given a lot of thought as to what we really want to know about rebounding rates.

In an ideal world we could measure a rating for each player that would allow us to parameterize rebounding rates to determine the percentage of rebounds we would expect each player to obtain under any set of conditions. More importantly than that, we want to know how each player affects his team’s probability of gaining an offensive rebound (even if he doesn’t actually get credit for the rebound).

I certainly don’t live in this ideal world (not yet, at least), so I’m stuck trying to figure out the important things that affect rebounding rates.

**The Model**

Borrowing from the multilevel model for 3pt shooting, I’ve fit similar models for offensive rebounding rates by grouping players based on position. These models will allow us to estimate some situational effects on player rebounding rates.

**The Data**

To fit these models I parsed play-by-play data from the 06-07 to 08-09 seasons to determine how many offensive rebounds a player obtained and missed while on the court. I ignored dead ball team rebounds and rebounds off of missed free throws, and I grouped the data by the following categories:

**Game Location**: home vs away**Shot Location**: low paint (from <= 6 ft), mid-range (all other 2pt shots), 3pt**Shooter**: did this position shoot the ball?**Height Difference**: number of inches taller (shorter) than the player at the counterpart position

I constrained both lineups such that they consist of a single player at each position. In the event of a “tie”, e.g. two PGs on the court, the players were assigned to the position alphabetically by first name. This is hardly ideal, so it’s something that could certainly be improved in future work. The hope is that this simplicity in choosing positioning doesn’t change our conclusions.

**The Fits**

Here are the model fits for each position:

**PG**: logit^{-1}(-3.57 + 0.04(home) – 0.62(lp) – 0.52(mid) + 0.02(hdiff) – 0.66(shooter) + 2.44(lp*shooter) + 1.2(mid*shooter))

**SG**: logit^{-1}(-3.44 + 0.02(home) – 0.5(lp) – 0.39(mid) + 0.03(hdiff) – 0.52(shooter) + 2.72(lp*shooter) + 0.76(mid*shooter))

**SF**: logit^{-1}(-3.03 + 0.03(home) – 0.53(lp) – 0.34(mid) + 0.04(hdiff) – 0.89(shooter) + 3.24(lp*shooter) + 0.83(mid*shooter))

**PF**: logit^{-1}(-2.57 + 0.05(home) – 0.35(lp) – 0.14(mid) + 0.05(hdiff) – 1.44(shooter) + 3.49(lp*shooter) + 0.78(mid*shooter))

**C**: logit^{-1}(-2.45 + 0.05(home) + 0.02(hdiff) – 1.66(shooter) + 3.52(lp*shooter) + 0.71(mid*shooter))

**Interpreting The Fits**

Based on the predictors listed in the fits above, I have rated each player’s offensive rebounding rate (the so-called random effects) when estimated to be at that position by controlling for home court advantage, shot location, height difference, and “shooter?”.

The only coefficient that isn’t statistically significant at the 0.10 level is the coefficient for (home) in the SG fit. It’s plausible home court advantage doesn’t affect a shooting guard’s probability of gaining an offensive rebound, but the coefficient is reasonably close to the other fits and has the sign we would expect. Thus it seems appropriate to leave it in the model for prediction purposes.

**Home court advantage**: We estimate that home court advantage (versus playing on the road) increases the odds of obtaining an offensive rebound by 5% for point guards, 2% for shooting guards, 4% for small forwards, 5% for power forwards, and 6% for centers, controlling for player ability, shot location, height difference, and “shooter?”.**Not the shooter**: We estimate that the odds a point guard obtains the rebound when the shot is taken by another player from the low paint are 0.54 times the odds a point guard obtains the rebound when another player takes a 3pt shot. These estimated odds factors are 0.61 for shooting guards, 0.59 for small forwards, and 0.70 for power forwards. For mid-range shots versus 3pt shots, these estimated odds factors are 0.59, 0.68, 0.71, and 0.87.**Was the shooter**: We estimate that the odds a point guard obtains the rebound when he takes a shot in the low paint are 6.2 times the odds a point guard obtains the rebound when he takes a 3pt shot. These estimated odds factors are 9.2 for shooting guards, 14.9 for small forwards, 22.9 for power forwards, and 33.6 for centers. For mid-range shots versus 3pt shots, these estimated odds factors are 1.97, 1.45, 1.64, 1.91, and 2.04.**Shooting vs not shooting**: We would also like to know what these odds factors are when the player shoots versus does not shoot from these various locations. For shooting from the low paint vs not shooting from the low paint, these estimated odds factors are 5.96 for point guards, 8.96 for shooting guards, 10.44 for small forwards, 7.70 for power forwards, and 6.36 for centers. For shooting from mid-range vs not shooting from mid-range, these estimated factors are 1.72, 1.26, 0.94, 0.52, and 0.34. Shooting from 3pt vs not: 0.52, 0.59, 0.41, 0.24, and 0.19.**Height Difference**: We estimate that each one inch increase in height difference increases the odds of obtaining the rebound by 2% for point guards, 2.9% for shooting guards, 4% for small forwards, 4.6% for power forwards, and 2.4% for centers, controlling for player ability, home court advantage, shot location, and “shooter?”.

**Points to Takeaway**

Some of these are obvious, but here are the quick points to takeaway from the jumble of odds factors listed above:

- Home court advantage increases our expectation on offensive rebounding rates.
- When not taking the shot, we expect the player to have a lower rebounding rate on shots taken in the low paint and mid-range compared to those taken from 3pt range.
- When taking the shot, we expect the player to have a higher rebounding rate on shots taken in the low paint and mid-range compared to those taken from 3pt range.
- When shooting versus not shooting, we expect a player to have a higher rebounding rate on shots taken in the low paint. We expect higher rebounding rates for guards and lower rebounding rates for all other positions on shots taken from mid-range. We expect lower rebounding rates for all players on 3pt shots.
- Height matters.

**Player Ratings for Offensive Rebounding Rates**

The following spreadsheet lists the player ratings for offensive rebounding rates:

Player Ratings: Offensive Rebounding Rates

These ratings are in the form of point estimates and confidence intervals for offensive rebounding rates on shots the player does not take. These shots are grouped by location: low paint, mid-range, and 3pt. These estimates are the same at all locations for centers, as we did not measure shot location effects for shots centers did not take.

These point estimates and confidence intervals also took height difference information into account by estimating the mean height of all players listed at each position and subtracting this mean from the height of each player to arrive at our estimated height difference. Although we want to control for players that may be feasting on the undersized, it seems only fair that we give them their height back when making predictions.

Because of the naive way I estimated positioning, you will see players that don’t seem to belong (such as Hassan Adams with point guards). These players aren’t the norm and tend to have a wide range of uncertainty associated with them. So hopefully they are not having a huge impact on the results for the players we care about.

**Summary**

This model has allowed me to measure the effects of some game situations on offensive rebounding rates, such as shot location and “did the player shoot?”. By using this model we can smooth out the results to give us more realistic estimates than what the 07-08 effective rebounding rates showed.

Major things of interest are controlling for player age, teammate and opponent ability, coaching strategy, and measuring the “other things” that a player may be doing to increase his team’s probability of gaining an offensive rebound. This model assumes none of this matters, so we must think about this sorta stuff when trying to compare player ratings from this model.

The next step I’ll probably take with rebounding is to look at defensive rebounding rates in a similar fashion before trying to come up with a way to model some of the important things mentioned above.

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