# Defensive Ratings: Estimation vs. Counting

Defensive rating, a box score calculation, is an estimation procedure that attempts to identify the points per 100 possessions that an NBA player yields in a game. In this calculation, a player’s defensive rating is effectively eighty percent of their team’s defensive rating plus twenty percent of defensive points per scoring possessions when on the court. In terms of equations, this is written as

This requires construction of a Team Defensive Rating, a Defensive Points Per Scoring Possession, and a Stop Percentage. In this article, we take a look at the construction of defensive rating. But more importantly, as it is a box score calculation, we look to see how it compares to truth by using play-by-play data.

## Stop Percentage

The first calculation, stop percentage, attempts to identify the percentage of possessions that result in no points: blocks, steals, defensive rebounds. Since blocks do not necessarily end possessions, there must be some form of estimation to identify the percentage of blocks that result in termination of a possession.

Stops, as defined by Dean Oliver, is a two-part process. The first part is the individual part. This portion attempts to identify stops generated explicitly from the player through their blocks, steals, and defensive rebounds. The second part is the team part. This portion attempts to identify stops generated by the team when the player is on the court.

### Individual Stops

Individual stops is calculated as

Holy cow… That’s a lot…

Note that this is a three part equation for steals, blocks, and defensive rebounds; in that order. Let’s break down this somewhat intimidating equation through each of these three parts.

The first part is steals. If a steal occurs, the possession ends. This is the primary reason the possession has ended.

The second part is blocks. In this case, blocks do not necessarily ends a possession. In fact, a block also may end a possession as a defensive rebound that may or may not be obtained by the player who obtained the block. So how do we break down a block?

The first parentheses deals with rebounding relative to shooting attempts. This can actually be written down in terms of a tree diagram of conditional probabilities.

Conditional probabilities that identify rebounds relative to shooting attempts. Probabilities are placed on the branches as the leaves are the results. Yes… there’s an estimated probability of defensive rebounds on made field goal attempts…

This term looks for only two of the four instances: defensive rebounds when field goals are made and offensive rebounds when field goal attempts are missed. The latter condition identifies possessions that continue after missed field goal attempt. The former term of defensive rebounds on made fields goals should never happen. Right? Wrong. They do happen, but require free throws.

The last term for blocks takes an opponents rebounding percentage and increases it by seven percent. This percentage increase is to correct for team rebounds. Therefore, one minus this corrected offensive rebounding percentage yields a defensive rebounding percentage. Therefore, the blocks calculation identifies the number of blocks that result in either defensive rebounds (second term) or continuation of play that result in made field goals (first term).

The defensive rebound portion identifies rebounds when field goal attempts are missed. We again see the continuation of play with made field goals percentage from the blocks calculation. This time, we find the opposite values, which are missed field goals. Multiplying the missed field goal percentage, relative to continuation and made FG, we obtain the number of field goals terminate in defensive rebounds.

Piecing these together, we have steals, field goals missed and defensively rebounded, blocks that are defensively rebounded, and blocks that eventually lead to baskets.

### Team Stops

Next, we focus on the team contribution of stops when a player is in the game. It is given by the formula

Even more craziness than the individual stops… Some terms look the same as above!

Again there are three terms. The first term focuses on field goal attempts that are not made nor blocked. Due to the inclusion of missed field goals and blocks, we have the same correction with eventually made field goals and defensive rebounds. As we look explicitly at missed field goals, the made field goals inclusion come from possessions that terminate on defensive rebounds despite having a made field goal.

The second term focuses on turnovers that are not generated by steals. These are bad passes out-of-bounds, shot clock violations, traveling, double-dribbling, et cetera. The first and second terms are scaled by minutes played.

Since these are box score calculations, there is an assumption that a uniform distribution of field goal attempts per second is upheld. The scaling by minutes played leverage this uniform distribution.

The third and final term is the percentage of free throws off of fouls that result in zero points. The squared term is two consecutive misses. There is an assumption of two free throws on average as one free-throw possessions are either continuation of possession free throws on made baskets or empty-possession technical fouls. The value of 0.4 is the 15-year old constant of percentage of free throws that are possession ending. This value has since been updated to 0.44 in some cases; or learned to a random value near 0.43 through the use of play-by-play data.

This time instead of scaling by minutes, we scale by fouls.

### Piecing this together…

Adding Individual and Team Stops, we obtain stops for when a player is in the game. This is the most complicated portion of identifying defensive rating. We can then calculate stop percentage for a player. This is given by

This formula calculates the number of stops per possession and scales by the minutes played. Think of the formula as the following: Stops per minutes played for an individual divided by the possessions per minutes played by the team. This yields the estimated stops per possession when a player played.

Recall that possessions is a complex computation that is found in offensive ratings.

## Defensive Points Per Scoring Possessions

Defensive points per scoring possession is as it sounds. We compute the number of points scored and divided it by the number of terminating possessions with points scored. This is also known as chances by other folks (thanks Seth :P).  In this case, we have the estimated scoring possessions given by field goals made plus free throws that result in at least one point. The defensive points per scoring possession is given by

I use the term DPpSP as defensive points per scoring possession because it saves space on the formula graphic.

## Team Defensive Rating

Team defensive rating is simple to compute. In this case, it is merely

This identifies the points given up per 100 possessions.

## Defensive Rating

We are finally able to calculate defensive rating. Recall that formula to start the article? If not, here it is…

If we substitute in our short-hand terms, we obtain the exact same equation:

So we can work in reverse and use the above formulas to compute defensive rating. In order to compute this explicitly, we can simply identify these box score elements:

• Steals
• Blocks
• Defensive Rebounds
• Opponent Offensive Rebounds
• Team Defensive Rebounds
• Opponent Field Goals Attempted
• Team Blocks
• Team Minutes Played
• Opponent Turnovers
• Team Steals
• Minutes Played
• Personal Fouls
• Team Personal Fouls
• Opponent Free Throw Attempts

## Application to a Game: Philadelphia 76ers versus Houston Rockets

Let’s consider the October 30, 2017 game between the Philadelphia 76ers versus the Houston Rockets. This game resulted in a 115-107 victory for the Philadelphia 76ers. The box scores for the game are:

Philadelphia 76ers box score versus the Houston Rockets

Houston Rockets box score versus the Philadelphia 76ers

Let’s look at Joel Embiid’s defensive rating. Here, we will not count anything other than box score statistics. In this case, we have the following:

• 2 Steals
• 1 Block
• 7 Defensive Rebounds
• 10 Opponent Offensive Rebounds
• 41 Team Defensive Rebounds
• 33 Opponent Field Goals Made
• 83 Opponent Field Goals Attempted
• 4 Team Blocks
• 240 Team Minutes Played
• 15 Opponent Turnovers
• 10 Team Steals
• 24 Minutes Played
• 5 Personal Fouls
• 31 Team Personal Fouls
• 38 Opponent Free Throw Attempts
• 28 Opponent Free Throws Made

### Total Possessions and Team Defensive Rating

First, we will use the common possession estimator, Possessions = FGA + 0.44FTA – OREB + TOV. For the Houston Rockets, this is 83 + 0.44*38 – 10 + 15 = 104.72 possessions. Of these 104.72 possessions, a total of 107 points were scored; resulting in 1.0218 points per possession. This gives us a team defensive rating of 102.1772 points per 100 possessions.

### Defensive Points per Scoring Possession

Computing the defensive points per scoring possession, we get DPpSP = 107 / (33 + 0.4*(1 – (1 – 28/38)^2)*38. This results in 2.269478 points per scoring possession.

### Stop Percentage

In the game, Joel Embiid recorded 2 steals, 1 block, and 7 defensive rebounds. This results in 4.465804 stops in the game. Computing the team stops portion of stops, we obtain 3.323866 stops. Combining these we obtain 7.789670 stops.

This results in a stop percentage of

That’s a stop percentage of 74.3857 percent. Note that this does not mean that Embiid stops 74% of possessions. This is contribution scaled by a factor of five and includes team effort. This is the rationale for the 80%-20% split in defensive rating.

### Defensive Rating for Joel Embiid

We can now finally calculate the defensive rating for Embiid. We obtain DRTG = 0.8*TDR + 0.2*100*(1-STOP%)*DPpSP = 0.8*102.1772 + 0.2*100*(1-0.743857)*2.269478 = 81.74176 + 11.626218 = 93.367978.

This indicates that Joel Embiid obtained a 93.37 defensive rating, which is significantly better than the 102.18 team defensive rating. We interpret this as Embiid improves team defense by an estimated total of 9 points per 100 possessions. Explicitly to this game, this indicates that Embiid has saved the 76ers roughly 4.5 points against the Houston Rockets.

## What Really Happened…

Through the use of play-by-play data, we are able to walk through the entire game and see all actions occur. The first order of business is to look at the number of possessions. Through counting all possession termination actions, we obtain a total of 204 possessions. This results in 102 possessions for both the Rockets and 76ers. Recall that the estimated possessions was 104.72 possessions.

Of the 204 total possessions, we find that Embiid participated in 105 total possessions. Of these 105 total possessions, Embiid played on 51 defensive possessions while playing in 54 offensive possessions.

Getting into foul trouble with five fouls, Embiid found himself playing in six spurts throughout the course of the game.

### Spurt One::  12:00 – 8:17 First Quarter.

Embiid started the game, playing the first 18 possessions of the game. During these 18 possessions, Embiid played 9 defensive possessions that resulted in 7 points for Houston. The playing time resulted in 3 minutes and 43 seconds of playing time, an average of 12.39 seconds per possession.

Of the 9 defensive possessions, Embiid recorded one defensive rebound only as the Rockets scored on three of the nine possessions. Robert Covington was the star of this stint, recording a defensive rebound and two steals of these nine possessions.

At the end of this stint, Embiid’s defensive rating is 77.78.

### Spurt Two:: 3:52 First Quarter – 10:25 Second Quarter

Coming back in to close out the first quarter and start the second quarter, Embiid participated in 23 possessions over the 5 minutes and 27 seconds of playing time. This resulted in 11 defensive possessions that resulted in 14 Houston Rockets points. The games slowed pace a little, resulting in 14.21 seconds per possession.

Of the 11 defensive possessions, Embiid recorded 1 defensive rebound and 1 steal; terminating two of Houston’s 11 offensive possessions. Of the 11 possessions, Houston converted on seven of the possessions, two of which were 1-for-2 on free throws. One of the scoring possessions, Embiid committed a foul that resulted in an extra free throw made. This was a particularly weak stint for the Sixers defense as Houston left off a couple points due to free throws and still managed 1.27 points per possession.

At the end of this stint, Embiid’s defensive rating is 105.00.

### Spurt Three:: 3:48 – 1:44 Second Quarter

For one last stint during the first half, Embiid participated in 9 total possessions, 4 of which were defensive possessions. During these four possessions, the Houston Rockets picked up 3 total points on a single three point attempt by James Harden.

Despite the one scoring possession in four attempts, Embiid had little to directly attribute to the stops. Rebounds by Jerryd Bayless, Dario Saric, and Ben Simmons terminated the other three possessions.

The average possession was 13.78 seconds over Embiid’s 2 minutes and 4 seconds of actions. At the end of this stint, Embiid’s defensive rating is 100.00.

### Spurt Four:: 12:00 – 8:15 Third Quarter

Embiid started the second half for his fourth stint, which lasted 3 minutes and 45 seconds. During this time, Embiid played in a total of 14 possessions; 7 of which were on defense. During these seven defensive possessions, Embiid recorded nothing on the defensive end. Up to this point Embiid managed two defensive rebounds and one steal.

Houston managed to score only six points over the seven possessions, converting only three possessions. A fourth possession, Philadelphia was bailed out by two consecutive misses from the foul line by Clint Capela. The other three possessions were terminated by Covington (steal, rebound) and Simmons (rebound).

The average possession lasted 16.07 seconds. At the end of this stint, Embiid’s defensive rating is 96.77.

### Spurt Five:: 2:24 – 0:48 Third Quarter

Embiid entered the game late in the third quarter for a short one minute and 36 seconds for a total of six possessions. With an average possession of 16 seconds per possession, Embiid played in three defensive possessions.

Houston converted on one of the three possessions, again bailing out the 76ers by missing both free throws after an Embiid foul. With the three points coming on another Harden three point attempt, the Rockets only mustered one point per possession during this stint.

At the end of this stint, Embiid’s defensive rating is 97.06.

### Spurt Six:: 8:37 – 0:26 Fourth Quarter

Embiid closed out the game with a significant eight minute and eleven second stretch of time. This stretch witnessed 35 total possessions, which started to speed up, thanks to free throws late. The average possession was 14.03 seconds. Of the 35 possessions, Embiid participated in 17 defensive possessions.

It was during this time that Embiid collected many of his stats. During this stretch Embiid picked up 5 defensive rebounds and one steal. Embiid also picked up his only block of the game. Unfortunately, Houston retained possession as the block went out of bounds.

As Houston only converted on seven possessions, one of which Embiid sent Houston to the line for two free throws. Fortunately for Philadelphia, Houston failed to convert on two free throws, leaving two points on the line. Due to this, Embiid’s defensive rating improved and finished with 47 points over 51 defensive possessions for a defensive rating of 92.16 points per 100 possessions.

Compare this to the team defensive rating of 104.90, we find that Embiid’s presence indicates an improvement in defense; however, his individual stats only seem to appear in his final stint. This implies that Embiid is not the sole reason for the defensive improvement. Seeing the actual statistics from when Embiid is on the court, Robert Covington turns out to be the premier defender. This in turn indicates that the combination of Covington and Embiid identifies a solid defensive tandem for the 76ers.

## Comparison of Truth versus Estimated

As we have seen that the 76ers’ defensive rating is actually 104.90, we had an estimated team defensive rating of 102.17 from the Oliver equations above. That’s not a terrible estimate by any stretch. However, this comes from estimation of possessions.

In turn, Embiid’s defensive rating was estimated to be 93.37 points per 100 possessions. In reality, Embiid managed a 92.16 points per 100 possessions.

This shows that the estimation process varies about the truth, and while it is a method for approximating points per 100 possessions, we are able to compute the actual defensive rating by performing play-by-play calculations.

The reason why the estimation process manages to miss by roughly 1-3 percent is due to many factors. First, possessions are estimated using coefficients that are not proper for the estimation process. Second, possession times are assumed to be uniform. This means that if 104 possessions are estimated for both teams, then every possession is estimated to be 13.85 seconds long. We see that this is not the case. Third, points per possession are assumed to be uniformly distributed over possessions. This again is not the case.

For the possessions issue, we have seen that possessions are grossly over-estimated in the past. For the uniform distribution assumption, if we are able to obtain thousands of possessions per game, then we may have a chance to argue uniformity assumptions. However, in small sample games… yes 100 defensive possessions is a small sample relative to the possible ways to terminate possessions… any deviation from uniformity will violate uniformity. And this explicitly happens here to induce these variations in points per 100 possessions.

Despite these flaws, if the user only manages to have box score data, we see that defensive rating is not egregious in estimation. Instead, it’s a carefully thought out process that leverages assumptions of uniformity to get close to the truth.

## That Said…

If we are to compare two players using defensive rating, we must perform a test of hypotheses. We cannot simply sort the players by defensive rating using Oliver’s defensive rating. This is because we are using an estimation process using the uniform distribution. Therefore, if a player has a defensive rating of 92.65 and another player has a defensive rating of 94.01; can we say that the first player is better? Most likely not.

Instead, we may say that the players are the same, as the uniformity assumptions may lead to large enough variances such that both scores are realistic for both players.

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