Introducing DAVIS: a holistic and transparent defensive stat

Introduction

Defensive stats suck. Basic counting stats like steals or blocks vary with luck and context. To combat this, we have new defensive stats that add context. For example, DWS (defensive win shares) weights a player’s defensive rating relative to league points per possession. DBPM (defensive box plus minus) takes this further, giving an estimate for a player’s points added relative to league average per 100 possessions. These stats present more information than raw stats. But, they are complex and have their own flaws, too.

Andre Drummond led the league in DWS last year. Jokic came in 8th, above players like Marc Gasol and Joel Embiid. Though neither Drummond nor Jokic are poor defenders, others seem more deserving of their spots.

In DBPM, Kawhi ranks 67th. Yes, Kawhi is not the same back-to-back DPOY level defender he was a few years ago. However, it’s hard to argue Kelly Olynyk – who ranks 64th in DBPM – is a better defender than Kawhi. Olynyk ranks above not only Kawhi but also Paul George in DBPM.

To combat this issue, ESPN invented RPM (real plus minus). RPM uses play-by-play data to give players’ ratings adjusting for context. RPM measures how many points per 100 possessions a player would add if he played with 4 other average players. It’s split up into offensive and defensive RPM, so it can rank players defensively. While this still produces funky results, it’s a big improvement. Still, RPM is a sort of “black box.” ESPN calculates it using private data with their private models (a ridge regression). So, RPM is not easy to replicate.

DAVIS (Defensive Average Value below Ideal Stats) uses simple stats in an interpretable way to create a better defensive metric. DAVIS combats the lack of public, easy to understand, and good defensive stats.

The premise

Because DAVIS relies on public and easy to understand stats, we’re still using the problematic stats we described before. Though they’re not perfect when evaluated individually, the stats can provide some insight when combined. For example, a player who leads the league in steals might not be the league’s best defender. However, if a player leads the league in steals, blocks, DWS, DBPM, defensive rating, etc., it would be hard to argue that they’re not among the best defenders in the league.

DAVIS computes the distance from each player’s defensive stats to the league-leader in that stat. We then invert the ratings such that higher is better. So, if a player leads the league in every stat used, they earn a DAVIS of 1, as their distance to the highest stats is 0.

DAVIS uses the following defensive stats:

• Steals
• Blocks
• Steal % (percentage of opponent possessions that end with a steal by the player while he was on the floor)
• Block % (same as steal % but with blocks)
• Defensive field goals attempted (DFGA, field goals attempted where the player was defending)
• DWS (described before)
• DBPM (described before)
• Defensive rating (DRTG, points allowed per 100 possessions while the player is on the floor)

The closer a player is to the league lead in all the above stats (highest possible values for all stats except for DRTG, where lower is better), the better his DAVIS score is.

Though we’re using problematic stats, together we can get a more complete picture. Furthermore, one concern is that there is collinearity between factors. This means one factor predicts another factor. For example, a player with more steals will also have a higher steal percentage. We included both steals and steal percentage to give some sense of minutes-scaling. A player could earn a high steal percentage by playing very little and getting lucky. Having rate stats along with cumulative stats gives us a balance between efficiency and sample size.

Methods

First, we collected the above stats for every player in the 2018-19 NBA season. DFGA and DRTG are from NBA.com/Stats. All the other stats are from Basketball-Reference. We limited the data set to players who played at least 41 games and 10 MPG.

After collecting these stats, we normalized them between 0 and 1. This is because each stat has a different range, so without normalization, we would be weighting some stats more than others. For example, DRTG varies more than steals, so players will have a larger distance from maximum DRTG, giving DRTG a greater weight on DAVIS. Normalization allows us to keep a constant weight across all stats.

Then, we calculated the distance between the stats and the best possible stats. DAVIS uses 3 different distance metrics:

1. Euclidean distance. This takes the square root of the sum of the squared differences between our stats. If we have two lists, p and q, of 3 elements, their Euclidean distance will be the square root of (p_1 – q_1)^2 + (p_2 – q_2)^2 + (p_3 – q_3)^2 where p_n and q_n are the nth elements the vector.
2. Manhattan distance (or cityblock/taxicab distance). This takes the sum of the absolute values of their differences. So, the only difference between this and Euclidean distance is that Euclidean distance squares these differences then takes the square root, giving us some different values. The Manhattan distance of two lists, p and q, of 3 elements will be |p_1 – q_1| + |p_2 – q_2| + |p_3 – q_3|
3. Wasserstein distance (or earthmovers distance). This is not as simple as the other metrics. Wasserstein distance measures the minimum amount of “work” required to transform one vector into another. “Work” here means the amount of the distribution weight we must move, multiplied by the distance it is moved. It’s often called earthmovers distance because many imagine it as moving one pile of dirt into another shape, where each pile of dirt is a vector.

Wasserstein distance is a bit harder to visualize than the other two. However, for the difference between Euclidean and Manhattan distance, we can create a simple example. Let’s say we have 2 points in a 2-dimensional space. For the sake of simplicity, suppose the points are (0, 0) and (1, 1). Now, let’s form a right triangle between them. The Euclidean distance is the hypotenuse (diagonal leg) connecting the two points. So, the Euclidean distance between the two points is sqrt(2). Meanwhile, the Manhattan distance is the sum of the two legs (2).

For higher-dimensional problems like this (8 stats used implies 8 dimensions), it’s harder to visualize. But, we can generalize this right triangle example.

After calculating the distance for each player to the best possible stats, we took the average of the metrics to arrive at the DAVIS score.

Why DAVIS isn’t perfect

While DAVIS presents a new and simple defense metric, it isn’t perfect. Several factors contribute to this:

1. Though DAVIS combines metrics to give a holistic score, we know the metrics composing DAVIS are imperfect. So, while DAVIS can adjust for some weird outliers in a given metric, if all metrics overvalue a player’s defense, DAVIS will still overvalue the player’s defense. This will be clear with a couple of the top guards.
2. DAVIS tends to rate big men highly. This is not much of a problem, as big men do often impact defense more than other positions. However, DAVIS suffers from the same issue as DWS in overvaluing bigs. The top 10 players in DAVIS are all bigs, and 18 of the top 20 are bigs. This is because DWS favors bigs, and DWS is a factor in DAVIS. Furthermore, bigs contest more shots – another factor in DAVIS. Though this is not ideal, another reason it’s not a big problem is that bigs play a different role on defense than guards or wings. So, the fact that DAVIS tilts towards bigs is not ideal, it does not present too big of a problem.
3. DAVIS does not know position. DAVIS compares each player to the best possible values, not the best possible values for the player’s position. This removes some sensitivity in DAVIS, but this trade-off may be worth it. If we calculated DAVIS relative to position, we’d run into a couple problems. First, positions in defense are not well-defined. There’s no clear way to separate it; even Basketball-Reference’s positions aren’t always perfect. Second, separating by position makes it harder to compare similar players who may be classified in different positions. So, while Ben Simmons would be considered a guard, he may play a very similar role to some wings.
4. DAVIS has no way of quantifying the difference between off-ball and on-ball defense. For example, an excellent off-ball defender in a free safety-type role might get lots of steals and blocks. This would lead to a high DAVIS regardless of their on-ball defense. A similar problem goes the other direction, too.

Results

The three graphs below show the top 10 players by DAVIS for guards, wings, and bigs. Note that DAVIS is not separated by position, so players are not compared to others in their position. Players are only compared to the best possible values in the required stats. A player’s position does not impact his DAVIS score. Also note that the positions are taken from Basketball-Reference (not my own classification).

For guards, we see that the top 2 are not what you’d expect. This is because of what we discussed before; imperfect advanced stats contribute DAVIS, so if all metrics overvalue a player, DAVIS will overvalue him too. In this case, Westbrook and Harden are #1 and #2 in DWS among guards. Westbrook is #1 in DBPM among guards. Furthermore, Harden ranked #2 in steals, and Westbrook ranked #5. So, that’s already three stats where Westbrook and Harden seem great. This isn’t to say that Westbrook and Harden are bad defenders – they’re not – but many would rank others #1 and #2.

Among wings, the results are what we’d expect. George ranks #1, as expected given his incredible defense last season that lead him to a third-place finish in DPOY voting. Following George, we have two-way superstars Jimmy Butler and Kawhi Leonard. These results also follow most expectations. Later on, we get more “defensive specialist” types like Mo Harkless and Andre Iguodala.

Among bigs, the results are about what we’d expect with a couple differences. Turner, Gobert, and Davis are often listed as the three best bigs. Drummond and Whiteside seem a bit overrated, but they did place #8 and #7 in blocks, respectively.

To see full results for DAVIS rank for every player who played at least 41 games and 10 MPG, scroll down to the table at the end of the next section.

DAVIS vs. other stats

Now that we’ve defined and calculated DAVIS, let’s compare it to other stats. We’ll look at not just how it correlates to other stats, but also how different DAVIS ranks are from other stat ranks.

First, we’ll compare DAVIS to DWS and DBPM. The two charts below show the correlation between DAVIS and DWS, and the correlation between DAVIS and DBPM.

We see that DAVIS is highly correlated to DWS and DBPM. This is expected given that DAVIS considers DWS, DBPM, and other stats correlated to these metrics. However, we’ll notice that DAVIS has some differences from DWS and DBPM among certain players. So, while they’re correlated, the large corrections in some players make DAVIS better.

Looking at rank-specific differences between DWS and DBPM is meaningless. Basketball-Reference’s DWS and DBPM go to one decimal place. So, our ranking of them isn’t ranking players from 1 to n_players, as there are several ties. Specifically, DWS ranks players from 1 to 49 (Sexton’s -0.5 ranks 49) and DBPM rank players from 1 to 77 (Crawford and Dellavedova tied at 77). Meanwhile, because DAVIS goes into further decimal places, it has no ties.

This tie problem will not happen for DRPM, as DRPM goes into two decimal places. The graph below shows the correlation between DAVIS and DRPM.

We see that DAVIS is also highly correlated with DRPM. This indicates that DAVIS may capture some defensive trends, as DRPM is often viewed as the ideal public defensive stat. The only issue with DRPM is that it’s not easy to replicate given that it requires possession data and a specific model. DAVIS solves this.

The graph below shows the correlation between DAVIS rank and DRPM rank.

There are still some ties in DRPM, making the scale go up to 250 instead of 350. Nevertheless, we can interpret the differences between DAVIS and DRPM.

The players with the top 3 differences between DAVIS and DRPM rank are Bradley Beal, Justin Holiday, and Montrezl Harrell. DAVIS likes these players much more than DRPM. For Beal, though this difference is large, DAVIS doesn’t love him; DRPM just hates him. He ranks 217 in DRPM, compared to 87 in DAVIS.

There are several other notable differences between DAVIS rank and DRPM rank. Klay Thompson ranks 87 spots higher in DAVIS than in DRPM (134 vs. 221), Kawhi Leonard ranks 82 spots higher (39 vs. 121), and Ben Simmons ranks 66 spots higher (31 vs. 97). The common trend here is that DAVIS favors excellent on-ball defenders more than DRPM.

The players with the bottom 3 differences between DAVIS and DRPM rank are Semi Ojeleye, Lance Thomas, and Ryan Broekhoff. DRPM likes these players much more than DAVIS. The common trend here is a lack of sample size. While DRPM tries to regularize possessions to limit the effect of small sample sizes, small sample sizes still sway the rating. Meanwhile, because DAVIS includes both rate-based and cumulative stats, it dislikes these players.

Among players who played a lot, it seems that DRPM favors players on better defensive teams. Though DRPM tries to estimate a player’s effect with 4 other average players, the player’s impact inevitably also depends on coaching and defensive scheme. These will be better among better defensive teams, leading them to have a high DRPM. Furthermore, the player could happen to always be in with another player due to rotations. So, players like Eric Gordon and Patrick Beverley rank lower in DAVIS than in DRPM.

The table below shows the DAVIS ranking for all qualifying players, along with their DRPM, DWS, DBPM, and difference in ranks.

Conclusion

No public defensive stat is perfect. Even when watching games, people debate what a “good” defender is. Naturally, many look at on-ball defense as an initial indicator and enjoy seeing highlights of players like Kawhi ripping apart players with their on-ball defense. However, even this is a flawed approach, as team defense also plays a big role in being a good defender. A famous case of this is Avery Bradley. Many deemed him a great defender due to his superb on-ball defense. Yet, others called him overrated due to his lacking team defense.

As such, even a pure eye test doesn’t give a great picture of defense. So, it’s even harder to create a good defensive stat. Nevertheless, DAVIS gives us a better idea of defense in a public and simple way.