Introducing Shot Difficulty: Comparing Game-Winning Shots in the Playoffs

May 10, 2015; Chicago, IL, USA; Cavaliers Cavaliers forward LeBron James (23) celebrates with teammates after scoring the game winning basket in the second half of game four of the second round of the NBA Playoffs against the Chicago Bulls at the United Center. The Cavaliers won 86-84. Mandatory Credit: Dennis Wierzbicki-USA TODAY Sports

The NBA playoffs have been filled with fantastic shots and this past weekend we were treated to three separate game winners. In fact, we’ve already had more game winning shots in this postseason than we did in the entire postseason last year. Each game winner was a low percentage shot with an absurd level of difficulty, but which was the most difficult shot? To determine this, I developed a metric that measures Shot Difficulty by running a logistic regression where the dependent variable is if the shot was made or missed. If you’re into the numbers, you can see the results here.

As you can see, I included many[1. A more parsimonious model is not necessarily always better.] different variables[2. Shot Distance, Closest Defender distance which was capped at 10, number of dribbles, Shot Clock, Location- whether the game was at Home or on the Road, Height Differential- the height of the shooter minus the height of the defender and finally, Game Time- which is the time in the game the shot occurred. So for example, a shot occurring in the 2nd quarter with 6 minutes left in the quarter would have a Game Time of 18.]—shot distance, defender distance, number of dribbles, time left in the game, time left on the shot clock, home or away and the differential in height between the shooter and closest defender. Each of those variables was determined to be statistically significant.[3. The diagnostics for logistic regression are not simple but I tested the significance of the model using the likelihood ratio test and the model was statistically significant. Additionally, all of the predictor variables were significant. Using the stepwise selection procedure, all of the variables were retained except for Assists, which did not improve the model. Additionally, Touch Time was removed from the model due to it’s 0.9 correlation with the # of dribbles. And finally, the area under the ROC curve was 0.635, which indicates that the model is better than predicting at random (where the area under the ROC curve would be 0.5).]

One variable that was significant but was removed was Shot Number. The theory for it’s inclusion would be to attempt to capture the hot hand effect[5. If a shooter is “hot”, theoretically shots become less difficult for that player.] but while Shot Number is significant, it was more likely capturing the effect of better shooters. For example, when the Shot Number is 30, it’s more likely that the player is a high usage player such as LeBron versus a James Jones, who will basically never have 30 shot attempts in a game. So the Shot Number variable was removed.

As a final note, Shot Difficulty will approximate FG%. So the higher the Shot Difficulty number is, the more likely the shot is to go in. So lower numbers will be more difficult shots.

So which of the game winners was the most difficult shot? Let’s look at each one:

So Bayless’ Shot Difficulty[7. Both Shot # and Touch Time are included in the table above but neither are included in the Shot Difficulty model.] was 0.496[8. Meaning he could expect to hit that same shot about 49.6% of the time], which is above average[9. The average Shot Difficulty is 0.449, which was also the average FG% during the regular season] but given it’s closeness to the basket, it’s surprising that it’s not an easier shot according to the metric. It certainly looks easier. But a few things factor in—one, it’s at both the end of the game and with the Shot Clock turned off. These shots are typically harder as defense gets tighter in the clutch and towards the end of the shot clock. Additionally, the closest defender—Rose—is just 1.6 feet away and while Rose didn’t get the greatest contest, he did pressure the shot so that Bayless had to go higher off the glass at an awkward angle. Still, this was by far the easiest game winner and the only shot that had an above average Shot Difficulty.

Chris Paul’s game winner had a Shot Difficulty of 0.397 and when watching the shot, it’s not hard to see it as an extremely difficult shot. As you can see in the table above, if we run the numbers for the same shot but with zero dribbles instead of six, the shot becomes a bit easier by about two percentage points. This isn’t surprising as it’s easier to shoot off the catch. However what made Paul’s shot really difficult was the fact that Tim Duncan was the defender. If we remove the effect of Height Difference entirely[10. In my initial version of the model, I did not include height difference and Paul’s shot was one of the primary reasons for wanting to add some sort of height variable. Credit goes to Seth for suggesting height differential.], we see that his shot is about five percentage points easier and essentially has an average shot difficulty.

Unfortunately, I think if one shot does illustrate the weakness of the model[11. Or rather, the weakness of the input variables.], it would be this one. While adding height differential helps, I’m still not convinced that this shot is only five percentage points tougher than your league average shot. Part of the reason it’s not harder is that the shot only occurred 9.7 feet away from the basket with the closest defender—Duncan—4 feet away. However, when you watch the shot, it looks like Duncan is a lot closer. Part of the reason for this is that he’s really tall while Paul is short but also because Paul is fading backwards. And the degree of difficulty is aided by where on the court the shot is taken—it’s at a tough angle where Paul has to bank[12. Another potential variable to add if the data becomes easily available.] it in[13. A way to improve the model would be to use the x, y location as opposed to the shot distance. I did run a model using these variables but it was scraped because of the difficulty of obtaining x, y coordinates immediately after a game. Additionally, merging the SportVu data with the play by play data is no easy task. But in the future, I may eventually add in x, y coordinates instead of shot distance. And of course even better would be to add the x, y coordinates of the defender as well plus if there was a secondary defender.]. So while I am generally happy with the results of the model, it’s not perfect and I do believe this shot was probably more difficult than the model indicates. But regardless, this was the second easiest game winner.

https://www.youtube.com/watch?v=Ht1y9T14EPE

Rose’s shot was the farthest away from the basket and he also hit it over the much taller Tristan Thompson, which made his shot easily the most difficult shot of the game winners. Of course, you’ll also notice something is off with that table after watching the game winner above. In the table, Rose is taking 0 dribbles before his game winner but when you watch the play, you can see he takes two dribbles. Unfortunately, there seems to be a small error in the data I pulled from the SportVU’s shot logs where it lists Rose taking zero dribbles. We can manually correct this and input two dribbles instead but the Shot Difficulty doesn’t drop much at all- 0.186 now. So Rose dribbling a few times didn’t make the shot that much harder since it was already a very difficult shot to begin with. If we remove the height effect so that the height differential is just zero? The Shot Difficulty goes up to 0.203, which would still be the most difficult of the game winners. The primary reason for this is because Rose’s shot came by far the from the farthest distance away from the basket—over three feet more than LeBron’s game winner.

Speaking of LeBron’s game winner, how difficult was that shot?

https://www.youtube.com/watch?v=-QWeQB5tEMs

LeBron’s game winner was pretty difficult too, just not as tough as Rose’s because of the shorter distance. However, the model may slightly underrate the difficulty of the shot because of the type of contest Jimmy Butler got on the shot which forced LeBron to have to fade backwards a bit. LeBron does get a slight boost (or penalty depending on your point of view) in the difficulty of the shot because his game winner occurred on the road. What would the Shot Difficulty look like if it had occurred at home instead? 0.269 so about a 0.7 percentage point difference. So as you can see, while playing at home does give you a boost in the model, it’s a very, very small boost.

https://www.youtube.com/watch?v=Hel7lbHRv7A

The Truth’s game winner was basically as tough as LeBron’s game winner[14. Very very slightly easier. And if we remove the home-road effect, Pierce’s shot actually becomes very very slightly tougher.]. Why is that the case? Pierce’s shot comes from a shorter distance but he’s also more tightly guarded on his shot.

In fact, there are two defenders who get a contest on Pierce and so the closest defender distance of 2.9 might actually underrate how tightly contested the shot was. In addition, because Pierce is shooting over the shorter Dennis Schroder, he actually gets penalized for shooting over a smaller defender. If we remove the effect of height[15. By setting height differential to zero], Pierce’s Shot Difficulty is 0.246, making it more difficult than Lebron’s shot. This could be one issue with the metric—should someone like Pierce be penalized for shooting over a smaller defender? On the one hand, it is easier to make the shot if the defender who is contesting the shot is smaller but on the other hand, if the smaller defender is up in your face and does contest the shot versus a larger defender who does not but remains the same distance away from the shooter, should that penalty really be there? Of course, this is ultimately the problem with the metric—the ability to get your hand up and contest is not factored in. But with regards to Pierce’s shot, I do think the Shot Difficulty shown above is a fairly good representation of it’s difficulty. And I think the difficulty of Pierce’s shot versus LeBron’s shot is fairly similar.

Let’s look at one last shot. This one did not win the game but it did tie the game and ultimately led to a win in OT.

How can we leave out the MVP? Curry’s shot is fairly similar to LeBron’s. Both shots were on the road with the closest defender distance about the same. Although, one difference is that Curry probably did get fouled. But in terms of difficulty, they were fairly similar. Curry’s shot was slightly closer but he also attempted his shot over a taller defender—Tyreke Evans—who is three inches taller than Curry.

So which shot ended up being the most difficult? As mentioned earlier, Derrick Rose was the ultimate winner.

Finally, I’d like to mention a few thoughts on the Shot Difficulty metric I developed. There are many potential applications where we can use this—whether it is to look at game winners or to potentially look at which players or teams are attempting the hardest shots. We can even use this metric to look at more SportVU On-Off data.[16. Stay tuned, I’ll have something on this soon.] But more importantly, this could be a precursor to a potential Defender Adjusted Shot Difficulty metric. And as SportVU continues to release more data, we can hopefully continue to tinker with and improve the metric[17. The biggest potential improvement would be the addition of x, y location data for multiple defenders.]. At the moment, there is still quite a bit that is not factored in which I think will help improve the accuracy- such as whether the shot was contested or not, how many defenders are “bothering” the shot, etc. However, I think what we do have is a pretty good approximation of Shot Difficulty.