## The Nature of Mathematical Models

Many people think of Newtonian physics as an outdated model. After all, how can a 350 year old theory built upon an apple falling on someone’s head be relevant to today? In fact, the concepts that Newton laid out in his Principia in 1687 remain remarkably robust in explaining most of the physical phenomena that we experience in our day-to-day lives. It’s only at the extremes of macro and micro scale that classical mechanics fail to explain observations from the natural world.

While not quite at the “Newtonian” level, the Wins Produced model has proven to be a very robust model for understanding NBA player performance and in predicting team performance. Wins Produced has many other positive virtues that I look for in determining if a mathematical model is useful and, most importantly, actionable. In fact, I’ve outlined these values in a previous post.

Nevertheless, as we are wont to do in the blogosphere, the hyper-critical among us will focus entirely on the inevitable limitations and flaws of a model, no matter how useful and reliable that model may be.   In many ways, this is a good thing. It helps us to constantly rethink and refine the model, improving both its descriptive and predictive power. Still, if we fall into the trap of focusing exclusively on the flaws, we incur the proverbial risk of throwing out the baby with the bath water.

## The Nature of the Wins Produced Model

Like classical mechanics, Wins Produced begins to show its limits at the extremes. When a 7-footer like Dirk Nowitzki plays a game that is closer to that of a swing forward than that of a power forward/center, many have (legitimately) argued that the Wins Produced model undervalues him by assessing him exclusively in light of his listed position of PF.

But don’t get me wrong; there is some logic to the current approach. If a coach plays Nowitzki at PF, it’s unlikely that he is using the other forward slot for another PF-type player. This leaves the rebounding and shot-blocking production we’d expect from a power forward to a player who is actually not playing that role. The counter-argument to this suggests that blame should be put on the coach, not the player. After all, we shouldn’t penalize Nate Robinson if Coach Thibodeau decides to slot him in at center on the scorecard.

It is also important to emphasize that, on the whole, the WP model focuses more on the “what” than the “how”. Until now, we’ve been happy to say “we don’t care whose fault it is, the player is under-rebounding.” Whether the player or the coach gets the blame has been treated as secondary.

However, as it turns out, the position adjustment in the WP model is just about the last and the most superficial of the steps in the model. It comes after all of the real analysis that translates the box score stats into raw wins produced. Therefore, it may be reasonable to suggest that the position adjustment could use some tweaking around the edges.

## Disclaimer

My purpose in this post is not to introduce a revised methodology for the position adjustment. Rather, my goal is to start the thinking process for what a revision might look like. I’ll leave it to Prof. Berri and WoW’s Chief Analyst, Arturo Galletti, to figure out how exactly to incorporate this into their latest revisions of the model (coming soon!). Conveniently, this also shields me from the inevitable hyper-specific criticisms that usually come along with a post like this. I only intend to outline the thinking and to lay out an example for how this might be approached. If you’ve got a different idea on the specifics, heck, it might be better than mine. Comment away!

## Limitations of the Listed Position

As I mentioned before, a player’s listed position in a game may not always be reflective of the role that he was actually expected to play. If a coach wants to play “small ball”, there is still one player who is listed at center. Clearly this player isn’t playing center and, indeed, his team may be suffering as a result of the roles being played and the problem with the small-ball strategy. However, the player may still be playing his assigned role. It’s up to the coach to ensure that a lineup is sufficiently balanced to cover all of the roles that are necessary to win the game.

Let’s go back to our Dirk example. Nowitzki may be listed at center or power forward, but his role on the team seems to better approximate a small forward. He takes a lot of threes, distributes the ball a bit more than is typical for a big and pulls in fewer offensive rebounds (probably because he spends more time further away from the basket).

Now, instead of taking the listed position as gospel, what if we were to calculate his position based on his style of play (as approximated by his box score stats)? One would imagine that Nowitzki’s position would change under this scenario. In fact, this is indeed the case — not only for Dirk Nowitzki, but for some other prominent players as well.

## Calculated Position

As an initial approximation, I treated position as a continuous variable going from 1-5 (actually, as you’ll see later on, I let it stretch down to 0.5 for the truest PGs). Using 2011 data, I regressed a bunch of the box score statistics against the traditional position listings. As it turns out, the linear approximation seems to be quite strong for many of the statistics associated with big players:

An approximation using statistics typical of smaller players is, while less obviously linear, still statistically significant:

After throwing all of these together using a stepwise regression, I got a first approximation of how one might calculate a player’s position, rather than just taking the official listing as a given. Now, I’ll save the purists some time. Yes, it would have been better had I looked at ratios of “big play” to “small play” and yes, this would be enhanced looking at detailed shooting charts with distance to the basket. I leave those corrections to Arturo and you can trust that he’s already on the job. Nevertheless, here’s the output from this first approximation:

Using this model as a basis for a new “Calculated Position”, I looked to see which players would move the most from their officially-listed positions. As you can see below, Nowitzki isn’t the only player to “change positions” in this model. LeBron and D-Wade effectively switch positions, with LeBron looking more like a shooting guard while D-Wade looks more like a small forward. Meanwhile, Andrea Bargnani gets slight redemption for not being anything close to a center, while Landry Fields is held to a higher standard as a small forward rather than a shooting guard:

After calculating new positions for each player, we can now consider a new position-adjustment for the Wins Produced stat. To start, we look at how the players’ raw performance (AdjP48) is distributed among the new calculated positions.

A 2-degree polynomial approximation shows PGs at one peak and Cs at the other peak. The nadir of the curve is at an AdjP48 of approximately 0.24:

I am of the opinion that the curve should remain flat after reaching its nadir. After all, why should we punish point guards for being so good at their jobs? The answer that many will offer is based on the belief that PGs can only be played one at a time. Therefore, there is a strong opportunity cost of playing a PG since he is the only PG that the coach can put on the floor at that time. I would argue that 2013 presented a direct challenge to this assumption. When they played together, Calderon and Lowry as well as Felton and Kidd were productive pairs and their teams played very well as a result. While time with the ball is certainly a limited commodity (unfortunately, as it is currently calculated, the Usage stat is not a very useful control for this), I contend that it is a less limited commodity than space in the paint. Still, I’m open to alternative approaches here.

We can get something close to the minimum that we saw above through an alternative approach. I divided the players into 3 groups; PGs (blue), Swings (red) and Bigs (brown), and created a linear trend line for each grouping. In this case, the minimum emerged as the intersection of the slopes for the swing players and the slope for the bigs:

In the end, for this first approximation of an alternative position adjustment, I used the minimum AdjP48 line (0.24), with a linear increase in the adjustment based on the calculated position that follows the slope of the brown line.

## Changes at the Edges

The net change resulting from switching from a calculated position to the new position adjustment is rather small. However, there are some (modest) systemic changes. The new adjustment favors PGs because it uses the same adjustment for them as the other smalls while penalizing SGs by holding them to the same higher standard. The slope for the big players is also steeper than that used in the current WP model, so centers don’t look quite as good.

Looking at the net effect of the two changes, LeBron and Dirk again stand out in terms of positive change while Kris Humphries, Tyson Chandler, Landry Fields and Serge Ibaka are among those most diminished by the changes. Overall, the major changes in WP48 that emerge seem to be for the players who effectively change positions using the new approach:

## Net, Net

In total, I think anyone looking at the top 2011 players in each position would not be surprised by who is near the top of each list:

As I stated at the beginning: when you have a model that is already pretty darn good, the tweaks are at the edges.

– Ari