Posted by Neil Paine on November 30, 2010
This post is a follow-up to this morning's piece about LeBron James & Dwyane Wade's current slumps, so should probably read that one first, if you haven't yet.
In response to the hand-wringing about Wade & James' sub-standard production thus far, some have suggested it's merely a pair of slumps that just happened to coincide with the duo joining forces in South Beach. How legitimate is this theory? Well, thanks to the magic of Monte Carlo simulation, I can test exactly how likely that explanation actually is.
Specifically, I'm going to simulate 10,000 18-game samples based on the career distribution of James & Wade's Hollinger Game Scores. (Yes, there are countless other, better metrics, but hey, this is a quick-n-dirty study.)
How do you do this? First, you start out with Hollinger's Game Score formula:
Calculate that for every game of James/Wade's careers. Then examine a histogram of the game-by-game distributions of Game Score, making sure the data is approximately normal (it was, for both players). Find the per-game average and standard deviation of each player's career Game Scores through the 2010 season:
Also, find the players' probabilities of playing in any given team game through 2010:
|Total Tm Games:||574||574|
Finally, find the per-team game average of James & Wade's game scores so far in 2011:
Now we have all the tools necessary to run the Monte Carlo simulation. For every game of an 18-game sample, I used a random number generator to determine whether James/Wade played (using the probabilities above), and if so, what their game score was (taking a random number from a normal distribution with the mean and standard deviation listed above). Average those numbers over 18 simulated games, checking if both players were at or below their actual 2011 averages. Then repeat this process ten thousand times, and count how many simulations contained 18-game averages at or below the real-life 2011 marks (this will be the probability that James/Wade's declines have been due to random fluctuations in performance).
The result? Out of 10,000 simulations, only 276 contained 18-game stretches where both James and Wade simultaneously put up average game scores as bad as they've posted so far in real life. This means there's just a 2.76% chance that Wade & James currently have the same inherent "Game Score skills" as their 2004-2010 averages suggest, but have merely gone through two simultaneous slumps.
Instead, it seems far more likely that either one or both players' inherent Game Score skills have declined from the 2004-10 period. This doesn't explain why those skills have declined (certainly, contextual effects like the new scheme and new teammates seem much more probable than any physical drop-off), but it does mean it's extremely unlikely that the 18-game sample we've seen out of Wade and James so far in 2011 represents two players with the same inherent game score ability as in 2004-10 simply having a bad stretch of games.