APBRmetrics

**jsill** · Joined: 19 Aug 2009 Posts: 73

I now have standard errors for most of the regularized adjusted plus minus numbers available on my site. Since I'm using regularization and interpreting things in a Bayesian way, strictly speaking these numbers are the posterior standard deviations of the estimated paramaters given the prior distribution and the data. In practice, though, you can think of them in pretty much the same terms as you think of the standard errors presented with standard APM.

This is not the most thrilling stuff, of course, but a few different people have asked me for it and it can be an important input to the decision-making process.

I don't yet have standard errors for the "explained" overall offense and defense RAPM numbers that I got by regressing the original offense and defense numbers vs. the four factors. I have to think about how to do that.

At first, it may seem odd that the original offense and defense RAPM standard errors are the same, but this is a consequence of me using the indirect method which Eli Witus described for getting the offense and defense numbers, where offense and defense are the sum and difference of the overall number and a second number he called DiffOD which measures the player's contribution to points being scored by both teams. There's a symmetry between the offense and defense formulas which leads to the standard errors being the same, the way I calculated them.

deepak · Joined: 26 Apr 2006 Posts: 664

Ryan J. Parker · Joined: 23 Mar 2007 Posts: 708 Location: Raleigh, NC

It's actually more dire than you originally anticipated, since the standard error is actually sqrt(1.484^2 + 1.373^2) = 2.02 instead of 2.857. This gives a p-value of 0.03 when performing a one-tailed test for Farmar having a lower mean DRAPM than Kobe.

Are you asking if it is crazy to say with statistical significance that Farmar has a lower DRAPM than Kobe? Under the specification of this model, apparently not.
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deepak · Joined: 26 Apr 2006 Posts: 664

Ryan J. Parker · Joined: 23 Mar 2007 Posts: 708 Location: Raleigh, NC

I wouldn't exactly say that Farmer played better defense than Bryant, I'd rather prefer to say that his mean 3-year equally weighted DRAPM is lower than Bryant's. (These are the 3-year equally weighted numbers, correct?)

It would not be correct to treat these as being independent. I imagine that Joe could construct a covariance matrix for this model that you could then use to measure the uncertainty using simulation. This would allow you to construct a 95% confidence interval for the difference in the players' mean DRAPM.
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I am a basketball geek.

**jsill** · Joined: 19 Aug 2009 Posts: 73

Actually, if you wanted to answer a question like "what is the probability that Farmar truly rates better than Kobe on defense", the proper way to do it is more complicated. The standard errors (or posterior standard deviations) are not independent. You can actually get a whole covariance matrix for the standard errors, so to answer that question you would want to take into account the covariance of the standard error for Kobe and Farmar. I'll have to think about exactly how to do that. I haven't checked the numbers yet, but I would guess that the standard errors are more correlated when you're looking at players on the same team, such as this case, so it would be more important to take the correlation into account here than when comparing guys on different teams.

Certainly, Kobe coming out as a below-average defender is one of the more counterintuitive results I got. Let me emphasize that I don't consider my results to be anything close to gospel- they are not even close to the final word on a player. All I would claim is that adding regularization represents a significant improvement over standard APM. My results do have Kobe as one of the league's best players overall, but according to my stuff it's mostly because of his tremendous offensive impact, where he simultaneously does a great job of helping his team make shots and a great job of reducing team turnovers.

I do think Farmar is generally considered to be a strong defender, so his defensive rating doesn't seem too crazy to me.

**jsill** · Joined: 19 Aug 2009 Posts: 73

I wrote my previous reply before the last 2 comments from deepak_e and Ryan. They were obviously already on the right track regarding correlation and covariance matrices...

deepak · Joined: 26 Apr 2006 Posts: 664

**jsill** · Joined: 19 Aug 2009 Posts: 73

I don't want to overstate my case regarding Farmar- it's not as if he's universally considered a stellar defender- but if you read Hollinger's ESPN profile (for instance) you'll see he's described as generally pretty good on defense.

TheUntouchable · Joined: 30 Jul 2009 Posts: 12

I wonder how much Kobe's defensive numbers improve if you take out the 2006-2007 season. From what I remember, there was a general notion that he was sacrificing defense that year to preserve energy for offense. He played 40.8 mpg and had a 33.6 USG% that year.

I know that taking out the third year raises the variance and kinda defeats the purpose of what you wanted to do, but it might be more fair to him in this case. There's almost definitely some middle ground between him being better than your numbers indicate and him being worse than he gets credit for.

deepak · Joined: 26 Apr 2006 Posts: 664

Caron Butler rates as one of the worst defenders in the league by this adjusted +/-.

kjb, if you're reading, would you mind providing some additional insight into Butler's defense? Is it as bad as these numbers suggest?

Mike G · Posted: Sun Dec 06, 2009 6:52 am Post subject:

Earlier this year, kjb wrote:

deepak · Joined: 26 Apr 2006 Posts: 664

themojojedi · Joined: 03 Dec 2009 Posts: 4

jsill, if you were to estimate the model using MCMC it would be straightforward to estimate the posterior distribution of the difference in DRAPM between Kobe and Farmar (or any pair of players) by storing/calculating for each iterate in the sample.