Overview and Analysis of FIRST Stats

[araniaraniratul]

Originally Posted by Citrus Dad

We're digging into your stats paper. It's quite useful for thinking about this issue.

However you wrote on p. 9:

Defense - Average of Opponent’s Match Scores
One simple way to measure a team’s defensive power is to simply average their opponent’s match scores.

That's not correct--it measures the average offensive output of the opponents in those matches. This is the error also of the assertion that OPR-CCWM=DRP as a defensive metric. The DPR is actually the OPR of the opposing teams in those matches played. A measure of defense will measure how much that average output differs from the average offense of those opponents in other matches.

(I'll have other thoughts as I go through this paper.)

I felt like the intention was to take the differences from the opposing team's average scores? That would make sense and is fairly obvious. Otherwise, I'm fairly lost there too!

[Citrus Dad]

Originally Posted by araniaraniratul

I felt like the intention was to take the differences from the opposing team's average scores? That would make sense and is fairly obvious. Otherwise, I'm fairly lost there too!

Yes, that's the added step that's needed. As mentioned below, look at the equations on p. 15 and p. 19 to see how to do this.

I also like the MMSE adjustment. However, one important point. This is a Bayesian estimation method. The errors terms around initial "guess" is normally distributed (or log normally since scores are constrained at zero), and presumably so are the regression results errors. But adding together 2 normal distributions does not result in a normal distribution. It's been 20 years since I looked at this issue, but there is software out there that correctly computes the resulting distribution.

[wgardner]

Originally Posted by Citrus Dad

But adding together 2 normal distributions does not result in a normal distribution. It's been 20 years since I looked at this issue, but there is software out there that correctly computes the resulting distribution.

??? A random variable produced by summing two independent normally distributed random variables is also normally distributed. See, for example, this.

[araniaraniratul]

Originally Posted by Citrus Dad

Yes, that's the added step that's needed. As mentioned below, look at the equations on p. 15 and p. 19 to see how to do this.

I also like the MMSE adjustment. However, one important point. This is a Bayesian estimation method. The errors terms around initial "guess" is normally distributed (or log normally since scores are constrained at zero), and presumably so are the regression results errors. But adding together 2 normal distributions does not result in a normal distribution. It's been 20 years since I looked at this issue, but there is software out there that correctly computes the resulting distribution.

Is it because they're not independent? (Unless my understanding of random variables is way off) Otherwise I feel like two independent random variables (normal) in summation still leaves you with a new normal random variable. In terms of software, I know that numpy does a ton of those corrections, but not sure about this specific one.

[wgardner]

On a tangentially related note: I came across "Regularized Adjusted Plus/Minus" stats for NBA basketball players today. They are essentially exactly the stats we're using in FIRST! See for example, this link.

In our terminology, they view each basketball possession as an "alliance" of 5 players versus a defensive "alliance" of the 5 players on the other team. They count each possession as a "match" and then compute the stats just like we do. Every time a new player is subbed in, the alliance changes. "Raw plus/minus" is just like the averages discussed in the paper. "Adjusted plus/minus" is like sOPR and sDPR (compare the first equation in the link above with the equations on page 6 of the paper). "Regularlized adjusted plus/minus" is just like the MMSE version.

This link shows ORPM, DRPM and RPM for NBA players which are essentially just like sOPR, sDPR, and sCPR.