Back in high school, when we had a really hard test all the smart kids would get together and produce a cheat sheet.

Over time, we learned there were a couple of hard and fast rules for doing this:

- It had to be simple and organized
- It had to be small (preferably fitting in the back of your sliding calculator or in the palm of your hand).
- It had to be readable (good bless laser printing)
- It only as good as the data ( take a preemptive bow Andres Alvarez, as the forthcoming post is
).*All Powered by Nerd Numbers*

It became a fun little game to make them better and better and cram more information into each one. Now this kind of thing might sound like cheating to the uninformed but we’re talking about AP math and physics. The formulas helped but really you needed to know how to do the problems (and it was graded on a curve 🙂 ). It was really a handy dandy reference but you had to bring the brains.

Now I’ve done similar things before in this space for many of the reader questions (and for my own useI’m pretty sure I hit the Basics more than basically anyone for reference) .For this one, I’m taking on one of the big ones : how do changes in the Box score stats affect a Teams Win Produced? Now keep in mind I just showed that on a game to game basis Wins produced correlates at a 99.8% with point margin (Point Margin for a game = 0.0377 + 15.5 Wins Produced for that game) and for the season a 95% correlation has been shown repeatedly (the difference is down to blowout) . So a mapping of the box-score stats to WP will tell us the expected effect on both wins and point margin from a shift in a single box-score stat.

First let’s do Box score stats to Win Produced. For this mapping I’m looking at changing each stat by itself in increments of 1 (per game). That mapping looks like this:

Cool looking , isn’t it. This assumes an increment 0f 1 of any stat is done while keeping all other stats the same (and improtant point that we’ll come back to later). Now all these coefficients come from regression and regression does a really good job at assigning value. But sometimes it can obscure what’s happening. Fortunately, we have a handy dandy little equation available remember?

**Point Margin for a game = 0.0377 + 15.5 Wins Produced **

**Note: I had a mistake when I put this up. To convert WP to expected Point Margin (and vice versa) for the team I have to account for the fact that for a single game half the win credit goes to the victor and half get charged to the loser so the equations for conversion become:**

**Expected Avg Point Margin for Team (season) = 31*(Wins Produced (team for the season) -41 )/82**

**Wins Produced (team for the season)**** = ****(Expected Avg Point Margin for Team (season)*82)/31 +41**

**Marginal Wins Produced (team for the season)**** = ****Expected Avg Point Margin for Team (season)*82/31 = PM*2.645**

**Point Margin = 31/82 * Marginal WP = .378 *WP
**

**So:**

** +1 Points = 2.645 wins over .500 (43.645 wins)**

** +10 Points = 26.45 wins over .500 (67.45 wins)**

**+1 WP = +.378 Points**

**+10 WP = +3.78 Points**

**And I’m resetting the tables from this point on to reflect this.
**

What if I used this to map and increase in boxscore stats to point margin increases? What if I did that and called it Point Margin Produced? Let’s find out. We take all the numbers in the above table and adjust them accordingly we get:

That really clear’s it up doesn’t it. Now before you start with the snarky comments, I see it too. And additional Three pointer assuming everything is held constant is only worth two points, and additional 2 point field goal is only worth a point if everything else is held constant. Let parse this before we go crazy. A field goal does not happen in a vaccumn . Multiple things have to happen for a field goal to be made . If we break it down:

We can get to a +1 on the number of 2PT FG for the same Number of Shots by getting :

+1 on a 2FG

-1 on FG Misses (there are some additional possibilities but this is the simplest one)

A similar story can be told for the Three pointer and if we add it up it looks like this:

And that looks right. The effect of increasing a FG adds up to 2 or 3 points (there’s some marginal effects i.e. assists,rebs etc. I’m not accounting for here, remember keep it simple and understandable) but regression splits up the value of those points among all the contributing stats.

So the numbers prove that making shots in basketball is really only part of the story.

One more table before we leave:

Oh hell. Let’s look at stats for this year (for everyone >100 MP) in WP48 converted to Points Margin produced per 48. This calculation is done as follows:

**PM48= (WP48-.099)* 31.1**

This should keep you amused for a few days 🙂

Reminds me of this:

http://www.basketball-reference.com/blog/?page_id=4122

Austin,

Yeah. But the coefficients are all set up using regression. In essence I’m detangling the WP numbers back to the point margin.

Of course. The difference is that SPM regresses to adjusted plus-minus, and this regresses to team point differential.

I think they’re both interesting because being able to predict the change in expected points based on actions taken seems to me to be the ideal of basketball statistics. Suddenly, if you can measure good screens, you can see if a good screen is worth +.1 PPP or something like that.

Alright, so I’m guessing this answers my question from the last post. My thing was on “double counting”. Using what you’ve made, I’d say that a rebound adds a marginal value of 1.0, because as it gives the team a rebound, it prevents the other team from getting one, so for a 100 misses, if your team gets 58 rebounds, your team has a marginal value of 29. At the same time though, that causes the other team to have gotten 42 rebounds which gets them a marginal value of 21. So although, you have have ((58-50) / 2) 4 marginal wins above an average rebounding team, you also cause the opponent to have ((50-42) / 2) 4 marginal wins less than an average team would allow, so that’s 8 marginal wins. 4 for the possessions you got and 4 for reducing the opponent’s possessions. Is this right?

so 58-42 is + 16 rebs * .5 = +8 point margin

8/15.5 = .51 of a win versus that opponent

58-50 = +8 rebs *.5 = +4 point margin /15.5 = .25 of a win * 82 games =20.5 wins (or if you use the first table which is more accurate +11 * 2 for +22)

Wait, I worded that wrong. That makes it seem like I’m trying to adjust the value of rebounding. What I was trying to say before was that at the team level, rebounds have a bigger effect than the percentages would make it seem, because a team rebounding at above 50% is also reducing the rebounding of the opposing team, and maybe I’m having a hard time wording this or getting my thoughts across.

Oh, thank you for answering the question. I think I get it now. So at the team level, rebounds do have a big impact, for the reasons I thought. oh ok.

No, 58-50 (must be overtime) means 4 rebounds above average, which is 4 extra points, which is about 10.8 wins. You’re basically saying it also means 10.8 extra losses for my opponents, which of course is true, but still doesn’t make it 22 wins.

Guy,

I actually did the calc for a +8 rebounding advantage over an avg opponent (which is the full twenty two)

Not sure what you mean. There are only X rebound opportunities in any game. If one team is 8 rebounds above average, then the other team can’t be average — it must be 8 rebounds below average. (Controlling for pace, of course).

If there are 82 missed shots, and one team gets 45 and the other 37 (8 rebound differential), the first team is 4 points above average, with an expected win% of .632. Over 82 games, that’s 10.8 wins.

for a Point differential of 4 I get 4/(15.5)*82 = 21.16

This is win differential right? Not wins above average? If a team average s +4 points a game over 82 games I would expect them to win about 52 games.

Yes, let’s talk W-L record, to avoid confusion. A team that outscores it’s opponents by 4 points will go 52-30 average.

Whoops, good catch.

For a single game the WP equivalent to PM is

PM= .0377 + 15.5 WP but each team gets credit for half

so the actual equation for the team for a +4 in points in terms of wins is:

(+4)/(15.5*2)= +4/31 = .129 increase in Winning % or +10.58 games (and that’s 52-30.

Need to update the tables to reflect that.

Oh wait (sorry for the triple post. Everything’s starting to click right now. At least I think) I remember Dberri saying that a player getting a rebound prevents the other team from getting the ball. I’m guessing that good rebounders not only increase the rebounds of their teams, but they also decrease the rebounds of the opposing teams. So it’s not about how many rebounds you add to your team, but rather the change in rebound differential you add, which I’d guess is big too.

Could that be the big answer? Great rebounders not only add rebounds to their teams, but also reduce the rebounds from opposing teams?

What if you increase your 3 pt by 1 by increasing pace by 1 possession (and everything else held constant)?

and the rest, of course.

SD,

Tricky. The possession has to come from somewhere. The simplest thing is an additional Oreb. Baseline is taking the shot and missing it (-.5 PM, -2.8 WP). The value would be:

Oreb ( +.5, +2.8)

3PT (+1, +5.6)

The opponent not getting the rebound/shot (+.5)

The opponent allowing the 3PT field goal (-1 for them and it would should up in the Team adjustments in the Team defense adjustment)

~~So you get credit for the full +3 ( about (3/15.5)*82=15.87 wins).~~The simplest thing to do is think (Point Margin/15.5)*82= Wins Produced

I screwed this up the first time it’s:

+(3/31)*82= 8 games

(Point Margin/31)*82= Wins Produced

missed a two there.

oh right, the possession has to come from somewhere. didn’t cross my mind for some reason. This makes sense. Either an additional rebound or a turnover must occur.

And how exactly does Moon Knight become a zombie? He’s already undead, basically.

Fred,

I think for Moon Knight it’s just a diet plan change.

Conceptually it seems like there’s some problems with “FT missed”. Let’s say opposing team gets a technical foul. There’s no negative outcome possible for the team shooting the foul, so the negative value for a missed free throw seems wrong. Ditto with an and-1; it’s strictly better than a normal basket.

Fred,

The math all works out though. I agree that in an ideal world we’d like to differentiate between 2 FT, And1 and Techs.

awesome Batman/GL picture at the end.

Thanks for the cheat sheet as well

So if we take PM/48 and multiply is by the minutes each player played and add that to every player on the team, we should get the team point differential?

sd,

🙂 It’ll be a table in the next post . I’m going to be really mean to Bargniani again.

cool, just wanted to make sure i got it right. As far as Andrea is concerned, you’re more than welcome to skewer him. 😀

Getting it right for what? Do I need to put up a paypal button for your kind donations.

getting it right i mean that i understood the data.

Somehow, I remember reading this in the morning and seeing all the marginal values had 1/2 their current value and thought, hey, this is odd.

BTW, I think this model overpenalizes for missed field goals. See here:

http://thecity2.com/2010/12/07/debate-which-shooting-performance-is-better/

Evan,

Cool post. Remember that you have to think of the whole action in the scenario. So if you did make the the additional field goal it would show up. That said, 3Pts have higher value than 2 pointers (so 2 3 pointer on three shots are better than 3 2 pts on 3 shots)

But WP values both those scenarios the same (for the shooter):

2X3-pt = 2*2-1=3

3X2-pt=1*3 =3

I just wanna re-ask this question since this has been a big thing. So here it is. So a for a 100 misses, if a team gets 58. It’s not plus 8 fro a team that would get 50, but rather 16, because you take 8 away from the opposing team.

I’m guessing that good rebounders not only increase the rebounds of their teams, but they also decrease the rebounds of the opposing teams. So it’s not about how many rebounds you add to your team, but rather the change in rebound differential you add, which I’d guess is big too.

Could that be the big answer? Great rebounders not only add rebounds to their teams, but also reduce the rebounds from opposing teams?

of course, but they also reduce rebounds from their teammates.

Ok well, that’s it then. I just wanted to make sure. All those studies that Guy (and/or some dude) on rebounding showed the increase in team rebounds, but never the decrease in opponents’ rebounds. Those studies failed to see the double effect of a rebound.

If I recall, they accounted for only the team’s rebounds. They should’ve checked for the increase in rebound differential. Yea high rebound guys take away from their teammates (diminishing returns), but because of the double effect of a rebound, you can see that effect of diminishing returns is very small.

So that ends that.

EA: I’m afraid this is all wrong. If you follow the discussion above, what we discovered is that Dr. Berri is correct: each extra rebound by a team is worth one point, or .033 wins. So each rebound per game means 2.7 wins. If there are 82 rebounds in a game, the results look like this:

41 reb = 41 wins (differential =0)

42 reb = 42.7 wins (diff = 2)

45 reb = 52 wins (diff = 8)

……

This is the valuation of rebounds Dr. Berri uses, and also what I used in every example or anlysis I have cited here. This is one issue on which everyone already agrees (or at least did), so let’s not start a debate on that!

So, for example, a team that is at the 83rd percentile in rebounding gains about +3.2 wins. The top winning teams in the NBA (best point differentials) typically are about .8 rebounds per game above average, which contibutes about 2.1 wins.

Sorry, should say:

“42 reb = 43.7 wins, and (differential = 2)”

“45 reb = 52 wins (differential is 8 rebounds)”

Obviously, we know the value of a team rebound, but we’re talking about the value of what that rebounder brought to the table. Isn’t your arugment that you can’t assign a team value to a player?

Also, how did you arrive at the .8 number? Glancing back at the last few years it looks like the top 5 teams (point idfferential) are consistently in the +1-+5 range in rebound differential.

You may know the value of a rebound, but EA is still working it out. And no, I don’t think the value of a rebound changes, the question is how many additional rebounds does a player with an XX Reb48 actually generate?

You measure teams’ rebound advantage by taking Dreb% – League% (about .73 usually) * (Dreb + OppOReb). Repeat for Oreb%. +.8 rebounds is what I recall getting over past couple of seasons, but I won’t swear to it. The differentials are larger because good teams have more defensive opportunities (which are about 3x easier), and because using the differential effectively doubles the number.

EA,

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