## The City’s Advanced Statistical Primer

this is an excerpt

I’m putting together this advanced stats primer for GSOM. Figured I post it here and solicit input from the more specialized crowd that frequents my personal blog. Most of  you know this stuff, but it never hurts to hear it from someone who might have a slightly different take. Any comments are welcome, except those I don’t like. 😛

## Does Your Team Optimize Its Offense?

Your better offensive players should use more possessions, right? The number of possessions a player “uses” is usually defined as follows:
$PU = FGA + TOV + 0.44FTA$

With that definition, “usage” is simply the number of possessions the player uses divided by the total number of possessions while the player is on the floor: $USG=PU/POS_{team}$. The ezPM model, which uses play-by-play data, makes it easy to calculate all these stats. Since I also track assisted field goals, I think it makes sense to slightly re-define possessions used according to the model:
$PU_{ez}=0.3AST+0.7AFG+UFG+0.74*MISS+TOV+0.44FTA$
where:

• $AST=text{assists}$
• $AFG=text{assisted field goals made}$
• $UFG=text{unassisted field goals made}$
• $MISS=text{missed field goal attempts}$

I define “offense” as the scoring/assist/turnover components of ezPM (i.e. everything on offense involved with using possessions):

$OFF_{ez}=0.7*(2-1.06)*AFG2+0.7*(3-1.06)*AFG3+(2-1.06)*UFG2+(3-1.06)*UFG3+0.3*(2-1.06)*AST2+0.3*(3-1.06)*AST3-0.74*1.06*MISS+FTM-1.06*(0.44*FTA')+AND1-1.06*TOV-0.2*1.06*TEAM_TOV$

The numerical factors represent league average points-per-possession (1.06) and defensive rebounding rate (0.74).

So, with offense and possessions used defined as above, we can calculate the offense generated per possession used (which we’ll just call OFFENSE):

$OFFENSE = OFF_{ez}*100/PU_{ez}$

For reference, here are the top 50 OFFENSE players currently (as of 1/04):

## Top 50 OFFENSE Producers According to ezPM

 RANK NAME TEAM POS OFFENSE USG POSS 1 Chris Paul NOH 1 25.16 24.36% 2417 2 Nene Hilario DEN 5 24.48 16.53% 1877 3 Steve Nash PHX 1 23.40 24.10% 2199 4 Tyson Chandler DAL 5 20.83 10.44% 1919 5 J.J. Redick ORL 2 19.46 16.57% 1241 6 Luke Ridnour MIN 1 18.98 18.77% 1766 7 DeShawn Stevenson DAL 2 18.98 14.43% 1032 8 Deron Williams UTA 1 17.32 27.28% 2717 9 James Jones MIA 3 16.72 10.22% 1587 10 George Hill SAS 1 16.41 16.21% 1697 11 D.J. Augustin CHA 1 16.34 19.13% 2087 12 Dirk Nowitzki DAL 4 16.11 24.20% 2177 13 Danilo Gallinari NYK 4 15.72 15.57% 2411 14 Matt Bonner SAS 4 15.53 9.89% 1307 15 Paul Pierce BOS 3 15.31 20.36% 2376 16 Kevin Martin HOU 2 14.89 26.39% 2236 17 Manu Ginobili SAS 3 14.57 22.52% 2419 18 Al Horford ATL 5 13.85 19.65% 2384 19 Tony Parker SAS 1 13.65 24.98% 2407 20 Chauncey Billups DEN 1 13.15 22.49% 1910 21 Brad Miller HOU 5 13.14 16.50% 1236 22 Ty Lawson DEN 1 12.97 20.08% 1773 23 Jameer Nelson ORL 1 12.47 24.41% 1655 24 Ronny Turiaf NYK 5 12.10 9.89% 1066 25 Raymond Felton NYK 1 11.88 22.83% 2773 26 Arron Afflalo DEN 2 11.74 13.11% 2398 27 Grant Hill PHX 3 11.59 18.23% 2134 28 Beno Udrih SAC 1 11.40 19.98% 1884 29 Lamar Odom LAL 4 11.22 17.62% 2537 30 LeBron James MIA 4 11.22 30.45% 2779 31 Jose Calderon TOR 1 10.93 20.80% 1587 32 Paul Millsap UTA 4 10.81 18.84% 2490 33 James Harden OKC 2 10.51 18.12% 1818 34 Hakim Warrick PHX 4 9.75 18.42% 1350 35 Wilson Chandler NYK 3 9.61 17.87% 2507 36 Dwyane Wade MIA 2 9.58 28.78% 2595 37 Devin Harris NJN 1 9.55 28.56% 1969 38 Pau Gasol LAL 5 9.28 20.24% 2757 39 Vince Carter ORL 2 9.08 23.11% 1312 40 Ray Allen BOS 2 9.05 18.01% 2406 41 Richard Jefferson SAS 3 8.95 14.46% 2327 42 Stephen Curry GSW 2 8.80 25.50% 1786 43 Kirk Hinrich WAS 2 8.70 17.91% 1966 44 Rodney Stuckey DET 2 8.34 25.46% 1839 45 Reggie Williams GSW 3 7.82 18.61% 1509 46 Eric Gordon LAC 2 7.79 27.06% 2442 47 Mike Bibby ATL 1 7.49 15.48% 2102 48 Daniel Gibson CLE 2 7.44 21.46% 1767 49 Chris Bosh MIA 5 7.36 20.70% 2661 50 Thaddeus Young PHI 4 6.99 18.96% 1661

Now, if you plot USAGE vs. OFFENSE for the entire league (for players who have over 500 total possessions), you will find a big mess:

No discernable relationship, right? I wouldn’t even pretend to say there’s a statistically significant result there (there isn’t, trust me, I checked). But let’s look at the results for each team:

Now, it’s a lot more interesting, right? Again, I wouldn’t pretend to claim statistical significance at this point. I think that requires a fairly sophisticated multi-level regression analysis. There are very few data points per team. But…if there were no relationship, one would expect about an equal number of positive and negative slopes. Judging by eye (my eye, of course), it seems apparent that most of the slopes are actually positive or slightly positive, suggesting that players with greater offensive efficiency do generally tend to have higher usage. Or is it the other way? Well, that’s certainly a good debate to have for another time.

What I will say is that one can look at his or her favorite team right now and judge whether the coach is doing a good job of “allocating” usage, which I would argue is one of the main responsibilities of a head coach. Perhaps, it is the major function. Teams like POR, OKC, and even GSW have nice positive slopes, indicating the best offensive players are using the most possessions. So, how does your team look?

## Regressing Point Differential on The “Four Factors” (Part 2)

﻿﻿There are four factors of an offense or defense that define its efficiency: shooting percentage, turnover rate, offensive rebounding percentage, and getting to the foul line. Striving to control those factors leads to a more successful team. (Dean Oliver, “Basketball on Paper”)

How well do these four factors predict point differential (and thus, winning)? How important are each of the factors relative to the others? The first question was the subject of Part 1 — now would be a convenient time to read Part 1, if you haven’t already done so (don’t worry, Part 2 will still be here when you get back, thanks to the magic of the interwebs). Today we will address the second question.

How important are each of the factors relative to the others? In Part 1, we found the following model for predicting point differential (p.d.) as a function of the four factors (well, eight factors, including offense+defense):

$$p.d. = 10.41 + 1.49 * eFG(own) – 1.63 * eFG(opp) + 0.187 * FTR(own) – 0.213 * FTR(opp) -1.51 * TOR(own)+ 1.37 * TOR(opp) + 0.327 * ORR(own) -0.365 * ORR(opp)$$

where,

• effective FG% (eFG): $eFG=(FG+0.5 *3PT)/FGA$
• foul rate (FTR): $FTR = FTA/FGA$
• turnover rate (TOR): $TOR=TOV / (FGA + 0.44 * FTA + TOV)$
• offensive rebounding rate (ORR): $ORR=ORB / (ORB + Opp DRB)$

Recall that positive coefficients (own eFG%, own FTR, opp TOR, own ORR) mean that terms add to point differential, while negative coefficients (opp eFG%, opp FTR, own TOR, opp ORR or own DRR) subtract from point differential.

Upon inspection of the model, one is, perhaps, initially tempted to conclude that the most important terms are the ones with the largest coefficients (in terms of absolute value) — eFG% and TOR. The problem with that logic is twofold: 1) It should be obvious that the means for each stat vary over a wide range (i.e. eFG% is typically around 50%, whereas TOR~13%). Therefore, even though the coefficients for eFG% and TOR are similar, eFG% is larger overall, and would appear to dominate. 2) The variation for each stat may vary. In other words, even if a parameter appears to be a large contributor based on its coefficient and mean, in practice, if there is little variation (i.e. between teams), it will not have a large effect on winning.

Fortunately, there is a straightforward way to deal with both issues and get at the truth. Specifically, we can use the model, itself, to calculate the variation in total wins due to a normalized change in each parameter. Here’s how it works. First, I will temporarily take over David Stern’s role as NBA Commish (thank you, thank you), and create a new franchise in Las Vegas (VEG — Vegas, baby!). Next, we will magically skip forward to next pre-season — yes, there was an expansion draft, no, VEG did not get LeBron James, although I will not rule out James having taken his talents to Vegas on several occasions. Before the season begins, we would like to predict how many wins VEG might have. How do we do this?

Oh, right, the model! Let’s start out by assuming (optimistically) that our new franchise is average in all eight of the four factors (you know what I mean). How many wins would such a team produce (if you’ve already guessed around 41, eat a cookie or something)? Take a look at the table below. I’ve calculated the NBA average value and standard deviation (STD) for each category. Next, I varied each parameter by one standard deviation (in a direction that increases wins for that category), and used the model to predict point differential (P.D.) and wins (uh, Wins). Wins are related to point differential by the following formula (see here for explanation):

$W = 2.54 * p.d. + 40.9$

 eFG% FTR TOR ORR Wins Team P.D. Own Opp Diff Own Opp Diff Own Opp Diff Own Opp Diff NBA 0.2 49.6 49.6 0.0 31.4 31.4 0.0 13.8 13.8 -0.0 26.2 26.3 -0.1 STD 6.1 2.3 2.0 3.5 3.3 3.3 5.2 0.9 1.1 1.4 2.8 2.7 3.1 UNCH VEG0 0.1 49.6 49.6 0.0 31.4 31.4 0.0 13.8 13.8 0.0 26.2 26.3 -0.1 41.2 eFG(own) VEG1 3.5 51.9 49.6 2.3 31.4 31.4 0.0 13.8 13.8 0.0 26.2 26.3 -0.1 49.9 eFG(opp) VEG2 3.3 49.6 47.6 2.0 31.4 31.4 0.0 13.8 13.8 0.0 26.2 26.3 -0.1 49.5 FTR(own) VEG3 0.7 49.6 49.6 0.0 34.7 31.4 3.3 13.8 13.8 0.0 26.2 26.3 -0.1 42.8 FTR(opp) VEG4 0.8 49.6 49.6 0.0 31.4 28.1 3.3 13.8 13.8 0.0 26.2 26.3 -0.1 43.0 TOR(own) VEG5 1.4 49.6 49.6 0.0 31.4 31.4 0.0 12.9 13.8 -0.9 26.2 26.3 -0.1 44.7 TOR(opp) VEG6 1.6 49.6 49.6 0.0 31.4 31.4 0.0 13.8 14.9 -1.1 26.2 26.3 -0.1 45.0 ORR(own) VEG7 1.0 49.6 49.6 0.0 31.4 31.4 0.0 13.8 13.8 0.0 29.0 26.3 2.7 43.6 ORR(opp) VEG8 1.1 49.6 49.6 0.0 31.4 31.4 0.0 13.8 13.8 0.0 26.2 23.6 2.6 43.7

As expected, if VEG is totally average across-the-board (case VEG0), the model predicts 41.2 wins (no surprise, eh, that’s just about 50%). And if you’re complaining that the prediction is not exactly 41.0 wins, well, get a life. (And curl up with a good statistics book that can tell you about the nature of error and uncertainty in model predictions.)

Next, we change eFG%(own) by +1 STD from 49.6 to 51.9 (VEG1). The result is that VEG is now predicted to win 49.9 games. Wow! That’s an increase of almost 8 wins, just by varying eFG% by 1 STD. What happens when we do the same thing to the other categories? Ok, alright. You get it by now…just look at the table.

Having varied each factor by +(-) 1 STD, we can now rank the factors in terms of wins produced over average. We see that the ranking goes:

 Rank Factor Case Prediction Wins Delta % 1 eFG(own) VEG1 3.5 49.9 8.7 26.8% 2 eFG(opp) VEG2 3.3 49.5 8.3 25.5% 3 TOR(opp) VEG6 1.6 45.0 3.8 11.8% 4 TOR(own) VEG5 1.4 44.7 3.5 10.6% 5 ORR(opp) VEG8 1.1 43.7 2.5 7.6% 6 ORR(own) VEG7 1.0 43.6 2.4 7.3% 7 FTR(opp) VEG4 0.8 43.0 1.8 5.5% 8 FTR(own) VEG3 0.7 42.8 1.6 4.9%

The last category (%) takes the wins produced above average (Delta) and divides that amount by the sum of the Deltas for each case. This is what we were looking for to begin with: the relative weight of each factor. Note that shooting efficiency (producing it and defending against it) accounts for about 52% of the extra wins. Shooting efficiency is followed by turnover ratio, rebounding, and foul rate.

To bring this back to reality a bit, now let’s look at the current season and how teams at the top and bottom of the league are doing with respect to the four factors. The top and bottom rows represent hypothetical teams that are +1 or -1 STD relative to the mean in all 8 factors.

 eFG% FTR TOR ORR Rank Team P.D. Wins Own Opp Diff Own Opp Diff Own Opp Diff Own Opp Diff +1 12.7 73.2 51.8 47.6 4.2 34.7 28.1 6.6 12.9 14.9 -2.0 29.0 23.6 5.4 1 BOS 12.4 72.5 54.3 46.7 7.6 31.4 32.7 -1.4 14.5 15.5 -1.0 21.3 23.1 -1.8 2 MIA 11.5 70.1 51.7 46.1 5.6 38.0 31.3 6.7 12.8 13.5 -0.7 24.8 24.5 0.3 3 SAS 8.9 63.5 52.3 49.4 2.9 31.9 24.5 7.4 12.9 14.8 -1.8 26.4 25.7 0.7 4 LAL 7.4 59.6 50.7 47.5 3.2 29.8 25.2 4.7 12.6 13.3 -0.8 30.0 29.8 0.2 5 DAL 7.2 59.1 52.1 47.4 4.7 30.5 27.7 2.7 13.9 13.6 0.3 23.6 25.3 -1.7 26 SAC -6.7 23.9 46.8 51.0 -4.1 29.9 35.1 -5.2 13.8 13.5 0.4 29.9 26.5 3.4 27 NJN -7.0 23.1 46.8 49.2 -2.4 31.4 34.3 -2.9 13.7 11.6 2.1 24.6 25.0 -0.4 28 MIN -8.0 20.5 47.7 51.2 -3.5 28.5 36.0 -7.6 15.3 13.0 2.2 30.9 24.9 6.0 29 WAS -9.5 16.9 47.9 52.1 -4.2 29.4 33.1 -3.8 14.9 14.5 0.4 28.6 32.6 -4.0 30 CLE -10.0 15.5 46.5 53.0 -6.6 30.1 28.2 1.9 12.7 12.7 -0.0 22.0 23.7 -1.7 -1 -12.0 10.5 47.5 51.5 -4 28.2 34.6 -6.4 14.7 12.8 2.1 23.5 28.9 -5.4

I’ve highlighted in green (red) the values that are above (below) 1 STD from the mean (in a direction that produces more or less wins, respectively). Lastly, since this is a Warriors-centric blog, let’s take a (sad and unfortunate) look at my favorite team with respect to the four factors:

 eFG% FTR TOR ORR Rank Team P.D. Wins Own Opp Diff Own Opp Diff Own Opp Diff Own Opp Diff GSW -4.7 29.0 49.8 51.4 -1.7 24.3 36.9 -12.6 14.4 15.1 -0.7 29.7 30.5 -0.8

Interestingly, the Warriors are not terrible in shooting, although they are just below league average in offensive efficiency, and well below in defensive efficiency. The Warriors are absolutely terrible in FTR. In fact, they are about 2 STD below the league average in going to the line. Surprisingly, considering the off-season acquisition of Lee and the return of Biedrins, the defensive rebounding is really bad. However, the offensive rebounding is actually very good. So, that’s a push. The Warriors are good at forcing turnovers, and this is helping them from dropping to the very bottom of the league. Make no mistake about it, though, the Warriors will continue to be cellar dwellars until their offensive and defensive shooting efficiency improves.

## Summary

I have shown that offensive and defensive shooting efficiency are by far the most important of the four factors, accounting for over 50% of wins alone. In comparison, offensive and defensive rebounding account for about 14% of wins. For reference, we can compare my results to Dean Oliver’s estimates for the weight of each factor:

 Factor DeanO Regression Shooting 40% 54% Turnovers 25% 22% Rebounding 20% 15% Foul Rate 15% 10%

I like that the results of the regression are consistent with what Oliver found, and it is especially comforting since I haven’t been able to actually track down his studies that show how he derived these weights. I assume he performed a similar analysis, but there may, of course, be other ways to arrive at the same conclusion. Lastly, it should be clear that the challenge for player valuation models is attributing credit for each of the factors to individual play. It is important to think about this team level analysis when you are considering models like Wins Produced, PER, Win Shares, etc.

## Regressing Point Differential on The “Four Factors” (Part 1)

﻿﻿There are four factors of an offense or defense that define its efficiency: shooting percentage, turnover rate, offensive rebounding percentage, and getting to the foul line. Striving to control those factors leads to a more successful team. (Dean Oliver, “Basketball on Paper”)

How well do these four factors predict point differential (and thus, winning)? How important are each of the factors relative to the others? The first question is the subject of today’s post. The second question will be covered in Part 2.

How well do these four factors predict point differential? To answer this question, we want to construct a model. The inputs (independent variables) to the model are the four factors (well, eight factors, since we should consider offense and defense separately), and the output (dependent variable) is point differential (p.d.). The (linear) model looks like this:

$p.d. = \beta_0 + \beta_1 * eFG(own) + \beta_2 * eFG(opp) + \beta_3 * FTR(own) + \beta_4 * FTR(opp) +\beta_5 * TOR(own) + \beta_6 * TOR(opp) + \beta_7 * ORR(own) +\ beta_8 * ORR(opp) + \varepsilon$

Here, $\beta_0$ is a constant term and $\varepsilon$ is an error term. The $\beta_1..\beta_8$ are model coefficients that will be determined by performing a multiple linear regression using the (free!) statistics software package R. The four factors are defined as follows:

• effective FG% (eFG): $eFG=(FG+0.5 *3PT)/FGA$
• foul rate (FTR): $FTR = FTA/FGA$
• turnover rate (TOR): $TOR=TOV / (FGA + 0.44 * FTA + TOV)$
• offensive rebounding rate (ORR): $ORR=ORB / (ORB + Opp DRB)$

It is important to note that the opponent ORR (also called ORB%) is simply the defensive rebounding rate of the team (DRR or DREB%). In other words, your DRR plus the opponent ORR should add to 1 (100%). And vice-versa.

The training set used to calculate the regression contains four factors data for each team over the past four seasons (2006-2009) — available from hoopdata.com. The current season is then used to test the model. Here are the estimated model coefficients (standard error):

$\beta_0 = 10.41 (3.69) \quad \beta_1=1.49 (0.039) \quad \beta_2=-1.63 (0.049) \quad \beta_3=0.187 (0.024) \quad \beta_4=-0.213 (0.021) \quad \beta_5 = -1.51 (0.074) \quad \beta_6 = 1.37 (0.072) \quad \beta_7 = 0.327 (0.029) \quad \beta_8 = -0.365 (0.041) \quad \varepsilon = 0.664$

The $R^2$ for the model is 0.985, and all coefficients were found to be statistically significant (meaning they contribute to outcome — how much so, we will examine in Part 2).

Note that positive coefficients (own eFG%, own FTR, opp TOR, own ORR) mean that terms add to point differential, while negative coefficients (opp eFG%, opp FTR, own TOR, opp ORR or own DRR) subtract from point differential.

Now that we have determined the model coefficients, we can test the model using the four factors stats for the current season. Here is a summary in table form:

 eFG% FTR TOR ORR Team OEFF DEFF P.D. Prediction Own Opp Diff Own Opp Diff Own Opp Diff Own Opp Diff BOS 107.2 96.3 10.9 12.4 54.27 46.66 7.61 31.4 32.7 -1.39 14.46 15.5 -1.04 21.33 23.09 -1.76 MIA 109.1 97.2 11.9 11.5 51.67 46.07 5.6 38 31.3 6.67 12.79 13.5 -0.71 24.75 24.47 0.28 SAS 110.3 99.7 10.6 8.9 52.34 49.4 2.94 31.9 24.5 7.37 12.93 14.75 -1.82 26.35 25.7 0.65 LAL 109.4 101.8 7.6 7.4 50.71 47.51 3.2 29.8 25.2 4.69 12.55 13.31 -0.76 29.99 29.75 0.24 DAL 106.9 99.6 7.3 7.2 52.13 47.39 4.74 30.5 27.7 2.73 13.87 13.59 0.28 23.61 25.31 -1.7 ORL 104.5 98.7 5.8 6.7 52.3 48.22 4.08 33.8 29.4 4.38 15.06 13.94 1.12 25.28 22.14 3.14 CHI 102.5 98.2 4.3 6.2 49.55 47.11 2.44 30.9 29.9 1 14.37 14.86 -0.49 29.24 25.3 3.94 UTH 106.4 103.2 3.2 3.9 50.24 47.66 2.58 31.7 36.7 -5.07 12.79 14.46 -1.67 25.27 30.07 -4.8 PHI 102.6 101.8 0.8 2.8 49.3 47.03 2.27 29.7 35.2 -5.55 13.2 13.33 -0.13 24 24.83 -0.83 NOR 101.3 99.5 1.8 2.7 49.05 48.17 0.88 31.5 29.7 1.75 13.6 14.31 -0.71 21.69 22.97 -1.28 ATL 105.6 103.6 2 2.1 50.95 49 1.95 29 28.3 0.71 13.22 12.7 0.52 23.82 25.63 -1.81 NYK 109.5 106 3.5 1.5 52.38 50.92 1.46 33 31.7 1.31 13.73 13.67 0.06 25.17 27.39 -2.22 DEN 108 105.4 2.6 1.1 51.35 50.32 1.03 39 30.9 8.16 13.53 12.41 1.12 23.88 25.42 -1.54 IND 100.9 100.5 0.4 0.67 49.84 46.47 3.37 26.3 34 -7.66 14.62 12.97 1.65 22.66 25.8 -3.14 MIL 97.6 99.7 -2.1 -0.47 44.49 48.71 -4.22 34.3 32.1 2.14 12.95 15.24 -2.29 27.33 21.93 5.4 MEM 102.4 104.5 -2.1 -0.81 49.35 50.64 -1.29 28.3 30.2 -1.94 13.87 15.83 -1.96 27.22 30.46 -3.24 OKC 105.9 104.2 1.7 -0.81 48 50.05 -2.05 38.3 29.8 8.54 13.13 13.91 -0.78 25.69 27.47 -1.78 PHO 109.5 110 -0.5 -1.1 52.96 53.33 -0.37 32.6 27.5 5.03 13.5 14.01 -0.51 25.8 31.25 -5.45 HOU 106.6 106.9 -0.3 -1.2 50.53 50.29 0.24 33.4 30.5 2.85 13.6 12.38 1.22 25.85 27.17 -1.32 POR 102.4 103.5 -1.1 -2.3 46.58 50.51 -3.93 28 35.2 -7.15 13.2 15.91 -2.71 31.23 27.43 3.8 CHA 100.1 103 -2.9 -2.4 49.34 49.53 -0.19 32.7 29.7 3.03 16.34 13.83 2.51 26.9 24.21 2.69 TOR 104.4 108.2 -3.8 -3.6 49.64 52.58 -2.94 32.1 32.1 0 14.38 14.23 0.15 30.1 25.89 4.21 GSW 102.8 108.9 -6.1 -4.7 49.77 51.44 -1.67 24.3 36.9 -12.59 14.4 15.07 -0.67 29.66 30.47 -0.81 DET 101.5 108 -6.5 -5.7 48.18 51.53 -3.35 29.5 30 -0.48 13.14 13.2 -0.06 25.94 27.87 -1.93 LAC 99.7 107.2 -7.5 -6.3 48.44 51.01 -2.57 34 34.6 -0.59 15.54 13.18 2.36 28.75 25 3.75 SAC 99.6 107.3 -7.7 -6.7 46.84 50.98 -4.14 29.9 35.1 -5.24 13.82 13.47 0.35 29.93 26.54 3.39 NJN 99.4 105.2 -5.8 -7.0 46.79 49.22 -2.43 31.4 34.3 -2.9 13.7 11.6 2.1 24.62 25 -0.38 MIN 100.5 108.4 -7.9 -8.0 47.68 51.15 -3.47 28.5 36 -7.55 15.28 13.04 2.24 30.86 24.88 5.98 WAS 100.2 109.2 -9 -9.5 47.9 52.14 -4.24 29.4 33.1 -3.79 14.87 14.51 0.36 28.59 32.61 -4.02 CLE 97.7 108.5 -10.8 -10.0 46.49 53.04 -6.55 30.1 28.2 1.92 12.72 12.73 -0.01 21.95 23.65 -1.7

Here is a plot showing the relationship between observed point differential and p.d. as predicted by the model:

So, there  you have it. The four factors (eFG%, TOR, FTR, & ORR) explains about 96% of point differential. Next time, we’ll explore the relative weights of each term in the model, which will enable us to understand how the factors truly contribute to winning. Stay tuned…