My principal method for evaluating the overall effectiveness of players is to look at their individual win-loss records. I calculated these for Jordan and Olajuwon all the way back to the '86-87 season (before that, I haven't entered into the computer all the data I need):

Jordan Olajuwon W L % W L % '87 17.0 2.4 0.877 12.3 2.2 0.850 '88 19.4 0.7 0.967 11.9 2.1 0.850 '89 18.0 0.7 0.961 14.9 1.7 0.898 '90 18.0 1.1 0.942 14.2 1.8 0.888 '91 17.5 0.4 0.980 9.3 0.5 0.945 '92 15.9 0.6 0.966 10.6 1.5 0.873 '93 17.0 0.8 0.953 17.3 0.8 0.957 '94 16.8 1.1 0.937 '95 2.8 0.9 0.757 14.0 2.3 0.859 '96 5.8 0.2 0.964 5.4 1.1 0.834 Total 131.4 7.8 0.944 126.7 15.1 0.894(1996 season through games of 1/7/96) As is obvious, Jordan has consistently done slightly better than Olajuwon in winning percentage and overall wins. There is no doubt that both are among the top in the league. By being so good and such exceptions to the average, both players defy the use of standard statistical methods for analysis. This is just one reason (of many) why Barra's method didn't work. When players are this good, you have to get into the details and identify the small things that are important and exactly how they are important. That's what my methods do. And they consistently show that Jordan has been a better player than Olajuwon (and everyone else, for that matter).

(Late Note: Though I said that Jordan was only "slightly better" than Olajuwon above, the difference is actually a significant one. I have a method (from Bill James) for determining how often a team with one record beats a team with another record and it indicates that a team with a 0.944 winning percentage beats a team with a 0.894 winning percentage 66.7% of the time This is a rather significant edge and, though, there is no way of evaluating what this really means since we can't compare a team of Jordans to a team of Olajuwons, it indicates very strongly that Jordan contributes more to winning than Olajuwon, another great player.... For people in the statistics field, this represents one way of comparing probabilities close to one or zero, which can be useful in evaluating rare events, such as earthquakes, storms, or floods.)

Jordan Rtg Olajuwon Rtg Off. Def. Off. Def. '87 116.5 103.4 106.4 95.8 '88 123.2 100.4 105.4 94.9 '89 122.5 100.8 104.2 91.4 '90 122.5 103.5 101.9 89.9 '91 126.0 99.6 106.0 89.2 '92 121.8 99.4 106.7 94.9 '93 120.2 100.2 111.2 92.2 '94 107.8 91.5 '95 107.6 100.4 107.5 96.4 '96 121.4 99.4 102.9 93.3Jordan has consistently been the better offensive player, which we know, while Olajuwon has been the better defensive player, which is debatable to some people. I would affirm, however, that big men are generally more valuable than guards because they occupy the middle of the defense and are more likely to be involved in stopping the opposition. My defensive method reflects this (and I've managed to convince Doug Steele of it, too). The way we put these two ratings together to obtain a winning percentage is through the Pythagorean Method, a method that accurately predicts winning percentages from offensive and defensive ratings for teams. (Of course, we are assuming that the same method applies for individuals, but that is a good first assumption.) Hence, with accurate offensive and defensive ratings, we have an accurate measure of winning percentage. The next step (going backward from how I actually calculate it) is how we get offensive and defensive ratings for players.

An important rule I've held all my work to is that, when I sum all the individual points created and possessions used, I get the team values of points and possessions. With this in mind, we must do two things:

- Devise a way to evaluate how the points produced on an assisted basket should be split up between assistant and assistee. Also devise how that possession should be split up.
- Evaluate how offensive rebounds affect both a player's ability to indirectly "create points" and reduce his own number of possessions used.

My theory on weighting assists is based on two factors. First,
the player shooting the ball should get at least as much, if not
more credit than the assistant because making the shot is only a 50%
proposition (normally), while passing is safe at a much higher rate.
Second, when a potential assist is not converted, only the person
missing the shot looks bad, whereas the potential assistant has
no statistical record (though I've actually taken some). What my method
comes down to is weighting assists by the chance that the shot has
for going in. In other words, a pass to a wide open Shaquille O'Neal
under the basket is worth half of the two points that are guaranteed
to be scored because both the pass and the shot are 100% sure to happen.
However, a pass to O'Neal at the three point line that he throws up
at the end of a quarter and happens to go in the basket is worth
less to the assistant because Shaq did the more difficult, or less
likely, action. Since there are no numbers on this sort of thing,
I have had to take some myself. I have roughly found that
players don't shoot much better on average off a potential
assist than they do otherwise. Because of this, I have defined
the *assist figure*, or how much to weight the assist, to
be 0.25, or about half of 50%, a benchmark field goal percentage.

This then gives the first part of the formula for estimating an individual's scoring possessions:

Scoring Poss.=FG+ASTFIG*AST-ASTFIG*FG*(TMAST/TMMIN*5*Min-AST) ----------------------- (TMFG/TMMIN*5*Min-AST)First, we give credit for all field goals (FG), then for all assists (ASTFIG*AST), then we take away credit for the baskets a player scored that were assisted on (the final part of the formula). That final part of the formula estimates how many baskets were assisted on by determining roughly how many field goals were made while the player was in that he didn't assist on and how many assists by other people while the player was in. It's a decent approximation, though it has a slight bias that I won't discuss here.

We can add free throws into scoring possessions easily because no assists are credited on most free throws. Using the well established approximation (in this not-so-well-established field) that about four-tenths of all free throws end possessions and finding out how many of these possessions are scoring possessions, we add to the above formula the term:

(1-(1-FT%)^2)*0.4*FTAThis, plus the term above, make up the simple form of the scoring possession formula:

Scoring Poss.=FG+ASTFIG*AST-ASTFIG*FG*(TMAST/TMMIN*5*Min-AST) ----------------------- (TMFG/TMMIN*5*Min-AST) +(1-(1-FT%)^2)*0.4*FTAThis is essentially the same form as that seen in The Basketball Hoopla, which I did in 1988. I have modified the formula slightly since then to account for offensive rebounds, which I will now discuss.

Offensive rebounds are *offense*. They help the offense
by preserving a possession. This has been my stand since
the Hoopla. However, using my logic above for
giving credit to assistants, I should also give some credit for
scoring possessions to offensive rebounders. How much? It
depends on how well the team scores when you look at the offense
without offensive rebounds. If the team never makes any baskets,
obviously, getting an offensive rebound doesn't mean much. If
the team shoots well, getting a second shot is more advantageous
probabilistically. This is how I've chosen to weight offensive
rebounds in scoring possessions; it is reflected in the complex
form of scoring possessions below:

Scoring Poss.=[FG+ASTFIG*AST-ASTFIG*FG*(TMAST/TMMIN*5*Min-AST) ----------------------- (TMFG/TMMIN*5*Min-AST) +(1-(1-FT%)^2)*0.4*FTA]* [1-ORFIG*(TMFloor%-TMPlay%)]+ OR*TMPlay%*ORFIG ORFIG=0.1 TMFloor%={(FG+(1-(1-FT%)^2)*0.4*FTA)/Possessions}_Tm TMPlay%={(FG+(1-(1-FT%)^2)*0.4*FTA)/(Possessions + OR)}_TmThe Team Floor % (TMFloor%) and Team Play % (TMPlay%) are, respectively, estimates of what percentage the team scores on its possessions and what percentage the team scores on its possessions when offensive rebounds are looked at as starting a new possession. Basically, the difference between these two team values indicates how much an offensive rebounds helps the team's chance of scoring. By arranging them in the formula as I do, I take away some credit from the earlier formula (where OR's are not included) and add it back in through individual offensive rebounds at the end. The ORFIG says a player gets about one-tenth credit for each scoring possession that occurs this way. This roughly means that, after an offensive rebound, there is likely to be about two other people handling the ball before a scoring possession. This is a guess and it's very complicated as it is. So if you didn't get all this, don't worry about it. This is a new science. There were only a couple people in the world who understood quantum physics when it was developed. Now everyone understands it. (That's a joke.)

So that is scoring possessions, but we want to know how many points that leads to. Since we know that there are usually two points for every scoring possession, we can multiply scoring possessions by two as a first approximation. That is exactly what I do...as a first approximation. There are some extra things I do to make it more accurate:

Points Produced={2*[(FGM+0.5*3ptFGM)+ASTFIG*AST*(TMFG-3ptFGM) ------------- (TMFG-TM3ptFGM) -ASTFIG*(FGM+0.5*3ptFGM)*(TMAST/TMMIN*5*Min-AST) -----------------------] (TMFG/TMMIN*5*Min-AST) +0.4*FTA*(2*FT%^2+2*FT%*(1-FT%))+qmk*FTM} *(1-(TMFloor%-TMPlay%)*ORFIG) +ORFIG*TMPlay%*OR*TMPtsPerScPoss qmk = 0.06 TMPtsPerScPoss = Team Points Per Scoring PossessionThe term

`qmk`

comes from additional free throws resulting
from things like technical fouls and three point plays; think of
it as a glorified fudge factor. The term `TMPtsPerScPoss`

is evaluated using
TMPtsPerScPoss= Team Points ------------------------ FG+0.4*FTA*(1-(1-FT%)^2)The logic is not transparent from the equations. I weight the two point scoring possessions by two, the one point scoring possessions by one, and the three point scoring possessions by three. I ignored scoring possessions on which more than three points were scored because they are very rare.

*Whew!* Remember where we are. I'm trying (perhaps in
vain) to demonstrate where the offensive rating for individuals
comes from. What I just did was give the formula for the
points produced by an individual. Since the offensive
rating is just points produced per 100 possessions used,
I now have to show you the formula for how many possessions
a player uses. If you're still with me now, you can handle
the next part...

Possessions used include scoring possessions, of course, plus all those possessions a player ends without a score. These are possessions with a turnover and missed shots (field goals or free throws) that aren't rebounded by the offense. Easy enough. Here's the formula:

Possessions = Scoring Possessions + FGA-FG - (FGA-FG)*(TMOR/(TMOR+Opp.DR)) + (1-FT%)^2*0.4*FTA + TOTold you that was easy. Now divide points produced by possessions and multiply by 100 to get a rating. (Thank goodness for computers.)

Defensive stops occur every time the opposing offense ends
a possession without scoring. This can happen via a turnover
or a missed shot that the defense rebounds. I evaluate
every player on the team as being roughly even in man
defense. I do this by finding out how often (times per
minute played) the opposition
is stopped *not* through a steal or block, then
multiply it by the individual's minutes played. In
the formula, this is

Min*[(OppFGA-OppFGM-OppOR-TMBLK)/2+(OppTO-TMSTL)]/TMMINThe first part inside the square brackets is how many times the opposition misses a shot, then divided by a factor of two. I divide by two because half of a defensive stop is the missing of the shot -- the other half is getting the defensive rebound. (There is a slight error in logic in the previous statement. See if you can pick up on it.) The second part in the square brackets is the number of turnovers not caused by steals.

Now we've taken out the individual contributions of blocks, steals, and defensive rebounds to get an average estimate of man defense. It's time to add those back in for each individual. Just adding in an individual's steals and half of both his defensive rebounds and blocks gives an overall formula for defensive stops:

Defensive Stops = Min*[(OppFGA-OppFGM-OppOR-TMBLK)/2+(OppTO-TMSTL)]/TMMIN + STL + 0.5*(DR+BLK)

Again, remember where we are and where we want to go. We have an offensive rating from above and now we need a defensive rating. We have defensive stops, not a defensive rating. We need to estimate "how many points per possession a player gives up."

This, I'm sorry to say, is difficult to define. How do you evaluate the guard who lets his man drive past him, where his center saves him by blocking the layup attempt? Who gets credit for the stop? Who gets the blame if the layup isn't blocked and goes in? These are theoretical difficulties. Generally, a team plays a style of defense that either asks for help defense or asks for straight man defense or something in between. Depending on whether that guard was supposed to let his man go by determines whether he did something right. If he was supposed to let the man go by, then rotated to the big man's assignment to cut off the dump pass, he apparently did OK. If he was supposed to play straight, then any points scored should mostlly be blamed on him, not the big man covering the little guys' back when it's not his responsibility.

Given this dilemma, how do we proceed? Basically, I try to fit defensive stops into a framework that makes some sense if we ignore the above questions. Ignoring the difficulties sometimes allows one to see a solution. That is my guiding paradigm here.

If we can somehow evaluate how many stops per possession a player has, we essentially have the defensive equivalent of a floor percentage Going from this to a points per possession rating is fairly straightforward. This is the method.

To find the stops per possession, divide a player's total defensive stops by his minutes played, multiply by the team's minutes played, and divide by the team's total number of possessions:

Stops*TMMIN Stops/Poss= ------------- Min*TMPoss.This number turns out to be very high for some players. For example, Olajuwon consistently has a stops/poss value of 0.7 or so. This would mean that he stops seven tenths of all possessions he is responsible for. Or his opponents score only 30% of the time against him, for a rough rating of 60. Olajuwon is a good defender, but he's not that good.

In order to compensate for this flaw, I weight the team's defensive rating with the individual's stops/poss value. I weight them 80% to 20% because I figure that one player is 20% of his defensive team. It's a little hokey, logically, but it gives results that I'm fairly pleased with, as I mentioned above:

Def. Rtg = 0.8*TMDef.Rtg + 0.2*(200*(1-stops/poss))For completeness, the factor

`200*(1-stops/poss)`

is an approximation of the points per 100 possessions
arising from the given value of `stops/poss`

.
So that's it. Those are the methods. Their derivation was probably pretty dry, but science can be that way. The results, however, can be trusted and are often very enlightening because there is good (but dry) science behind them. Linear weights methods, on the other hand......

I should also note that linear weights are very subject to individual biases because there is no theory behind them. I have seen at least six different linear weights formulas that arbitrarily weight blocks, rebounds, assists, steals, etc. How you assign those weights determines your answer to questions like "Who is better?". If Barra thought that blocks were not very important he could reduce their weight from 2 to 1. It would certainly change Barra's conclusion. Because there is no theory to help determine what weights should be, Barra (and others) hint at equating blocks, rebounds, etc. to points. But this makes no sense. When a player blocks a shot, his team doesn't get a point. When a player rebounds a shot, he doesn't get a point. For example, say one player blocks an opponent's shot, gets the rebound, then misses the layup at the other end (and the opponent rebounds). According to Barra's formula, this player has accumulated 2.5 points in this sequence, but no points were scored! Maybe we should look at it another way -- his opponent has accumulated 0.5 points in this sequence by rebounding and missing a shot. Still, three Max Points (as Barra calls them) were scored, but no points that really counted were scored. By Max Points, our hypothetical player is also leading by two (2.5 to 0.5), but by real points, he is still even. Finally, there is no reason for linear weights to equate with points since it accounts for both offensive and defensive statistics (which prevent opponents' points). Even if it did better equate to points scored, where does it account for points allowed?