597Re: [APBR_analysis] Re: nice methods

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• Feb 6, 2002
On Wed, 6 Feb 2002, HoopStudies wrote:

[...]

> The only thing that is clear is that it wasn't that hard to make up
> for Brand's loss; they didn't drop to a 4 win franchise and it's hard
> to name any 15 win team that got worse by losing its "best player".

This is one of the extreme cases where the nonlinearities become
important. If we measure players by their individual wins and losses and
furthermore require that those sum to 15, then no Bull can appear to be
as highly productive as Jordan was with his 19 individual wins.

But in situations such as these where we're looking at won-loss
percentages, it's probably a better idea to look at odds instead of
probabilities, or even the logarithm of odds (known as a logit
transformation). Things may become linear with respect to odds or to
logits, which are non-linear with respect to probabilities.

Two examples: odds ratios are what's behind the log5 method for
predicting win probabilties that some mathematician friend introduced Bill
James to.

And here's an example of how logits could be applied: For the 15-67 Bulls
is ln(15/67) ~ -1.50. (Their odds of winning were 15/67 = .22, and their
probability of winning of course was 15/82 = .18.)

If we were lucky and life were relatively simple, Elton Brand's
contributions to the Bulls might be linear with respect to a logit, e.g.
subtracting him from the Bulls and replacing him with a pretty much
useless (for this year) high school player might hurt the Bulls to the
tune of -0.4 logits. For a team that had been 15-67, the new logit would
be -1.9, the new odds would be exp(-1.9) = .13, the new probability would
be .13/(.13+1) = .13, and the number of victories would be 10.7. So
losing Brand would cost the Bulls about 4 victories (which could of course
be counteracted by increased production from Artest, etc. -- another
non-linearity that we'd have to deal with).

Adding Brand to the Clippers, assuming that the other players' didn't
change (probably not a good assumption, non-linearities again), would help
them by +.4 logits. So their 31-51 2001 team, which had had a logit of
-.50, now has a logit of -.10, and therefore odds of .90, probability of
.475, and 39 wins. So Brand adds 8 wins to the Clips, in contrast to the
loss of 4 wins by the Bulls. (Obviously some of the Clips' wins would
therefore have to come at the expense of some team other than the Bulls,
non-linearity again.)

That's a nice simple yet non-linear model: Brand's quality measure stays
constant at .4 logits, but that translates into 4 marginal victories for
the Bulls and 8 marginal victories for the Clippers.

Unfortunately, this all assumes that (a) the logit function is the correct
functional form and (b) that the other players' production stays constant
(and of course there will be other roster changes which add even further
complications).

Life is undoubtedly not so simple, so I'm not claiming that that model
will actually work in terms of predictive value.

One thing which I've been meaning to try for years but never gotten around
to however is to use this kind of model to look just at rebounding. It's
a smaller, simpler task than trying to model offenses, defenses, or team
wins. It's clearly going to be a non-linear process: if Tim Duncan gets
added to the Spurs and replaces ... who'd he replace, Carl Herrera?
Anyway, if Herrera was getting 4.5 rebounds per game and Duncan gets 12
per game, it is clearly not correct to predict that the Spurs will gain an
additional 7.5 rebounds per game. Some of Duncan's rebounds will come at,
so to speak, the expense of teammates. Yet he clearly should cause some
improvement to the Spurs' rebounding. I wonder if odds or logit measures
could be used so that players' rebounding quality stays constant even
though their teammates' and team's rebounds may change.

Such a measure wouldn't meet the "David Wesley" test that alleyoop2
suggested: we know that players' rebound stats will change when they
change positions (centers get more than power forwards, thanks to their
inside position). But it might pass the Greg Anthony test: a good
rebounder going to different teams or having different teammates (maybe
Dennis Rodman, maybe Vin Baker) might end up with a constant rebounding
score using these models, even though his rebounds-per-48-minutes would
change.

--MKT
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