THE BOOK--Playing The Percentages In Baseball

What is the minimum number of outstanding PA’s to get to 8 SD’s?  How close would John Paciorek’s one game be?  Or would he have to go 50 for 50 before disappearing?

If I did the formula right I get a guy with a .950 Woba in 45 PA at the same SD as the .380 guy for 20 seasons.

A guy with 13,000 PA with a .380 wOBA is 9.0 SD from the .340 point. 

A guy would need 56 PA at .950 wOBA to match that.

(Note: for wOBA, you have to do wOBA*(1.1-wOBA), as noted in the Appendix.)

On the other hand, if the comparison point is a .260 wOBA (which I think probably makes the most sense if you look at the blog entry above), you need 395 PA at .950 wOBA.

So, a guy who plays for 4 months and is basically Barry Bonds on a hot streak to end all hot streaks… he’s demonstrated the talent required more than say Fred McGriff in his career.

I think that a Bayesian style analysis is the way to go here. Start with the distribution of true talent wOBA over all players, as a prior distribution. Then add 10,000 PA at .380 wOBA, or whatever, and get a posterior distribution.

This avoids the problem of having to choose a baseline.

That sounds the same as SD from the league mean, implying a .340 baseline.

Can you work out an example of say someone who has a career 100 safe per 300 PA and 1000 safe on 3000 PA, where the true league mean is .3333, with 1 SD = .033?

I’ve put this formula into the Lahman database to see how it works for seasons post WWII.  I’m not sure if I buy it as a measure of greatness yet.

Heres how some players stack up:

Rudy Pemberton’s 1996 comes in at 3 SD, for 41 AB of just over .500 hitting.  This is comparable to the best seasons a guy like Brian Downing put up.

Kevin Maas’s 1990 is a 2.4 Those are the big fluke partial seasons that come to mind.

The top seasons are by Bonds, Williams, and Mantle.  McGwire 1998 is up there (8.6).  Norm Cash 1961 also passes the 8 SD mark (8.5), the best year by a guy not thought of as a HOF type hitter.

Back to Edgar, his 1995 season is + 7.0

Over his first 64 PA plate appearances of 2004, Barry Bonds had a wOBA of .740, which translates to +6.74 SD above a .340 wOBA.  This is about equal to the SD of a .380 hitter over 6400 PA, or 10 years.  That is much higher than, for instance, Jim Rice’s entire career.

But what about if you only looked at Rice’s peak, since he obviously had alot of filler seasons in there that simply drag him down.

Also, as noted, I think I prefer the .260 baseline (or thereabouts), to handle the Kevin Maas issue.  In fact, ALL hitters will have a Kevin Maas issue, since everyone will have had say a 100 or 200 PA string in which their performance was 5 or 6 SD above the .340 mean.  That’s why lowering it to .260 might make more sense, in that it rewards longevity.

As you can see by the manipulation of the baseline, you kind of are fitting the model to the data.  But, that’s really what we are trying to do.  We know greatness (Koufax, not Shane Spencer) when we see it, and so you want a model to reflect that.

If you are happy with the model, then you must accept its conclusions.

I just wanted to highlight this fantastic post by Blackadder at BTF:

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I was thinking about this last night, and I believe when you actually examine the math there is much less difference between Dan’s and Tango’s positions are not that different.

What, precisely, is Tango’s method? He starts with a hypothetical player, say, Joe Smith, of a certain level, say average, or replacement, or AA. Then he “rates” the actual baseball player, Tom Awesome, by asking how likely it is that Joe Smith could have put up the performance that Tom Awesome did. Since the probabilities are obviously going to be incredibly small if Tom Awesome is even a marginal hall of fame player, it is easier to express them in terms of standard deviations. In other words, Tango rates players by asking how many standard deviations their performance is from the mean performance of some baseline player.

The stat Tango uses is wOBA, which is basically just a version of OBA that gets the relative weights of the offensive events right. It has the nice property that it is linearly related to the number of runs a player adds, hence the number of (offensive) wins. If a player has a fixed wOBA and has PA plate appearances, the standard deviation of his sample wOBA is

SQRT(wOBA*(1.1-wOBA)/PA)

(I am not sure why it is 1.1 instead of 1, but I think it is because wOBA is rescaled.) To make things readable, let’s assume that we use the baseline of .300 wOBA, which is a replacement level hitter. Then if Tom Awesome has a given wOBA in a given number of PA, his Standard Deviations above what the .300 wOBA guy does is

(wOBA-.300)/(SQRT(.300*(1.1-.300)/PA))

Now, since we are only interested in comparing players, we can ignore multiplicative constants in this formula (they won’t change any ordinal rankings). In particular, we can ignore the 1/SQRT(.300*(1.1-.300)), since we are comparing everyone to a .300 hitter. The formula becomes

(wOBA-.300)*SQRT(PA)

wOBA-.300 is proportional to batting runs above replacement per plate appearance. This, in turn, is proportional to batting wins above replacement per full season, assuming seasons of fixed length. By taking any player’s fielding, positional, and baserunning value and “reassigning” it to his hitting, wOBA-.300 can be taken to be proportional to WARr, the players WAR per season. PA is proportional, by definition, to the number of seasons. So the formula becomes

WARr*SQRT(N)

Where N is the number of seasons player. Squaring this, which won’t change any relative rankings, we get

N*(WARr)^2

As Tango’s definition of a players value: the number of seasons played, times the square of his rate of accumulating WAR. But this is REMARKABLY similar to Dan R’s old salary estimator! Indeed, if you apply Tango’s value formula to each individual season (so that N<=1) and add up the results, you precisely get Dan R’s formula. Applying Tango’s formula to a player’s entire career, or prime, is not exactly the same as applying it season by season and adding; the later method will, for example, favor players who were more inconsistent in their value. Still, the errors are not that great.

So what Tango’s method with the .300 baseline does is essentially rate each season by the square of its WAR. Weighting seasons non-linearly in this manner has been fairly common in HOM discussions as way to seamlessly blend peak and career considerations. Dan R, as I mentioned, used an exponent of 2, and now uses 1.5. Joe Dimino, in his Pennants Added method, uses something more like 1.25. Although no matter the baseline Tango’s methods are always at root quadratic, they are well approximated by other exponents.

For instance, consider Tango’s method with a .260 wOBA baseline. .040 points of wOBA is about two wins over a whole season, so running through the same calculation as before with the new baseline we get the formula

(WARr+2)^2*N

for a players value. In the range [4,11], which is the relevant range of WARr values for HOF contenders (anyone less is probably not a reasonable candidate, and there aren’t many people higher) the function

(x+2)^2

is very well approximated by

4.5*x^(1.5)

Thus, the .260 baseline is very close to using an exponent of 1.5 to weight seasons, which Dan R currently does (there are other differences here; Dan currently considers in-season durability, which this does not). The .220 baseline is about an exponent of 1.2, which is not that far from Pennants Added.

Thus, instead of thinking “what is my baseline”, you can think “with what exponent do I weight each season”? Personally, I like the 1.5 exponent, or the .260 baseline. But for people who want to rank players solely by career WAR, there is a baseline (in fact, infinitely many) that, in the [4,11] WAR range, are well approximated by a linear function.

So Tango is not really that much of a heretic after all!

"(wOBA-.300)*SQRT(PA) “

Would then become:

PA/1.15*(wOBA-.300) * PA^(1/3)*1.15

That is, I am removing PA from SQRT(PA) or from PA^(1/2), and splitting it into:
PA* PA^(1/3) = SQRT(PA)

This term:
PA/1.15*(wOBA-.300)
is WAR

So we get:
WAR * PA^(1/3)*1.15

We can drop the 1.15 constant and we have:
WAR * PA^(1/3)

So, we need to get it to the power of 3, so:

WAR^3 * PA

And then you continue with your process.

I think that’s right.  Haven’t had breakfast yet…

Someone at BTF said that my process implied that Kerry Wood’s one-game performance meant he was one of the best pitchers ever.  And, I’m sure some similar batting performance (I dunno… Mark Whiten?  somebody anyway you can can come up with) would lead to a similar conclusion.

If we compare against the .260 wOBA level for a hitter, 4 Barry Bonds seasons would have the same significance as someone with a 5.715 wOBA in 5 PA.  The maximum wOBA possible (all HR, no outs) is 1.950.  How many PA would you need to get that? 52.  If you can hit 52 HR in 52 PA, that would give you the significance you need to proclaim that player as just being above the threshhold for greatness (HOF).

The equivalent for a pitcher is being perfect all season.

Clearly, this process will not allow anyone, not matter how great a performance, to be considered great after one fantastic unbelievable season.