THE BOOK--Playing The Percentages In Baseball

Two interesting things:

1. While HR stabilizes faster “per denominator unit” when the denominator is FB than contacted PA, it stabilizes faster in terms of elapsed time with the contacted PA.

2. I don’t believe the BABIP result, as it certainly looks inconsistent with the 1B and 2B+3B numbers.  It would imply a negative correlation between 1B and 2B+3B (I think).

Anyway, great stuff.

While HR stabilizes faster “per denominator unit” when the denominator is FB than contacted PA, it stabilizes faster in terms of elapsed time with the contacted PA.

And FB/CON stabilizes slower than HR/CON, so breaking it down into FB/CON and HR/FB is a net loss in terms of predictive power.

As for 1B being uncorrelated with 2B+3B - it should be a little, shouldn’t it? I mean, there are some hits where depending on how the fielder reacts and the runner’s speed could be either a single or a double; in theory, if you leg out more doubles your singles will drop, all else being equal. Obviously, there are countervailing pressures, but it doesn’t sound totally outlandish.

It just seems that the relationship should be more positive correlation than negative correlation (all things considered). 

But I could very well be wrong…

I’m interested in seeing the pitching results.  HR numbers will shoot way up, and the BABIP will shoot way way way up.

Yeah, BABIP was one that looked a little off to me initially, but I checked and double-checked and triple-checked and couldn’t find anything wrong with how it was being run.

Tango, I just took a look at batters with 300+ AB from 2001-2010, and the correl between 1B/AB and (2B+3B)/AB was -0.22.

Colin: fascinating!

I’m not sure the negative correlation btwn 1B and 2B is because a pool of BIP could (somewhat randomly) become either a 1B or 2B.  I think it has more to do with the fact that players tend to have one skill (hit singles) or the other (hit XBH).  Singles are also negatively correlated with BB and with HR/AB, while 2B are positively correlated with both.  Singles are strongly negatively correlated with strikeouts, while 2B have little relationship and HR have a positive correlation.  I think you’re just seeing two sets of skills (hit singles vs. power/BB) that are in fact negatively correlated.

I updated the above with Derek’s pitcher data.

By the way, I was extremely excited to see Derek’s number here:

Stabilizes Years Stat Denominator
3729 8 1B+2B+3B (BABIP) PA-HBP-K-BB-HR-ROE

Why?  Because a few years ago I said:
http://www.insidethebook.com/ee/index.php/site/comments/career_dips_numbers/

The halfway point (r=.50) is with BIP = 3700 or so. 

Has there been any discussion whether the number at which R=.5 is the best way to present this type of information? 

I believe a credibility standard is less prone to the types of misuse noted in the linked article.  Using something like either of the following would acknowledge that 99 is almost as good as 100.
1) Z = N/(N+K)
2) Z = square root ( N / # of N needed for 100% credibility)

where
Z = credibility to give to the observations
N = number of observations (PA or AB, etc)
K = a ballast factor (not perfect terminology here)

I have been pushing for the r=.50 for several years.  There are two reasons for this.  If I say “ballast” = 300 PA, then it tells me TWO things:

1. r=.50, when PA = 300.
2. I can calculate the correlation at any given number of PA by doing ballast / (ballast + PA)

If on the other hand you do r=.707, then you only learn ONE thing.  In order to figure out the r at other points of PA, you need to do extra work.

Furthermore, by stating the ballast number as r=.50, we can use it conversationally, like “half the performance is noise when you have PA = 300”.

I should note that I am aware pretty much everyone in this blog uses #1 from my post #11.  Just not sure if the Buhlman credibility approach was selected for any particular reason.

Buhlman?  Don’t know what that is…

Buhlman is the name for the Z = N/(N+K) credibility approach (also called least squares).  Textbook Buhlman uses the prior mean as the complement instead of a larger group, but there are many considerations to selecting a complement.

There is also the Classical approach (also called limited fluctuation approach and uses the square root rule in #2 from post #11), and a Bayesian approach.

I guess the question I’m trying to answer, is why select K such that that is the number of observations for the total to be .5? 

Maybe I am thinking about this incorrectly.  But my thinking is the value of K should be a ratio of how much variance there is between players and how much variance there is within an individual player if the complement is the league average.  In other words, is this measuring how quickly variance is explained for a stat, or how quickly the stat differentiates between players?

(note here that I haven’t ran any numbers so all the following examples are purely for demonstration purposes) Ground ball rate stabilizes faster in general than BABIP does.  Would the 0.5 benchmark be met by just taking a random sample from all players of 116 classifications to compare it to?  It would in all likelihood take more than 116 (maybe 150, maybe 300), but would it take the 1,126 observations that BABIP takes to get to 0.5, even if Year 1 in the Year to Year correlation was pure random from everyone?

If K is uesd to get how much credibility is given to the one player compared to the average, shouldn’t it account for how much different that one player is from the league average?

#16: your 2nd paragraph sounds right.

Let me make it clear however: we do NOT look for number of trials such that r=.50.  We look at various levels of number of trials to IMPLY r=.50.

So, you’d find that if your sample has 100 PA each, r=.33. If your sample has 200 PA, r=.50.  If your sample has 300 PA, r=.60.  If your sample has 400 PA, r=.67.

Each of these implies PA=200 r=.50. 

Of course, it’s never so clean.  But, that’s what we are after, by looking at various PA levels to get an implied r=.50.

I am missing where there is a distinction for the variance of the true talent between all players and the variance within the individual’s performance in this study.

The sample is split into two groups, right? What would it look like if the second group had the playerID completely randomized? 

My hypothesis is the R would still increase as observations were added for half or more of these stat categories even though we know there “should” be zero correlation between the two because we made the second group completely random.  I am guessing that it will still be implied that r=.50 at some PA threshold using this design.

You would get r=0

Thanks for the responses.  I appreciate the effort you put into making your comment section a valuable piece of the site. 

I think I see now why would get a terribly low r with randomized payerIDs with enough observations(because I would be trying to find a correlation between a random number and essentially a mean).  So that line of thinking won’t help me.

I am still trying to make it make sense for me as to why that is the proper value to use for K, though.  I think I keep trying to get something resembling an intercept coefficient the same time as the variable coefficient.

Maybe I am thinking about this incorrectly.  But my thinking is the value of K should be a ratio of how much variance there is between players and how much variance there is within an individual player if the complement is the league average.

That is what it is.  For example, the SD between pitchers for BABIP talent is about .0075.  The variance for a single BIP for an individual pitcher randomly drawn from the complement distribution is about .21 [p(1-p), where p is about .300 (average BABIP)].  .21/(.0075^2) ~ 3729.

In other words, is this measuring how quickly variance is explained for a stat, or how quickly the stat differentiates between players?

It measures how quickly the observed variance is explained as much by the variance between players as by the random process variance within an individual player, which more-or-less tells you how quickly you can distinguish between players.

If K is uesd to get how much credibility is given to the one player compared to the average, shouldn’t it account for how much different that one player is from the league average?

K is dependent on the complement (namely how much variance there is in true talent between players-i.e. the variance of the hypothetical means- and how much random variance there is for a randomly selected player-i.e. the expected value of the process variance).  In this case, the complement is the pool of players in MLB.  That complement doesn’t change no matter which player you select from the pool, so whatever you observe from that player doesn’t change how much weight you give that observation. 

If you wanted to assess something like how confident you are that the player is truly above average, then an observation further from average would lead to a more confident result, but as far as determining how much weight to give it in estimating the expected mean going forward, it doesn’t matter.

I have taken credibility theory and have a working knowledge of it.  However, I have never used this type of year to year correlation to get an implied K before.  If someone gives me a VHM and EVPV (or the underlying distributions) I can get a K for the N/(N+K) formula.

But, I am struggling to get this year to year correlation process to reconcile that for me (yes, I know the correlation results give the amount of variance explained by separating players).  Do the results correspond to what is believed to be the true distribution in all cases?  I tried to test this.

Using hitter’s BABIP (because I have that from Fangraphs already in a spreadsheet) as an example gives .21/1126 = .00019 as the SD. 
10-90 percentile range.
BIP of 50or more = .233 to .339
151 most BIP* = .270 to .343
BIP of 1 or more = .000 to .349

*that is about 350 BIP, which was about equivalent to 500 PA I believe.

The MLB BABIP was .296, the BABIP for this pool of 151 was .306 (I don’t have/didn’t use reached on error).  I plugged those into a normal distribution to give me the percentile range from the implied distributions.

Mean K 10-90 Range
.296 1126 .279 to .313
.306 200 .264 to .348
.296 100 .237 to .355

Obviously there is noise in these numbers, but it would be nice for them to match as well as possible.

The group of players who reach 350 BIP in a year are presumably more homogenous than the entire MLB.  I presume this is also true for the group of players who accumulated the necessary 1126 given as the linked result.  The group changes to be more homogenous, but the complement didn’t change to match the new group.  The complement is for a different group than what the K was computed on.

If K is computed using p(1-p) as EVPV where p = BABIP, and I compute the VHM as the variance of observed individual BABIP I get these K results for the following BIP thresholds
BIP = 1 … K = 94
BIP = 50 …K=144
BIP = 350…K=206

Those K numbers match the implied normal distributions with the observed distributions better.  So I looked at 2010 David Wright (because he was near one of the 10-90 range end mark and had a nice round N).

N...... 400.......400......400......400......400
K...... 94.......144......206......1126.....1126
Comp. 0.296...0.300...0.306...0.296...0.306
Actual 0.343...0.343...0.343...0.343...0.343
Cred.. 0.810...0.735...0.660...0.262...0.262
Result 0.334...0.331… 0.330...0.308...0.316

Maybe 2010 was a fluke and I need to look at more years and larger data sets, but this is why I am having trouble understanding and embracing this process fully.

You might like this:

http://www.tangotiger.net/solvingdips.pdf

Especially the second-half of that…

For hitter BABIP, .21/1126=.00019 represents only the variance of the hypothetical means, not the total variance of the observed means.  Over 350 BABIP, the total observed variance would be estimated by:

variance of hypothetical means + random process variance = total observed variance

The random process variance over 350 BIP with p=.306 is about 0.00061 [(.306*(1-.306))/(350)].  The total observed variance then is .00019 + .00061 = .00080.  Using the normal approximation with mean=.306 and SD=sqrt(.00080), the 10th and 90th percentiles are .270 and .342, which is a pretty close match to the observed.

Setting the variance of hypothetical means to equal the total observed variance implies no random process variance (since total observed VAR = VHM + EVPV), so K would end up being zero if you did that no matter what the variance of hypothetical means is.

I’ll eagerly await the pitchers, but what are the points for platoon drag for individual components (singles against lefties versus righties) and for parks (singles against lefties in Fenway)?

The pitcher update is in the main post at the top.