Tango on Baseball Archives

Converted OBA (December 15, 2003)

A HR has a run value of 1.41 runs from 1999-2002. The out has a run value of -.30 runs, and a safe play (hit, HR, walk, reached base on error) has a run value of .575.

Now, what if you had 2 PAs, one of which was a HR and the other was an out. If you had 2 other PAs that were both a typical safe play, how would these 2 compare?

1 HR + 1 out = 1.41 - .30 = 1.11 runs
2 safe = 2 x .575 = 1.15 runs

That's pretty close. So, in terms of having a run-effective OBA, we can convert say a hitter's 50 HR season by matching it with 50 outs, and getting 100 safe plays. (This reduces the SLG, and increases the OBA, while maintaining the exact same overall production.)

The conversion essentially becomes 2 safe plays = 1 HR, while maintaining the same number of PAs.

If we repeat this for triples:
2 3B + 1 out = 2 * 1.06 - .30 = 1.82 runs
3 safe = 3 x .575 = 1.72 runs

So, that's a little off, but let's go with it. So, this time the conversion is 3 safe plays = 2 triples.

Repeating this with all the safe events, and making sure it adds up exactly, we get the following conversion rates:
HR 1.95
3B 1.56
2B 1.22
1B 0.88
NIBB 0.72
RBOE 0.97

Why is this so important? One of the great things about OBA is that we can use the probability distributions with it. It's not so simple with non-binary events like SLG. Therefore, by converting everything into an effective OBA, this opens up alot of avenues for us.

Shortcuts
What if we had only OBA and SLG? As noted elsewhere, the best-fit is 1.8*OBA + SLG. So, if the OBA is .340 and the SLG is .410, if you want to convert that best-fit equation into an effective OBA, you simply divide the equation by 3. That is: (1.8 * OBA + SLG) / 3 = effOBA.

Does this have any basis whatsoever? Well, look at the single. In the above equation, it adds 2.8 "points" to the numerator. 2.8 / 3 = 0.93. The conversion value is actually 0.88, so that's pretty close. For the double it's 3.8/3 = 1.27, which compares favorably to the actual 1.22. The triple (4.8/3=1.6, 1.56) and the HR (5.8/3=1.93, 1.95) are also pretty close. The walk (1.8/3=0.6, 0.72) is a little off. As well, since OBA and SLG don't have the same numerator, it's not that simple to make the comparison in this way. But, it gives you some basis.

Conclusion
I am certainly not suggesting its widespread use, as I think that having the run-based Linear Weights or the win-based Linear Weights to be much better to describe what you're trying to measure. As well, since the LWTS values themselves are dynamic, and changes for every run environment, it's alot easier to set this with the run values, than these conversion values.

What is being offered here is a way to convert the various safe plays (hits, HR, walks, errors) into a common single safe play to be able to use probability distributions. And, what is also being offered is to do the same using only OBA and SLG.

--posted by TangoTiger at 03:40 PM EDT

Posted 4:08 p.m., December 15, 2003 (#1) - AED
Tango, this doesn't quite work out the way you think. The problem is that the variance of your effective OBA does not equal effOBA*(1-effOBA)/PA, which is necessary to apply binomial statistics.

Consider two extreme examples that can be treated with binomial statistics. Player #1 never swings the bat, so his at-bat results are either walks or strikeouts (or HBP, but we're not counting those). Letting "x" equal the fraction of plate appearances in which he walks, the variance in the walk rate equals x*(1-x)/PA. Since a walk is given a weight of 0.72 in your system, the player's effOBA equals 0.72*x, which means the variance of effective OBA equals effOBA*(0.72-effOBA)/PA.

The opposite extreme example is a player who swings at every pitch, and either whiffs or hits a home run. Again, the variance in his home run rate equals x*(1-x)/PA. Adopting the weight of 1.95 for a home run, the variance in his effective OBA equals effOBA*(1.95-effOBA)/PA.

Giving both players typical effOBA values of 0.340, the variance in the walker's effOBA is a factor of four smaller than the variance in the slugger's effOBA. Obviously the example is an extreme case that would never occur in reality, but it does illustrate the point that the variance will be higher for power hitters than for singles hitters, which violates the necessary requirements for use in a binomial distribution. (Another more straightforward problem is that there is a nonzero chance that a player's effOBA could exceed one, which would really cause problems with the statistics.)

Something along these lines can be used to improve EqA, although of course one might as well do it right if making the effort to compute EqA.

Posted 4:27 p.m., December 15, 2003 (#2) - Arvin Hsu
Tango: Why do you want to convert everything into a binomial distribution? I can understand an ease-of-use argument, but the multinomial isn't that much harder to use. And the multinomial describes all the events precisely, without the loss-of-accuracy, as long as you agree with the premise (which you have to in order to use the binomial as well).

-Arvin

Posted 4:28 p.m., December 15, 2003 (#3) - tangotiger
I would prefer to consider the limits to be Barry Bonds and a pitcher. Can you repost your thoughts under those realistic limits?

Posted 4:36 p.m., December 15, 2003 (#4) - tangotiger
I'm doing this to use in future articles. I think it's alot easier to show, for example, that Thome's effective OBA is .420, and Maddux's is .280 and come up with the matchup effective OBA, than to list the full line of the players, along with the expected matchup line, and then convert it.

I can do in 1 number and 1 line (and for 30 players), what would take me 10 times as much data to get virtually the same level of accuracy.

Posted 6:37 p.m., December 15, 2003 (#5) - tangotiger
This is the Ben matchup method:

=============================
B=Batter rate
P=Pitcher rate
L=League rate
B*P/L = X

Sum X for all the events, and divide the X for each specific event by the sum.
=============================

Now, here's some homework for those who want to get their hands dirty.

Method 1, the long accurate way:
Take 2 realistic players, and calculate their matchup lines using the Ben method noted above. Take that resultant matchup line, and convert it into the effective OBA as I have shown it.

Method 2, the quick and dirty way:
Take 2 realistic players, and calculate their respective effective OBA as I have shown it.

Then, use the Odds Ratio method (log5) and calculate the resultant matchup OBA.

How far apart can you get these 2 methods?

Posted 6:42 p.m., December 15, 2003 (#6) - AED
Here are some examples. I've used a Monte Carlo test to compute the variances per AB+UIBB, so it's imperfect but should be pretty close. I don't have ROE data handy, so I've limited the test to the other categories (singles, doubles, triples, home runs, walks).

effOBA variance smp.var player
0.495 0.388 0.250 Bonds
0.324 0.201 0.219 Pierre
0.279 0.260 0.201 T Batista
0.162 0.136 0.136 average pitcher
"variance" denotes the actual variance in effOBA using last year's stats as the model; "smp.var" denotes the variance one would assume using binomial statistics and effOBA as the rate. Variances are per plate appearance, so the variance in N plate appearances equals the value divided by the number of appearances.

As you see, the relationship between actual and assumed variances is not a simple function of effOBA, but depends on how much of the player's contribution to effOBA is a function of extra base hits.

Posted 6:56 p.m., December 15, 2003 (#7) - AED
I seem to have interpreted your intention of using this number with "probability distributions" literally. In the way your example would use this - an application that involves only the mean, not the variance - I suppose it should work out OK.

Posted 7:31 p.m., December 15, 2003 (#8) - Arvin Hsu
Tango:

Your Ben matchup idea is interesting. I played around with a matchup similar to this for real OBP a while ago. It used the same system as Chess ratings and Sagarin power ratings.

Essentially, all batter/pitcher matchups are equivalent to a chess match. Eventually, you should come up with a rating for each pitcher and batter. The higher the rating, the better the ability to "win." Win is defined as keeping players off base for pitchers, getting on base for batters.

Wikipedia has a good intro to the ELO rating system.
http://en2.wikipedia.org/wiki/Elo_rating_system

Once the true values are determined, batter/pitcher matchups should be able to be predicted with precise % success rates (eg. probability of getting on base).

-Arvin

Posted 8:21 p.m., December 15, 2003 (#9) - tangotiger
I'm trying to run some tests, and I don't think I'm liking the Ben Matchup Method right now. Here's my data:

player PA outs 1B 2b 3b hr bb effOBA
League 600 390 100 30 5 15 60 0.342
Hitter 600 375 100 30 5 30 60 0.391
Pitcher 600 375 100 30 5 30 60 0.391
matchup 600 351 97 29 5 58 58 0.476

All I did was transferred 15 outs into HR for both the hitter and pitcher. Now, look at the resultant matchup. It's far higher than I expected for the HR. And the outs look too low.

When you look at the effOBA, the resultant effOBA should be .442, which certainly looks better (at this level, a differential process, as opposed to an Odds process, should work out to pretty much the same).

I'm actually pretty much liking this effOBA approach. Kid has my mouse... gotta go.

Posted 1:15 p.m., December 16, 2003 (#10) - Arvin Hsu
B=Batter rate
P=Pitcher rate
L=League rate
B*P/L = X

So you did (.391)*(.391)/(.342) = .447??

Then you take the .447 and back-derive the constituent HR,BB,etc?

-Arvin

Posted 2:23 p.m., December 16, 2003 (#11) - tangotiger
That is the Odds Ratio (log5) approach.

*****

For the Ben Matchup method, you do that process for each individual component. So, you have 1b/pa, 2b/pa, hr/pa, outs/pa, etc, etc, and you follow that process to get the matchup line I presented.

That matchup line seems unreasonable, and I think the Ben Matchup method is questionable.

Posted 2:52 p.m., December 16, 2003 (#12) - FJM
You're right, it is unreasonable. A minor change to your example will show that very clearly. Instead of transforming 15 Outs into 15 HR's, change 15 SINGLES into 15 HR's. Now both the batter and the pitcher have the same BA (.278) and OBA (.350) as the league average. So we would expect those stats to stay the same when they face each other. But when you apply the Ben Matchup method to each rate stat and add everything up, you get a BA of .310 and an OBA of .379! The problem is, reducing the batter's singles rate by an additional 15% (from 85 to 72) is more than offset by doubling his HR rate (from 30 to 60).

Posted 4:56 p.m., December 16, 2003 (#13) - tangotiger
The only way to prove this is to use empirical data. Just because you and I think it's unreasonable doesn't mean that we're right. It would be interesting to see that if the specific matchups does result in what Ben VL says, or what my model says. I'll get to this at some point within the next few months (along with 100 other things).

Posted 4:57 p.m., December 16, 2003 (#14) - Arvin Hsu
Why do people think the League average should be involved at all?

Assuming true rates for each event, the batter and pitcher abilities should be all that is taken into account.

-Arvin

Posted 8:32 p.m., December 16, 2003 (#15) - AED
The reason your finding looks odd is that no MLB pitcher who faced 600+ batters last season gave up twice the league average number of home runs. Helling was the only pitcher in 2003 who faced 600+ batters and gave up more than 150% the league average number of home runs per batter faced.

A more detailed ability analysis appears to confirm the result from the "Ben matchup" system of around 60 HR in 600 PA. In a nutshell, you assume that the odds of a batter with home run ability of "X" hitting a home run off a pitcher with home run prevention ability of "Y" equals:
max * (1+erf(X-Y))/2
where max is the maximum rate for a great slugger and a horrid pitcher, and erf(x) is the error function as defined in C; (1+erf(x))/2 is the normalized probability from negative infinity to x. X and Y are computed from the observed rates, accounting for the distribution of opposing pitching/batting skills faced, park effects, and so on. (I'm leaving out a lot, of course, since it's sort of off-topic.)

The problem that FJM brought up is because "Ben's method" doesn't handle the rates correctly. You actually need to convert the line to rates, combine the rates, and convert back to a line. In other words, all plate appearances can be divided into walks, strikeouts, and batted balls, so there are three rates there (BB/PA, SO/PA, batted balls/PA). Apply Ben's method to the three rates, normalize to one, and distribute the 600 PA among the three categories. Batted balls can be broken down into home runs, outs, and non-HR hits (or a two-step process involving the HR rate and then the non-HR hit rate). Again, apply Ben's method to the three rates, normalize to one, and distribute the batted balls among the three categories. Finally, for non-HR hits, divide among singles, doubles, and triples (or a two-step process with the XBH rate and then the triple/double ratio).

I don't have time to run through it now using the original example and FJM's example, but this will fix any problem. I should note that the batting average from the matchup need not equal the league batting average, even if the batter and pitcher both have the league averages.

Posted 10:28 p.m., December 16, 2003 (#16) - Patriot
Arvin is probably a lot more knowledgeable about probabilities and the like then I am, but how can you not take league average into account? If you have a pitcher who gives up a .260 BA and a hitter with a .260 BA and they play in a league with a .250 BA, both BAs were compiled against lower BA opponents, so you'd expect it to go up when they face. But if the league BA is .300, then you'd expect it to go down. No?

Posted 9:14 a.m., December 17, 2003 (#17) - tangotiger
AED: unless I missed something, this IS what Ben does.

***

Arvin: the .250 OBA means something only in context to the talent distribution of the opponent. To get a context-free metric (some "inherent" talent):

.250/.750 = player
.333/.667 = league

player's "true talent" = player/league = .667

Posted 9:48 a.m., December 17, 2003 (#18) - tangotiger
Let me go through this step-by-step.

Ben Method:
Assume you have 600 PA, 440 outs, 100 hits, 60 walks as your league. Your pitcher and hitter are both 600,380,100,120. What's the resultant matchup?

On a perPA basis, these are the rates of the lg,hitter,pitcher:

outs H bb
0.733 0.167 0.100
0.633 0.167 0.200
0.633 0.167 0.200

For each one, you do H*P/L, so you get:
outs H bb
0.547 0.167 0.400

You sum this to get: 1.114

You divide the 3 numbers by 1.114, and then multiply by 600 PA to get:
outs H bb
295 90 216

So, that's your matchup line.

The Tango (Odds Ratio) Method

Convert everything into a the effOBA, so we have: .219,.291,.291

Turn all those rates into RATIOS, so we have: .281,.411,.411

Calculate the matchup ratio as H*P/L, so we have: .602

Convert the ratio into a rate (.602/1.602) = .376

That's my effective OBA mathcup rate. Ben has it as .391.

*********

If I were to swap the BB and 1B numbers, Ben gets an effective OBA of .425, and I get .404.

So, there's something also to the various "weights" I give them that carry extra information (that may or may not be correct).

Posted 12:06 p.m., December 17, 2003 (#19) - Arvin Hsu
Patriot:

You are correct. That's why I said assuming "true rate."
IMO, the best way to do a prediction is to first regress the observed rates. IOW, what you predict batter A and batter B's true rate will be in their next matchup. Then, you do the comparison, and you shouldn't care about league. (park or D may matter, but that's a whole other can of worms).

-Arvin

Posted 1:55 p.m., December 17, 2003 (#20) - tangotiger
Arvin,

But then what do you compare against.

I agree that you don't need the league if you have the true rate. But, to go from the observed rate (which is true rate of hitter against the true rate of the opposing league of pitchers and luck), you need to account for the league (as I show in post 17), and regress as you mention.

Now, we've got everyone on an "inherent" true talent level.

Then, you can compute the expected "true talent" matchup. But, you then need to convert that back to the league level to give it meaning.

I can have Bonds as 2.00 and Pedro as 0.75 matchup against each other, and I'll get a 1.50 matchup. But, that doesn't mean anything. The expected OBA and SLG will depend whether they face each other at Coors or the Astrodome, in 1994 or 1968.

I think we are all talking about the same thing, but for some reason, we are not.

Posted 2:02 p.m., December 17, 2003 (#21) - tangotiger
For those who may not be following, think of this.

The success rate of the SB is dependent on:
- runner speed, lead, jump
- pitcher pitch type, speed, move, handedness
- catcher arm, release, accuracy

So, all these things are context-free. You can measure each of these things, without accounting for anything else (let's assume all runners are only on grass surface).

Then, you can create probability distributions for all of these variables, and you can figure out the chance of Raines stealing a base against Pettite and Bench.

However, for hitting, it's not quite so easy. If you can "scout" a player to figure out his strike zone judgement, power, acceleration, etc, etc, and the similar for the pitcher and fielders, and add in some game theory, THEN you can do the same. But, for now, we can't. (This would be the holy grail by the way.)

Anyway, you can try to simulate all this by using the actual OBA record of the hitter against the league OBA of pitchers he's faced and parks and fielders, and after regression, end up with a "true talent" rate, similar to say a runner's speed.

Once you do this for all players, you can then create probability distributions to see how they would matchup. Again, you need a park, and that's another probability distribution. Then, you get your resultant matchup.

These last 3 paragraphs I'm simulating using the Odds Ratio Matchup method (and Ben is doing with his Matchup method).

Posted 2:03 p.m., December 17, 2003 (#22) - Arvin Hsu
it sounds like we are on the same page... but still not quite.
Once you regress, IMO, you're done. You don't need league anymore.
Of course, you can use park effects and defense and such, but that's different.
And we have a lot of practice at regressing to find true talent level, so I would use the work that's been done there already. Once you have the regressed true talent level for both batter and hitter, you can throw league out the window.

-Arvin

Posted 2:06 p.m., December 17, 2003 (#23) - tangotiger
Arvin, you probably posted at the same time as I did, but let me ask you a question:

How do you figure out the expected OBA of Bonds v Pedro at Coors 1994, and Astrodome 1980?

Posted 5:42 p.m., December 17, 2003 (#24) - Arvin Hsu
Aaah... I see what you're saying now.
For any given performance, there are multiple factors at work:
alpha = player's true ability
beta = park effect
gamma = year effect

For a batter v. pitcher matchup like we have, there are two alpha's, but only one beta and one gamma.

For events that are in the field-of-play, you also have
delta = defense effect (subdividable by type of play, if you wish)

-------------------------
Bonds v. Pedro
alpha1 = Pedro's true OBP
alpha2 = Bonds' true OBP
beta = Coors/Astrodome
gamma = 1994/1980

============================
My initial confusion was that I was thinking of true talent level as a true talent level estimated for a particular year(specifically b/c of aging curve effects). Thus, my erroneous assumption was that
Bonds' true OBP for 1994 would be a function of alpha2+gamma, which I thought of as one effect, but which in fact, is two effects. In some ways, this translates into a very weird concept of "true talent level." As of 2004, we can predict Bonds' true talent level, but it shouldn't be the best prediction of Bonds' performance(ignoring parks and pitchers faced). The best predictor would be a function of both Bonds' true talent level and a predicted league-year effect for 2004.
In essence, the value which we assign to each player as "true talent," has no direct correlate in measurable stats.

Posted 7:49 p.m., December 17, 2003 (#25) - FJM
Arvin: It's even weirder than that. If "true talent level" exists only in some context-neutral sense, with all park, league, and year effects removed, than what does it really mean, anyway? And if you want to somehow talk about how Barry would have done against Pedro at Coors in 1994, you first have to specify which Barry and which Pedro you are talking about. After all, both Bonds and Martinez were playing in 1994, presumably in Colorado, although not against each other. It might make some sense to ask how such a matchup would have turned out if it had taken place. But that's not what is being asked here. We're asking how the 2003 Bonds would have performed against the 2003 Martinez if both had been sent back thru time and space to Coors 1994. True talent level is constantly changing, and can never be totally separated from the context in which it was observed. We can measure it many different ways, making dozens of adjustments. But in the end we can never know it's true value. This is the sabermetric equivalent of the Heisenberg Uncertainty Principle. Asking questions like how many homeruns the 1929 Babe Ruth would hit today, training on steroids instead of beer and babes, may be fun. But if we take the answers too seriously we make a mockery of our own analysis. We have made great strides in understanding this wonderful game since the days of Bill James' Abstracts. Let's be mindful of one of his later warnings: This time, let's not eat the bones.

Posted 8:31 p.m., December 17, 2003 (#26) - tangotiger (homepage)
Yes, the true talent level means "nothing" and "everything". It's just a number from 0 to infinity that captures the player's true talent level in some context-free sense (like Raines' speed from 1b to 2b).

If you go to the homepage link, I showed what the "true talent level", in a context-free sense, would look like for all people. I set it to so that "1" meant a current MLB player, and the function essentially is at "2" for a Bonds-type player, and "0.8" for a top minor leaguer.

********

FJM: I didn't mean to consider the time as a chaning variable, though it certainly is. I just meant to use it to establish a playing environment that we can all picture.

Posted 8:31 p.m., December 17, 2003 (#27) - tangotiger
chaning=changing