Equivalent Average - Unmasked

[Mephistopheles]

Hello there. You may have heard of me. My name is Mephistopheles. I post here at NSBB from time to time. My posts can generally be factored into three distinct types. The first are the posts where I am belittling a person without reason and not refuting my claim. Obviously, these are the posts that moderators hate. Obviously, these are the posts that da loungers love. Obviously, these are the posts that the idiots hate. The second type of posts are what I call the obscure baby-baseball posts. These are the posts about Japanese Baseball, MLB Draft, MiLB, Cuban Baseball and so on. Generally, you'll find a relatively tame Meph in these posts. A few moderators love them {1908, Dried Grapes, etc}. Loungers don't read them and neither do most of the idiots. Finally, there's the Mephistopheles that everyone loves. In these posts you'll find Meph teaching about all sorts of things. They tend to include things about sabermetrics, minor leagues, or both. In other words, in case you have not noticed they are generally epic and beautiful. The moderators love them. The loungers love them. The idiots don't read the because they have the attention span of a two year old. The group of others love them too, because they're smart enough to understand them. Generally these posts are not condescending. Being condescending while teaching isn't a really effective teaching method. That brings us to this topic.

 

Runs Batted In was created in the late 1800s. A few teams created the statistic to show how good they were. In fact, some sportswriters of the day realized it's inherent bias towards hitters in the middle of the order and disregarded it. The little guys with pointy hats and horse-drawn carriages knew what they were talking about. RBI would not surface as widely accepted statistic until after the dead ball era was over. Eventually it became THE way to grade an offensive players "production." We all know why it's a bad statistic.

 

Batting average has its flaws as well. If you go out on the street and ask someone what batting average is, they will respond with something sounding like this: How often a player gets a hit. Wrong. Batting average does not tell us how often a player gets a hit. It tells how often a player gets hit when while deciding to throw out some times he goes up to the plate for no reason other than we feel like it. It also fails to tell us to what type of a hit the player got. A single is not worth the same as a double. This is why we use on base average and slugging average. Then again is slugging average really any better? Well yes and no. It tells you the type of hit, but it still has the first problem of batting average. We're partitioning the times the player comes up to bat and excluding one for inherently biased reasons. Is on base average any better? It fixes the first problem, but fails to solve the second problem of batting average. It acknowledges all plate appearances, but it makes a walk and a home run equal.

 

We can sum on base average and slugging average for OPS, but then again who says that the relationship for that is better. Instead we can try to develop a system that solves both problems. Enter equivalent average. This post is going to describe anything and everything about EqA so you can come up with the exact EqA's BaseballProspectus comes up with. One of the criticisms for EqA is that BP develops it in a black box. No one knows how they arrive at it. They do spell out the method here. You can do all the things they do. You'll find out that the league leaders in EqA are generally around .300. BP's EqA leaders are generally around .350 or so. You can play around with the stuff in that article for days and never come up with anything remotely close to their EqA. Sorry. As TangoTiger put it: Opening up the black box will not cause a single dent on [baseballProspect's] bottom line.

 

What I am going to tell you is everything and why Baseball Prospectus is doing what they do. Unfortunately, Tango won't tell you why they're doing what they're doing, either - he says he does not know. It's rather simple. In fact it's essentially what people say mathematicians criticize sabermetricians for: Units. People who dislike sabermetrics generally say real mathematicians would hate their "work" because they shed units completely. This really isn't true. Everything in EqA is measured in relatively precise units that in the end cancel out leaving an answer in runs.

 

Now let's go on and attack the two major problems with oba, slg, and avg. We need to create some sort of rate statistic that includes getting on base and hitting for extra bases as well as stealing a base efficiently. The first thing that is calculated answers all of these problems in what they feel is the best way. We'll call this Raw:

 

Raw = (SF + SH + 1.5*BB + 1.5*HBP + 1.5*SB + 2*1B + 3*2B + 4*3B + 5*HR)/(SF+SH+BB+HBP+SB+CS+AB)

 

What is Raw measuring? It's essentially scaled bases per opportunities of moving up a base. Intuitively the idea that walks are worth more than sacs, but not quite as much as singles is good. Raw EqA addresses our two problems effectively, only adding in SB and CS, which can be described as a third problem with each oba, slg and avg. So in the end what does raw measure? Scaled Bases per PA+CS. It gives a numeric value of production. Now we can use Raw and convert it to runs. For a team we do this with this equation:

 

EqR = (Raw/LgRaw )^2* PA * LgR/LgPA

 

So what is EqR doing? It's measuring the relative production of the team divided by what an average team does squaring and multiplying it by PA and the runs per PA an average team scores. The squared term is based on the idea that the relationship between Raw/LgRaw and runs is not linear. This makes sense because when you add good hitters your other good hitters get more guys on base and each of their hits cause more runs. Now since we're looking at EqR on a team level and we want it on the player level let's look at that.

 

First, an assumption: The player in question is being analyzed by an average team in his home park. This assumption is needed to derive the equation most people see for EqR. Now, to look at the change in EqR for some change in Raw, take the derivative of EqR with respect to Raw. We get this equation:

 

dEqR = 2*Raw/LgRaw*PA*LgR/LgPA

 

Now we're adding some guy to this team, but a team only has nine slots it can play. So what are we doing? We're replacing an average player on this team and adding this players production. So basically we have our runs minus an average player's runs in the same PA. We're NOT measuring runs over an average player. We're measuring all of the runs created by a player. So our equation becomes:

 

dEqR = 2*Raw/LgRaw*PA*LgR/LgPA - PA*LgR/LgPA

 

Now we can factor out PA*LgR/LgPA resulting in the equation for EqR for a player you'll see at BP, only they drop the dEqR and call it EqR.

 

EqR = (2*Raw/LgRaw - 1) * PA* LgR/LgPA

 

Generally people look at that and say what the heck are they doing? Now you know why you're subtracting 1 and multiplying the ratio by two. Here is where we can multiply this by our park factor to normalize for parks, if desired. Now we want to scale EqR and to some rate statistic. What should we use? Outs of course. Why? Outs are the stopclock in baseball. We have 9 sets of 3 outs. We can bat as long as we want as long as we don't make those outs. So we decide to make our rate be something close to runs per out used. So then we get this equation, that you can find at BP, albeit not in the article I linked to regarding how to compute EqA (lol).

 

EqA = (EqR/Out/5)^.4

 

First let's analyze the "units". We have runs divided by outs, which is want we wanted. Pay no attention to the .4 right now. The thing that should cross your mind is what crosses everyone's mind: Why the hell do they divide by five? WHY? This is where (I think) Tango gets lost. This is where everyone gets lost. In fact if you follow the calculations done in this thread and divide by five you will won't get the EqA BP computes. This is the black box, so to speak. Remember, average EqA is supposed to be .260. If you plug all this in you'll get the league average to be about .266 or so, depending on the season. IT DOESN'T WORK. 5 is more or less a constant that forces the average to be equal to .260. How do we do that?

 

Well League average is going to be (LgR/LgOut/C)^.4. Since we want to "force" EqA to be equal to .260 for an average player, simply set that equation equal to .260 and solve for C. So C =(LgR/LgOut)/.260^2.5. This number tends to be around 5, ranging anywhere from 4.6 (Japan Central League) to about 5.6 (2007 AL). The 2007 National League was about 5.2.

 

And there, with the above information you can get the exact answers that BP gets for EqA and puts on their player cards. In fact, If you want to you can find out the park factors to extra digits. I've gotten to the point where the average "error" on the EqA I come up with is .000226 compared to their's. Remember that their EqA is the ring of integers divided by 1000. In other words: It's rounded after three digits. Theoretically, the average error in rounding then will be .00025, which is actually greater than the error I come up with.

 

So there you have it. EqA perfectly. Now go look up EqR on BP and you'll see this:

 

EqR = 5*Out*EqA^2.5

 

Oh and 1/2.5=.4, so solving that equation for EqA gives us the EqA=(EqR/Out/5)^.4. Look familiar? Oh, but now we're all smart enough to realize that the five isn't five.

 

So now if you have any questions or anything fire away. Nice Meph isn't eternal. This mood does not last forever.

 

 

And yes, in case you noticed LgRuns gets canceled out. If you plug in everything you get:

 

EqA = ((2*Raw/LgRaw - 1) * PA* LgR/LgPA) * Out * LgOut/LgR*.26^2.5)^.4

EqA = ((2*Raw/LgRaw - 1) * PA * Out * LgOut/LgPA * .26^2.5)^.4

 

When you ever want to scale EqA to some league average production based on runs, it's going to cancel out....which of course makes sense.

Edited February 13, 2008 by Mephistopheles

[seanimal]

Question #1: Why are free passes only x1.5 while singles are x2 in Raw EqA?

 

edit: higher opportunity?

[Mephistopheles]

A walk is not worth a single. For instance, runner on third two outs. Which is better? A single of course. The only time a walk is worth the same amount as a single is when there are no runners on base*.

 

 

*Unless it's the first inning, runner on third and your walk requires 104 pitches after fouling off so many from Johan Santana.

[seanimal]

A walk is not worth a single. For instance, runner on third two outs. Which is better? A single of course. The only time a walk is worth the same amount as a single is when there are no runners on base*.

 

I thought about that but it didn't seem right. I guess the reason I'm hung up is because the player doesn't contribute the runner that's on base, so if we're trying to mitigate environmental effects (park, league, team) as much as possible, a walk and a single are the same exact thing. But that's not right though, because we're pretending the team is average, not non-existent, so for all the situations where they're not worth the same, a single is more valuable than a walk. Am I getting there or am I still way off?

[Mephistopheles]

all players have ABs with runners on. in fact we assume that players have an equal proportion of their PAs as the leagues average for them. Ie if the league has 12 percent of PAs with a runner on first, we assume that our player has 12 percent of his PA's with a runner on first. So for those 12 percent of PAs, a single is worth more than a walk. Since we're mapping EqA to runs, we need to take in account for this.

 

In fact roughly a little under 50 percent of the PAs on average come with runners on base. So about half the time a walk is equal to a single, about half the time a walk is less than a single.

[haltz]

Since a HBP is worth slightly more than a BB which is worth more than an IBB, shouldn't it be separated out that way?

 

If so, have you worked on improving EqA? Is it just diminishing returns (or would it be a good thing at all) to make the values of the coefficients a little more precise?

 

Even though the 5 changes, how well does it handle different run environments considering, e.g., that the ratio of a walk to a single always stays the same?

[Mephistopheles]

Well the problem is what is the point of the stat? Is it measuring performance? Is it measuring value? Is it measuring true talent level? All three of these lead to different ideas on how to improve it. Adjusting for run environment for walks, singles, etc can be done through a park factor inside the Raw formula. If we're in coors, we'll see more runners on base so we'll see walk value fall. if we're in petco getting on base is at a premium so walks will become more valuable. if we're measuring true talent level, who cares? if we're measuring performance, it's kinda both. Yeah HBP/BB/IBB should also be factored out with different weights, but that's not what BP does. It can be improved on this way and probably by adding GIDP in the bottom.

[Mephistopheles]

dear vance

 

read this and fix my grammatical errors

 

signed meph

[seanimal]

a few typos, nothing that would make hardcorecubfan blush

[Rob]

a few typos, nothing that would make hardcorecubfan blush

 

i dnto thkin thsi wolud mkae hmi blsuh

[David]

This is awesome. Great job, Meph...

[vance_the_cubs_fan]

dear vance

 

read this and fix my grammatical errors

 

signed meph

 

I've read it one time through, but did not notice any grammatical errors. That was probably due to the fact that I was strongly focusing on the content. Excellent explanation, by the way.

 

In the first paragraph there are a few instances where your verb agrees with the predicate nominative rather than the subject. It's a common error and easy to fix. :wink:

[MSG T]

Dear Meph,

 

 

=D>

 

 

Thanks for the explanation. I always understood the stat in general, but not how they determined each of the values.

[Derwood]

I was told there would be no math

[Mephistopheles]

Dear Meph,

 

 

=D>

 

 

Thanks for the explanation. I always understood the stat in general, but not how they determined each of the values.

 

you can probably count the number of people who do on one hand.

[bukie]

Always seemed too convenient for the constant to normalize the data to be 5 exactly. Makes more sense that it isn't all the time.

 

Still don't like the way park effects are factored in, since in any player's home park, 90% of the league is facing the home team's pitching staff, which can throw off park averages. However, I'm not sure there's a better way to do it, since factoring in multiple years of home park effects is also flawed if the home park is altered in any way.

[Mephistopheles]

well coming up with acceptable park factors is another subject entirely. keep in mind that all of the analysis of statistics to runs dont include park factors (theyre using raw stats mapping to runs)

[wolf stansson]

Award winning post.

[jyoung55]

Forgive me if I made an error in understanding the math involved, but I have been curious about this since I looked up the raw formula for EqA the other day. If Raw = (SF + SH + 1.5*BB + 1.5*HBP + 1.5*SB + 2*1B + 3*2B + 4*3B + 5*HR)/(SF+SH+BB+HBP+SB+CS+AB), are you not actually ignoring Sacrifice hits and flies? Since SF and SH are both in the top and bottom half of the equation and are not affected by a multiplier, they seem to be surplusage. i.e. SF/SF = 1 and SH/SH = 1 and therefore do not effect the result.

 

Or is it that the bottom of the equation is actually SF and SH opportunities? If so, how is that calculated, since I don't believe its a stat that's regularly kept.

 

Thanks in advance for helping me understand this concept.

[bukie]

Forgive me if I made an error in understanding the math involved, but I have been curious about this since I looked up the raw formula for EqA the other day. If Raw = (SF + SH + 1.5*BB + 1.5*HBP + 1.5*SB + 2*1B + 3*2B + 4*3B + 5*HR)/(SF+SH+BB+HBP+SB+CS+AB), are you not actually ignoring Sacrifice hits and flies? Since SF and SH are both in the top and bottom half of the equation and are not affected by a multiplier, they seem to be surplusage. i.e. SF/SF = 1 and SH/SH = 1 and therefore do not effect the result.

 

Or is it that the bottom of the equation is actually SF and SH opportunities? If so, how is that calculated, since I don't believe its a stat that's regularly kept.

 

Thanks in advance for helping me understand this concept.

 

The mathematical reason that SF and SH don't cancel or factor out of the equation is that they are added into the numerator and denominator, not multiplied.

 

The logical reason...

 

The bottom of the equation (SF+SH+BB+HBP+SB+CS+AB) is just a fancy way of factoring in all plate appearances, and including all stolen base attempts. The top half of the equation is measuring the value of each possible outcome for those plate appearances/stolen base attempts. Negative outcomes (strikeout, hit-out, caught stealing) have no value, so they don't have a factor in the top half.

 

EDIT: It could be argued that hitting into a double play has a negative value, since it is a worse than zero effect, but it simply isn't factored into the EqA equation.

[Guest]

Nice write-up.

 

(Oh, and I like "Dried Grapes")

[DiamondMind]

Seriously, good job man. I read the entire post and have to say I learned quite a few things from it, which hopefully was your intention. Thanks for writing this!

[DenverCubs]

Forgive me if I made an error in understanding the math involved, but I have been curious about this since I looked up the raw formula for EqA the other day. If Raw = (SF + SH + 1.5*BB + 1.5*HBP + 1.5*SB + 2*1B + 3*2B + 4*3B + 5*HR)/(SF+SH+BB+HBP+SB+CS+AB), are you not actually ignoring Sacrifice hits and flies? Since SF and SH are both in the top and bottom half of the equation and are not affected by a multiplier, they seem to be surplusage. i.e. SF/SF = 1 and SH/SH = 1 and therefore do not effect the result.

 

Or is it that the bottom of the equation is actually SF and SH opportunities? If so, how is that calculated, since I don't believe its a stat that's regularly kept.

 

Thanks in advance for helping me understand this concept.

Someone already said this, but:

(SF + SH + 1.5*BB + 1.5*HBP + 1.5*SB + 2*1B + 3*2B + 4*3B + 5*HR)/(SF+SH+BB+HBP+SB+CS+AB) /=

(1.5*BB + 1.5*HBP + 1.5*SB + 2*1B + 3*2B + 4*3B + 5*HR)/(BB+HBP+SB+CS+AB)

In fact, since the numerator must be larger than the denominator in this case, adding SF and SH (essentially a constant) to both will always lower the value of Raw. Adding a constant to both the numerator and denominator will make your answer approach 1.

A simple example: 100 / 50 > (10+100) / (10+50)

I like that SF's and SH's lower your Raw, since you did after all make an out. It's a decent amount better than just any other out though, and this equation agrees.

Edited February 13, 2008 by DenverCubs

[Geech]

Huzzah! Now I can finally cancel my BP subscription!

[CubinNY]

EqA is like sausage, you don't want to see how it's made, you just want to bask in its goodness.