## How, in the name of Barry Bonds, does EqA work?

I was browsing through the hallowed pages of Baseball Prospectus when I started to ponder the meaning and validity of various statistics. Sure your everyday fan knows what AVG/OBA/SLG is, but what about some of the more esoteric sabermetric measures? Now BP has more stats than La Russa has wins, but there is one particular perennial favorite that is worshipped above all others. It's the ubiquitous Equivalent Average, EqA for short. What I want to do in this article is closely examine EqA to understand exactly how the different pieces fit together and to analyse how well it reflects reality. The techniques that I outline in this article can be used for myriad other stats; and in a couple of weeks I'll be giving the same treatment to DIPS & FIP.

Let's start by ringing up the definition of EqA. Enter stage right the ever helpful BP statistics glossary:

EqA is a measure of total offensive value per out, with corrections for league offensive level, home park, and team pitching. EqA considers batting as well as baserunning, but not the value of a position player's defense. EqA adjusted for all-time also has a correction for league difficulty. The scale is deliberately set to approximate that of batting average. League average EqA is always equal to .260. EqA is derived from Raw EqA, which is (H + TB + 1.5*(BB + HBP) + SB) divided by (AB + BB + HBP + CS + SB/3). REqA is then normalized to account for league difficulty and scale to create EqA)

Note: this is EqA in its rawest form. A slightly more complex version is actually used by BP which adds sacrifice hits to both the numerator and denominator. We needn't worry about that here.

Simple, right? Let's try to deconstruct this a little. The first clause in the BP definition says that EqA is a measure of total offensive value per out. What exactly is meant by "value per out"? Well, the out is the currency of baseball. To win a 9-inning game of baseball a team has to get the opposing batters out 27 times. Another way of thinking about it is that 3 outs equates to an inning. So EqA is also a measure of total offensive value per inning, which is a little more intuitive.

Reading on we learn that EqA has been adjusted for league difficulty. This means that it is normalized for the park and other league specific factors such as total scoring rate. This allows us to use EqA as a tool to compare players across divisions, leagues and even eras. Moreover, we are told that league average EqA is always equal .260, thereby anchoring it to batting average so the casual fan can understand the context of different EqAs. Simplistically, .300 EqA is good, .220 EqA is bad. However, the aforementioned casual fan must tread with some care. The variance in EqA is greater than for AVG - if you look at the formula there are a bunch of extra terms which have higher weights - so an EqA of .400, though rare, does happen, which is not the case for AVG (witness my persistent bleating over the last two to three weeks)

The definition then gives the mathematical equation for unadjusted or raw EqA, which is:

```              H + TB + 1.5*(BB + HBP) + SB
EqA =         ----------------------------
AB + BB + HBP + CS + SB/3
```

To work out how EqA works what I want to do is take each term in turn and see how it contributes to a batter's overall EqA. Comparing this to the value of each event as determined by linear weights will allow us to work out how well EqA reflects the run value of the events it purports to model.

Let's start with the denominator. The sum of the first three terms is just total plate appearances. Next we have CS (caught stealing). If a player is caught stealing he costs his team an out, and his team is 1/27th closer to the end of the game. So by including CS in the denominator we are adjusting for loss of run value from the out, and we attribute that loss of value directly to the player caught stealing: he has after all cost his team a potential run. The final term in the denominator is SB/3 but given that this also exists elsewhere let's leave this until after we have evaluated the numerator.

The numerator, again apart from the SB term, is a measure of the number of bases the batter has gained. Actually it is remarkably similar to OPS except that for EqA BB and HBP are weighted slightly more than they are in OPS (where the numerator is: H + TB + BB + HBP). At first glance this is a little strange because it would seem to suggest that EqA values a walk more than a single. We know from linear weights that this isn't so. A single is worth 0.48 runs while a walk is worth 0.33 runs. This is because on a single base runners can and will move over, which is only the case for a walk when someone is on 1st. The answer is contained in the H + TB term. When a batter hits a single the value of the numerator is 2, which is more than the 1.5 that it is for a walk. Now back to our linear weights and we note that 0.48/0.33 ratio is not too dissimilar to 2/1.5. It turns out that walks are valued correctly in the EqA formula. Take another example: the triple. We know from linear weights that this is worth roughly 2.1 times a single. The numerator puts a value of a triple at 4 (1 for H and 3 for TB). Dividing this by 2, the value of a single, and we get 2.0, which is a reasonable approximation.

It turns out that the only offensive event that EqA slightly undervalues is the long-ball. Linear weights tell us that, with a run value of 1.4, it is worth almost three times a single. Yet EqA yields a relative run value of 2.5 (HR = 5; 1B = 2). Perhaps adding a HR term in the numerator would adjust for this. Anyway the difference this would make to EqAs is small.

We still have the SB term to contend with. Stealing has a run value of 0.2. How well does EqA model this? We see that an SB adds 1 to the numerator and 1/3 to the denominator. To understand what is going let's do a brief thought experiment. Imagine that a batter has hit a single putting him on 1st, at this point his EqA equals 2. Now, on the next pitch suppose he successfully steals. Total EqA for that "at-bat" is 2.25 (3/1.33). Without the SB the EqA would have been 2, so the difference is 0.25. Indeed, we can see that EqA models base running pretty well.

Recall the BP definition of EqA as total offensive value per out. We tried to think through the logic of this earlier, but having delved a bit deeper in to the machinations of the formula it is worth taking a second look. A player who doesn't get on-base is out and so the numerator registers zero while the denominator ratchets a 1. Ergo you can see the formula is measuring offensive value per out. Likewise with the CS term. This simply increases the denominator by 1, which registers another out. When a player makes a positive offensive contribution (eg, hits a single) the numerator increases by more than the denominator, essentially giving the batter offensive credit.

Theoretically, EqA is a good model of baseball offense but how well does it do in practice? Those kind folks at BP have, not surprisingly, done the leg work to show what an effective stat EqA is. If we measure the correlation to runs and the RMSE* of a whole bunch of wannabe run predicting stats then we see that EqA puts in a good showing:

Stat Correlation RMSE
Batting Average .828 39.52
On-base Percentage .866 34.16
Slugging Percentage .890 31.56
On-base plus slugging .922 25.54
Equivalent Average .928 24.13
BaseRuns .930 24.38
eXtrapolated Runs (per PA) .920 24.83
Runs Created (per PA) .928 24.96
Total Average .926 25.33

EqA is more effective than the far more common OPS. From the treatment above, which compares the two, this isn't surprising (EqA better weights offensive events). In fact we can see that EqA is on-par with other, more complex sabermetric measure of offense like BaseRuns and Runs Created. This is also no surprise. The premise on which all these stats are built is similar. They each try to accurately weight the various values of different offensive events to reflect their run value. EqA turns out to be a good combination of simplicity and efficacy. No wonder the folks at BP persist with it. It works.

*RMSE stands for Root Mean Square Error. A regression analysis fits a linear equation to a set of data. Because of random variation, this equation will not be a perfect fit. In other words there will be an error between the regression line and the actual data points. If you work out the smallest distance between a point and a line, square this (so it is always positive), add all these distance across all the points, and take the square root gives a measure of the amount or error in the model. This is known as the RMSE.

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