February 24, 2004

Baseball Prospectus Basics

About EqA

by Clay Davenport

Dayn Perry explained why various statistics--like batting average (AVG) and runs batted in (RBI)--were not as reliable as you've always been told, and why we at Baseball Prospectus don't use them in our analysis terribly often. Today, we're going to look into one of the statistics we do use: Equivalent Average, or EqA.

In its rawest state, EqA is a simple combination of batting numbers, not so very different from OPS:

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

Compared to OPS, it counts walks and HBP a little higher (at 1.5 instead of 1), it has stolen bases, and hits and extra bases are counted a little less (since they are divided by plate appearances, not just walks). What, then, makes EqA different from the other statistics? Simply put, it's more accurate, it's unbiased, and it models the scale of batting average, so it's easy for a new fan to understand.

Let us start with accuracy. Accuracy is traditionally measured in one of two ways--by correlation, and Root Mean Square Error (RMSE) against runs scored. Correlation is a statistical tool that measures how closely one set of numbers tracks a second set. It is measured from negative-one (-1) to positive-one (1); negative scores mean that when one number goes down, the other number tends to go up; positive scores mean that both sets rise and fall together. The closer you get to either end, the more perfect the relationship is, while a score close to zero means that knowing the first number tells you squat about the second number. You'll sometimes see people use r-squared instead, but that is essentially the same thing (mathematicians use "r" to stand for correlation). RMSE is just a fancy way to say how much you missed by, on average--it's a form of standard deviation. Statistics that have better correlation (closer to +1 or -1) usually have lower RMSEs as well.

Consider the following table of fairly traditional statistics.

			Correl	 RMSE
Batting Average		.828	39.52
On-base Percentage	.866	34.16
Slugging Percentage	.890	31.56

This shows how well the statistics have done for every team in history, from 1871 to 2003. In each case, I have compared the statistic relative to the league (team batting average divided by league batting average, for instance) to the relative run rate (team runs per plate appearance, divided by the league RPPA). Batting average has, truthfully, a very good correlation...it is just that on-base percentage is even better, and slugging percentage is better still. Combine the last two elements into OPS and the results get better still:

			Correl	 RMSE
On-base plus slugging	.922	25.54

This is pretty much it for advanced methods, since they all represent only a rather small improvement over what OPS provides. Still, some improvement is better than none, that leads to a variety of stats that have been called "best" by one person or another, stats such as:

Metric				Correl   RMSE
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

Here you see a big reason why we use EqA: because its ability to estimate runs scored from team and league data is unsurpassed. What the chart does not show is how these errors change over time. If you only looked at the years from 1971 to 2003, eXtrapolated Runs would have a virtually identical RMSE to EqA (20.98 for EqA, 21.06 for XR), while BaseRuns actually does a little better (20.77). However, if you look at the 30 years from 1871 to 1900, the same XR and BsR equations are getting more than 60% worse--their RMSEs shoot up to 34.16 and 33.41, respectively. EqA, in contrast, "only" loses about 50%, scoring a 31.69. EqA is less sensitive to the conditions of the times than many of the other metrics which have been "tuned" to fit recent performances, so it's especially good for historical performances.

(Aside for the technically interested: In all of the above, the formulas are limited to the same set of input statistics: at-bats, hits, doubles, triples, home runs, walks and hit-by-pitch, steals and caught-stealing. These are the basic forms; most of them, including EqA, have more advanced versions that count in things like sacrifices and intentional walks, and these can generally squeeze another run or so out of the RMSE. The RMSEs have been calculated using a best-fit relationship of estimated runs equals team plate appearances times league runs per plate appearance times (A times relative statistic plus B).)

All of this is intrinsic to the equation. The rest of what goes into EqA is what we, the users, force onto it. The first thing we force on--what nobody does for OPS, for instance--is to actually establish how to move between the rate statistic and the number of runs that come from it. Equivalent Runs is simply the number of runs that you get from a given EqA and plate appearances, and it goes up twice as fast as the EqA does. In formula form:

EqR = (2 * EqA/LgEqA - 1) * PA * (LgR / LgPA)

Equivalent Runs is tied to the league average runs for two reasons. One, it serves to reinforce the idea that everything is relative--that you cannot say for certain whether any statistic is good or bad, unless you know the average value. Secondly, there is always information in the league total that is not part of the normal statistical line--things like reaching on errors, balks, wild pitches and...well, you get the idea. All of the statistics in the chart above had the same data available, and none of them are able to put that information to use as well as EqA.

The second thing we force onto the EqA/EqR figures goes to the second point I made, way back at the beginning: biases. The two primary biases are the league offensive level and the home park. When the league offense is high, players can put up astronomical totals--but since everything is relative, the numbers don't lead to as many wins as you think. It is the same with home parks: a hitter-friendly park, like any park in Colorado, lifts both sides up, so that once again you don't get the kind of winning results you would expect from the production.

Since winning and losing are what the game is all about, we have to adjust for this if we want to have a good, unbiased statistic. We already know how many EqR a player has; think of that as runs scored. We can easily calculate how many runs an average player would have produced, if he played in this league, with this home park, and made just as many outs as the player did. If you think of that as runs allowed, then you can use some form of the Pythagorean formula to estimate a winning percentage for that player.

The nice thing here is that, no matter what the batting conditions in the league may be, the winning percentage of the whole (and of an average player) will always be .500. We could rate the player's performance entirely by this number, except that virtually nobody has any comprehension of how good a .600 winning percentage is in player terms. Sure, it is better than average, but is it league-leading material? Top 10? For that reason, we re-map the winning percentage onto a familiar scale: batting average. For all its faults, anybody who is even a casual fan has a good feel for what is good, bad, or ordinary in a batting average. The final adjustment is entirely to make it easy to tell how good it is:

EQA (adjusted) = [ (winpct)/(1 - winpct) ] ^ 0.2 *.26

By this formula, an average player will have a .260 Equivalent Average--always. Compare that to the all-time major-league batting average of .262. A .300 EqA (.672 WPCT) is almost exactly as common, historically, as a .300 batting average; a .400 EqA (.896 WPCT) represents the top-14 seasons in history. The .600 winning percentage I mentioned above would be reported as a .282 EqA--good, but not overly impressive. Easy to understand--the third stump in EqA's wicket.