Beating a Dead Horse, pt. 4

In the (finally!) conclusion to this series, I’m going to look at a couple of other rate stats that while not nearly as common as OPS, are amazingly common in their “invention”, if not in their use.

I refer to the multitude of statistics based on the number of bases a player has accounted for, divided either by his plate appearances or by his outs made. I am going to present a list of proposed stats based on this concept. Almost all of the ones I will list have been published in books or in SABR publications; it would be impossible to document all of the times that these types of metrics have been floated on message boards and other similar outlets. What is incredible to me about the reoccurrence of this idea is that most of the “inventors” seem to be utterly clueless that their great idea has been proposed countless times before. Barry Codell is the only one out there who can take credit for actually coming up with something new.

1) Codell published a piece on his Base-Out Percentage in a late 1970s SABR Baseball Research Journal:

BOP = (TB + W + HB + SB + SH + SF)/(AB - H + CS + SH + SF + DP)

2) Thomas Boswell, a columnist for the Washington Post, published his Total Average in Inside Sports:

TA = (TB + W + HB + SB)/(AB + W + HB + SB + CS)

3) Thomas Boswell quickly revised his TA, making it for all intents and purposes a knockoff of BOP:

TA = (TB + W + HB + SB)/(AB - H + CS + DP)

4) Breaking out of chronological order, Paul Adel posted on the internet about his Offense Ratio:

OR = (TB + W)/(AB - H)

5) Leo Leahy in his book Lumber Men introduced Bases to Out Ratio:

BTOR = (TB + W)/Outs

6) Lawrence Tenbarge, wrote in the 1996 SABR Baseball Research Journal about his Earned-Base Average:

EBA = (TB + W)/PA

7) Apparently unrelated, John McCarthy had his own EBA in his book Baseball’s All-Time Dream Team:

EBA = (TB + W + SB - CS)/(AB + W)

8) Stephen Grimble, in a book called Setting the Record Straight: Baseball’s Greatest Batters, used Base Production Average as a secondary measure:

BPA = (TB + W + SB - CS)/(AB + W)

9) Bill Gilbert has posted on the internet for many years now about Bases per Plate Appearance:

BPA = (TB + W + HB + SB - CS - DP)/(AB + W + HB + SF)

10) In their 1994 book, Essential Baseball, Norm Hitzges and Dave Lawson published TOPR (thanks to "Karmadrome" for the info, and see his comment below for additional details on Hitzges and Lawson's work):

TOPR = (TB + W + HB + SH + SF + SB - CS - DP)/(AB - H + SH + SF + CS + DP)

11) In the 2000 Baseball Research Journal, Mark Kanter wrote about New Production:

NewProd = (TB + W + HB + INT)/(AB + W + HB + INT + SH + SF)

12) I wrote this article sometime in late December or early January, as I am wont to do. Two weeks ago, another one emerged from Greg Raleigh at Dugout Central, “Bases Per Out”. This is amusing in its own right since that is the site where Mike Pagliarulo throws out barbs at sabermetricians. Anyway:

BPO = (TB + W + HB + SB + SF + SH)/(AB - H + CS + SF + SH + DP)

I documented all that not because it’s important, but because I find it amusing. None of these measures (with the exception of TA, which was carried in Total Baseball, first for all players and in later editions just for season and all-time leaders) ever caught on, and so countless people “discovered” them. Boswell himself is delusional about the history of the idea, as evidenced in a 2005 chat:

“What matters is that ONE of the "new" stats created in the early '80--Total Average, Runs Created, OPS, etc--all of which were different versions of my TA (which was first) has gotten general acceptance.”

Since TA was preceded by BOP in the bases/outs ratio format, Boswell’s claim is absurd on its face, but of course Bill James introduced RC in 1979 (the same year as Boswell, and presumably in the spring as well), and Pete Palmer developed OPS sometime in the 1970s--although I’m not sure when he first published it, it was before 1979. Also, RC is essentially a repackaging of Earnshaw Cook’s Scoring Index (1964) or Dick Cramer’s BRA (mid-1970s), or Dick Cramer’s BWA (mid-1970s as well). Boswell didn’t even invent bases/out, let alone beat sabermetricians to developing advanced measures of offensive performance.

Is there really anything worth arguing about, though? Are there multiple people taking credit for developing New Coke? We’ll examine the two variations on the idea, one per PA and one per out. Since we are comparing with OPS, I’m only going to consider categories that OPS considers, which means we can define our two stats, which I’ll call BPA and TA, pretty easily:

BPA = (TB + W)/(AB + W)
TA = (TB + W)/(AB - H)

Just looking at them, you can see right off the bat that walks are weighted equally to singles, which we know is incorrect, and that a homer will be worth four times a single, twice a double, … Neither the faulty total base relationship nor the walk = single contention has a counterweight, which is not the case in OPS, where the faulty total base relationship present in SLG is offset by the equal treatment of all on base events in OBA, and the equality of OBA is offset by SLG.

It should come as no surprise, then, that when we look at equations to predict runs from these stats that they are not as accurate as those derived from OBA and SLG:

aR/P = 1.51(aBPA) - .52 RMSE = 27.12
aR/P = 1.19(aTA) - .20 RMSE = 24.95
aR/O = 1.78(aBPA) - .78 RMSE = 31.10
aR/O = 1.42(aTA) - .42 RMSE = 25.66

Total Average does better than BPA at predicting both R/PA and R/O, so we’ll remove BPA from the discussion at this point. When we use TA as a run estimator to predict R/PA, or as a stand alone rate stat (which in part one of this series I contended obligates one to consider the R/O relationship), what are we saying about the values of the offensive events?

Knowing that the TA for our data as a whole was .658, we can work on both equations, starting with R/O, which can be rewritten as:

Runs = (1.42*aTA - .42)*(AB - H)*.172 = .371*TA*(AB - H) - .072*(AB - H)

There is no need to differentiate this, since by definition TA*(AB - H) = TB + W. So we have Runs = .371(TB + W) - .072(AB - H), which can be expanded out to:

LW = .37S + .74D + 1.11T + 1.48HR + .37W - .072(AB - H)

So TA severely underweights singles, while overweighting homers and walks.

Looking at the PA relationship:

Runs = (1.19*aTA - .20)*(AB + W)*.117 = .212*(TA*PA) - .0234*PA

Differentiating, we get:

LW = .212*(pTA + PA*dTA) - .0234p

Where dTA = (O*b - B*o)/O^2, where O = outs, B = bases (TB + W), o = 1 for an out, b = total base weight of any event, and 1 for a walk as well. For our dataset, this results in:

LW = .43S + .74D + 1.06T + 1.37HR + .43W - .09(AB - H)

Here, the hit values are more reasonable, but the single is still undervalued, and the .06 run bump it received also is parceled out to the walk, compounding that issue.

Total Average and all of its cousins are just too simplistic to provide a good model of offense. The idea of working from bases is not a bad one; while runs are the most fundamental unit to work with, bases are highly correlated with runs. But the problem is that TA focuses only on the bases that are gained by the batter, and not by the impact that the batter has on the baserunners. There is no difference between a single and a walk if the bases are empty; it is the fact that singles often advance runners by two bases, or advance runners when walks do not force them, that makes singles more valuable. TA disavows this basic knowledge, and thus the weights used do not reflect the true value of offensive events. If you could account for all of the bases that a player is responsible for (or estimate them), then you would have a much better foundation from which to evaluate them.

So please, please, stop “inventing” these statistics, unless you are twelve years old, in which case you may have a pretty good future as a sabermetrician.

Power: Who Needs It? (Or so we hope)

This Friday, the Ohio State baseball team will open its 125th season. February 22 is an odd date to start a baseball season, but at least the NCAA has finally decided that all teams have to wait until then. When a group of OSU base ballers hosted their neighbors from Capital University on May 21, 1881, they would likely have though the notion of February baseball played in Florida downright odd.

Anyway, the situation is what is for the teams of the north, and the Buckeyes navigated it quite well last year, struggling to a .500 record in conference play, sneaking into the Big Ten Tournament, and then proceeding to sweep right through in four games. With a twelfth Big Ten crown for coach Bob Todd ensured, OSU went to College Station to play in Texas A&M’s regional, where they acquitted themselves nicely, losing to Louisiana-Lafayette, beating LeMoyne, and bowing out against the hosts.

However, the late surge did tend to mask a regular season in which expectations were not met. The injury to #1 starter Dan DeLucia did no favors, but the real culprit of OSU’s average campaign was the offense. The Bucks allowed 5.3 park-adjusted runs per game versus the average of 6.1, but scored just 6.2. Ohio collected hits and walks at slightly above average rates, but hit for very little power--a .089 ISO vesus the conference mark of .114.

As we look ahead to 2008, things do not look better on that front. The Buckeyes lost catcher Eric Fryer (+10 RAA) and center fielder and leadoff hitter Matt Angle (.458 OBA, +20) as junior draftees, while DH and #2 hitter Jacob Howell (.371 OBA, +2) and middle infielder Jason Zoeller (+14 and 9 of the team’s 21 longballs) were lost to graduation.

As a result, the offense lacks punch even more blatantly than it did last season. Junior Justin Miller will move into Fryer’s vacated backstop slot; he hits for a high average but puts up little in the way of secondary numbers. A high school catcher, he will be a plus hitter at that position. His backup will be freshman Dan Burkhart, with sophomore Shawn Forsythe next in line if something should happen to them.

Miller’s position switch leaves first base open, and it appears as if true freshman Ryan Meade will be expected to win the job. The double play combination will be a pair of sophomores. Cory Kovanda took over second base midway through the season and played solid defense, although he was -8 runs offensively with a SLG of just .325; he did post a .352 OBA. Cory Rupert started the season at shortstop, then lost his job as he didn’t hit at all and struggled in the field. When he was reinstalled late, his fielding was much improved, but his overall hitting was -10 with just a .283 OBA and .303 SLG. I expect them to be as good defensively as any recent OSU keystone combination, and I wouldn’t be surprised to see Kovanda emerge as a solid on base guy.

At third base, sophomore Brian DeLucia, younger brother of Dan, will step into the starting role. In his limited time last year, he showed a solid glove and average bat in 47 PAs. He apparently has some power potential. Tony Kennedy, who started at the hot corner last year, could get some time there but will likely play the outfield.

The infield corners will be backed up by senior Chris Macke, who has hit decently in limited opportunities throughout his career. Other reserve infield roles will be filled by sophomore Ben Toussant and freshman Tyler Engle.

In the outfield, Kennedy, who posted a .397 OBA and +5 runs last year, seems to be in line to start in left field. Junior JB Shuck will slide over to center; he was +5 runs last year despite having a rough go of it late, and may be the team’s best offensive player. Sophomore Ryan Dew will be in right field, where he played frequently last year; he is probably the team’s best hope for a real power threat, although he didn’t show much in that department last year.

The top outfield reserve (and a likely DH candidate along with Macke) will be junior Michael Arp. Sophomore Zach Hurley can backup center, while classmate Chris Griffin and redshirt freshman Brad Brookbank look to get in the mix.

On the hill, DeLucia will be the ace, when he is ready to pitch after Tommy John surgery. Whether he will pitch like an ace or not is of course an open question, but the fifth year senior was a leading preseason candidate for Big Ten Pitcher of the Year in 2007. Unfortunately, Cory Luebke, who wound up winning the award, was drafted by the Padres. That leaves Shuck as the #2 starter (+5 RAA and a 6.56 eRA after a better freshman debut aided by a low BABIP). Big junior righty Jake Hale will be the #3; he was a starter as a freshman, and moved to closer last year before being put back in the rotation late. He pitched well with a 4.64 RA and +11, improving on a solid debut.

The #4 spot will be up for grabs; junior Josh Barerra (+2 in 8 appearances) would seem to be the favorite, as he was in the mix before an injury shortened his 2007 campaign. Eric Best, a sophomore lefty who pitched well out of the pen last year, is listed by the official preview as another candidate for the rotation along with redshirt freshman Dean Wolosiansky, leaving sophomore lefties Josh Edgin (-2 in 19 games) and Theron Minium (-8 in 13 games) in the bullpen.

Senior Rory Meister is back in the closer role; if he can find his command, he can be a stopper as he was as a freshman. Recently, though, he has been less than automatic. Sophomore righties Taylor Barnes and Brad Hays will also be in line for some relief appearances. Four true freshman pitchers will also be on hand, although it is unclear if any is expected to make an impact.

The non-conference schedule looks a little tougher than it did a year ago. The opening weekend, starting Friday, will be played in Tennessee, and the Bucks will face Arkansas State, Memphis, and Seton Hall. A week later, OSU returns to College Station to face Texas A&M, Louisiana Tech, and Arkansas for a meat grinder of a weekend. March 7 sees the team in West Palm Beach to take on Air Face, Maine, and UConn. The traditional spring break trip to Bradenton runs from March 15-21 against Bucknell, Dartmouth, Northern Iowa, Bradley, St. Louis, Army, and Kansas.

The home opener is Wednesday the 26th against Pitt. Other midweek opponents (all home) will be Toledo, Central Michigan, Louisville, Akron, Eastern Michigan, Marshall, and Buffalo (the home finale on May 13). The Big Ten season opens Friday the 28th with the Bucks hosting Penn St. Traditional foe Minnesota is in a week later, then OSU travels to Michigan St. The ridiculous front-loading of home games continues with Purdue, then Ohio goes to Northwestern and Michigan. The Big Ten home slate ends the weekend of May 9 with Illinois, and a trip to Iowa closes things out. The Buckeyes will not face Indiana in the regular season for the first time since 1943.

In summation, the Buckeyes should be able to put out an above average defense as usual. The success of the team will depend on whether the offense can score enough runs. There is potential there, and I am bullish on the multi-year outlook for the program, but it could be a big struggle to score baserunners this year.

Beating a Dead Horse, pt. 3

So far we have considered several different means of adding OBA and SLG together to measure offensive productivity. Now we will look at other means of combining OBA and SLG, namely multiplication.

OBA*SLG (OTS) is a relationship that has been known in sabermetrics for many years. It has been independently developed at least three times. The first was by Earnshaw Cook, who in Percentage Baseball and the Computer presented a model that was essentially OBA*SLG. Then Dick Cramer developed his Batter’s Run Average, which was published in SABR’s Baseball Research Journal at some point in the 1970s; BRA actually was defined as OBA*SLG. Cramer later developed this into Batter’s Win Average, which was based around a model of BRA*PA. The third is the most famous, the Runs Created formula of Bill James, originally written as (H + W)*TB/(AB + W). Given the assumptions about calculating OBA that I have been making throughout this series, RC is precisely equal to OBA*SLG*AB.

I am not going to go too in-depth in discussing the flaws of these methods, because I and many others have noted the flaws of Runs Created elsewhere. Briefly, RC attempts to model run scoring, but fails to take into account many logical properties that such a formula should have (Base Runs addresses several of these problems, but while less flawed, is also not perfect). (Basic) RC underweights walks and overweights all types of hits but particularly extra base hits. As a team scoring model, it is inappropriate for application to individual players, and because of the design shortcomings, can boomerang out of control for extreme environments.

Another OTS-based method comes from David Smyth, who has advocated the use of OTS*34 as a quick estimator for runs/game. You may look at that and wonder how multiplying by a constant helps anything.

First, we have to recognize what estimated unit OTS is expressed in. As you can see, RC = OTS*AB; so OTS is an estimate of runs/at bat. So Smyth’s formula converts runs/at bat to runs/game, and is essentially assuming 34 at bats/game.

The average major league game does have something around 34 at bats (for the data I have used throughout this series, the average is within .15 of 34). Of course, we know that the number of at bats any team gets will depend on their own BA and OBA. By holding AB/game constant, the formula is attempting to counterbalance the fact that OBA*SLG goes overboard in predicting the run creation rate of good teams.

When you apply the R/G estimate to actual major league teams by assuming 25.2 outs/game, the RMSE is 25.25 (I used OBA*SLG*1.36*(AB - H)). This is actually more accurate than Basic RC (OBA*SLG*AB), which comes in at 26.09, and presumably the benefits are greater when applied to extreme players.

Another twist on the idea of multiplying OBA and SLG is a method posted by “dq” on the Inside the Book blog. In it, OBA and SLG are each raised to powers, with the SLG power being dependent on OBA. This is designed to alleviate the issues caused by applying OBA*SLG to extremely high offense situations. OTSE, for “Onbase Time Slugging, Exponential” is defined as:

OTSE = PA*OBA^.85*SLG^(1-OBA/2)*.652

For the data we’ve been working with here, a multiplier of .668 will be used, since our OBA does not include HB or SF. This will still throw off the entire equation a bit, though, since the OBA version we are using is not the same as the one it was designed to work with.

Throughout this series, I have not discussed what happens at extreme levels of performance too much. Do not think for a minute that this is because I feel that the extremes are unimportant; I am always concerned about theoretical accuracy in addition to empirical accuracy. However, in the case of the cruder metrics being considered (OPS, OPS+, OTS, etc.), we can see their flaws even at normal levels of offensive performance. It is the better constructed metrics for which a more thorough investigation is warranted.

Of course, for a formula like OTSE, it is at the extremes where it shows its superiority. As shown in the link above, OTSE matches our expectations for linear weights in extreme environments better than the more simple OBA/SLG combinations. However, when used with average teams using OBA = (H + W)/(AB + W), it leaves a little bit to be desired on the linear weight level:

LW = .52S + .78D + 1.05T + 1.31HR + .36W - .103(AB - H)

The big issue is that homers are underweighted by around a tenth of a run. However, OTSE is a lot more robust than other OBA/SLG combinations.

With that being said, though, is it really worth it? The appeal of OBA/SLG combinations is rooted in the idea that the two stats are readily available. But when you have to resort to two non-linear operations, I think that any claim of “simplicity” is dead on arrival. If you do not consider OBA and SLG to be known, and have to figure them and then plug into OTSE, it is far, far more complex than Base Runs. Furthermore, OTSE is not be applicable to individual batters for the same reasons that multiplicative run estimators like RC and BsR are not. So you may as well just figure Base Runs for the team and be done with it. I fail to see any practical application for which you would want to use OTSE.

I did not give the OTSE linear weight formula before the results that it generates, because I was hoping that someone would still be around to read those. The formula for the OTSE weights will scare everyone that’s left off:

LW = .668*((OBA*PA*.81*SLG^(-.19)*dSLG + SLG^.81*(OBA*p + PA*dOBA))

Beating a Dead Horse, pt. 2

Let’s examine how OPS and OPS+ actually weight offensive events when viewed as approximations of R/O. We had this equation for runs from aOPS:

Runs = (2.06(aOPS) - 1.06)*(AB - H)*.172

If you remember that LgOPS is a constant .715 for our sample, we can rewrite that formula as:

Runs = .495*OPS*(AB - H) - .182*(AB - H)

Then, we can differentiate to find the linear weights for each event:

LW = .495(O*dOPS + OPS*dO) - .182*(dO)

Where O is outs, dOPS is the derivative of OPS with respect to a given event, and dO is the derivative of outs with respect to a given event (1 for an out, 0 for anything else).

dOPS = dOBA + dSLG, with dOBA and dSLG defined similarily, and calculated thusly:

dOBA = (P*n - N*p)/P^2 where P = PA, N = H + W, n = 1 for an on base event, 0 for an out, p = 1 for all events

dSLG = (A*t - T*a)/A^2 where A = AB, T = TB, t = 1 for a single, 2 for a double, …, and a = 1 for all events except walks (0).

If you go through this process with an average team in our sample, you will find that the equation for runs can be rewritten for the average team as:

Runs = .45S + .82D + 1.18T + 1.55HR + .23W - .08(AB - H)

From this, you can see how OPS treats the offensive events, at least for an average player or team. It underweights singles a bit, while doubles get a pretty fair value, but it overvalues the triple and the home run while undervaluing the walk.

We can do the same thing with the OPS+ equation:

Runs = (1.06(OPS+) - .06)*(AB - H)*.172

Which reduces to:

Runs = .563*OBA*(AB - H) + .466*SLG*(AB - H) - .193*(AB - H)

Which can be differentiated to:

LW = .563*(OBA*dO + dOBA*O) + .466*(SLG*dO + dSLG*O) - .193*(dO)

Which, when applied to the average team, gives us this:

Runs = .47S + .81D + 1.16T + 1.50HR + .26W - .087(AB - H)

You can see that OPS+ is an improvement over OPS, as all of the weights come more in line to what we would expect--but it is still undervaluing the walks and overvaluing super extra base hits.

There is nothing novel about exposing the fact that OPS leaves something to be desired--this has been shown by others before, they are still not awful for a quick and dirty estimate, etc. I do think it is useful to document what exactly they are. But this whole series is rehashing old news, and I do not want anyone to get the impression that I am saying otherwise.

Let’s talk about another way that people combine OBA and SLG. One very common formula is 1.2*OBA + SLG. The reason that a weight of 1.2 is used is that on the macro level, SLG is generally around 20% higher than OBA. This obviously is a relationship that varies, but for the long-term sample I have been using, the SLG is .391 and the OBA is .324--a 1.21 ratio. This in essence is a way to get to the insight of OPS+ without bothering with league averages--OPS+ uses the exact ratio for any given year, and the denominators aren’t the same if you expand them out, but generally, 1.2*OBA + SLG can be considered as similar to OPS+.

I saw a post on r.s.bb from some years ago that referred to 1.2*OBA + SLG as OPS*, so for the sake of completeness, let’s figure the regression equations for aOPS*:

aR/O = 2.12(aOPS*) - 1.12
aR/P = 1.79(aOPS*) - .79

Focusing on the r/o equation, knowing that the LgOPS* is .780, we can rewrite it as:

Runs = (3.26*OBA + 2.72*SLG - 1.12)*(AB - H)*.172

Which can be differentiated to:

LW = .561(OBA*dO + dOBA*O) + .468*(SLG*dO + dSLG*O) - .193*(dO)

As compared to the equation we got for OPS+:

LW = .563*(OBA*dO + dOBA*O) + .466*(SLG*dO + dSLG*O) - .193*(dO)

Confirming our premise that OPS* is a close cousin of OPS+ for average league OBA/SLG combinations. Obviously, the closeness of the relationship will depend on the league SLG/OBA ratio. In our case, it is 1.21, so they are nearly identical when we convert to runs. In modern leagues, OPS+ will usually be “better”, simply because the OBA/SLG ratio is higher than 1.2, and the more weight on OBA, the better (to a point, a point which OPS+ is a long way off from).

Some people have of course tried to figure the best weight based on approaches other than just evening out the contribution of each component on the league level. I’m going to run a regression here to do the same. I prefer an approach to try to find the best match for LW values, but this series is focusing on regressions. For the 1961-2002 sample, this is the equation we find:

aR/O = 1.376(aOBA) + .899(aSLG) - 1.273

This one can be rewritten as:

Runs = (4.247*OBA + 2.299*SLG - 1.273)*(AB - H)*.172

You can see that OBA is weighted at 4.247/2.299 = 1.85 times SLG.

Continuing to break this down:

Runs = .730*(OBA*(AB - H)) + .395*(SLG*(AB - H)) - .219*(AB - H)

Which differentiates to:

LW = .730*(OBA*dO + dOBA*O) + .395*(SLG*dO + dSLG*O) - .219*(AB - H)

Which gives us this equation:

Runs = .51S + .81D + 1.10T + 1.39HR + .33W - .102(AB - H)

These weights match our expectations better than any we have yet seen. The equation has a RMSE of 24.05, which is better than the aOPS and OPS+ based R/O estimators but not as good as the versions that estimate R/PA. So let’s take a look at a R/PA estimator:

aR/P = .902(aOBA) + .894(aSLG) - .795

Which can be rewritten as:

Runs = (2.784*OBA + 2.286*SLG - .795)*.117*PA

Here, the relative weight of OBA against SLG is 2.784/2.286 = 1.22. Remember, though, that this formula is attempting to estimate only R/PA. To convert to R/O, you would have to divide by (1 - OBA), which gives a lot more weight to OBA.

The version above, estimating R/O, attempts to be a final rate stat in and of itself, and in that case, the 1.8 weight on OBA is justified. But it is really more logical to first estimate R/PA and then convert to R/O. If you want a pretty x*OBA + y*SLG + z model, though, that approach doesn’t fit the bill.

Continuing, we have:

Runs = .326*(OBA*PA) + .267*(SLG*PA) - .093*PA

Which differentiates to:

LW = .326*(OBA*p + dOBA*PA) + .267*(SLG*p + dSLG*PA) - .093*p

Which gives these weights:

Runs = .52S + .81D + 1.10T + 1.39HR + .34W - .103(AB - H)

At this point, we have basically reached agreement between the two approaches (estimating R/PA and estimating R/O). Note that I do not actually endorse using these formulas to estimate runs scored; I am only using them as a means of discussing OBA/SLG combinations. Linear Weights and Base Runs, depending on the question at hand, are the only run estimators I endorse (I’m using “run estimators” to refer to those methods that can be easily written as a simple formula, not approaches like markov modeling, simulators, and the like).

As I mentioned earlier, regression is by no means the only (or best) approach to determining what the weights for OBA and SLG should be. Other people have looked at this in different ways. I will point you to one such study, Tango Tiger’s “OPS: Begone” series (part one and part two).

Let’s deal with one more OBA/SLG combination, Aaron Gleeman’s Gross Production Average. GPA = (1.8*OBA + SLG)/4. GPA starts with the 1.8 weight on OBA, which as we have seen is pretty reasonable, and then rescales the result by dividing it by four, making the scale resemble batting average.

I have previously criticized the use of this scale shift with EQA, but I do find it a bit more palatable in the case of GPA. I always think that it is best to have a statistic expressed in terms of estimated units, with the unit being something fundamental (I would consider things like OBA, R/PA, R/O, and R/G in this category). Batting Average is fundamental in the sense that it is still the statistic most in use among the general baseball public, it has had great relevance throughout history, etc., but it is not fundamental to understanding baseball. I would consider hit frequency (H/PA) to be a much more fundamental measure than the batting average.

So while I’m not fond of the BA scale personally, I understand that some people like it. The reason why I don’t like EQA goes deeper; it is that it cannot really be used for any mathematical comparison between players without serious adjustment. In the case of GPA, 1.8*OBA + SLG is roughly equivalent to R/O in terms of scale. A team whose 1.8*OBA + SLG is 20% better than another team figures to score about 20% more runs.

Dividing by four does nothing to eliminate this property, which I would call “ratio comparability” (see here). Dividing everything by a constant does not affect ratio relationships.

There is also “differential comparability”. An example would be that an OBA of .400, when compared to an OBA of .350, can be taken to mean not only “on base 14% more often”, but also “.050 more times on base per PA” (or 5 per 100 PA, etc.). OBAs are comparably by both ratio and difference.

GPA, on the other hand, is not differentially comparable. A .400 GPA is not .050 more anything than a .350 GPA, other than GPA points. It is a unitless measure. However, the same is true for 1.8*OBA + SLG. So the transformation to GPA does not remove a property that would otherwise be present.

If you look at the same questions for EQA, the conversion that Clay Davenport makes from estimated RC/out to EQA destroys both multiplicative and differential comparability that previously existed. That is why I dislike the EQA scale, but have only the mild objection to GPA that it is not expressed in a meaningful estimated unit.