The Account-Form Scoresheet

Cross-posted at Weekly Scoresheet, for which I still want your scoresheets.

Most scoring systems endeavor to keep a chronological account of the game (at least in theory--only the Project Scoresheet system is purely linear, but one can reconstruct the game in order from a traditional scoresheet). However, what if one was only interested in recording the basic box score statistics of individual players, and not about preserving the game play-by-play?

For starters, you could certainly save a lot of paper. You could eliminate a lot of codes--there would be no need to record wild pitches, passed balls, balks, and the like (remember, I said basic box score statistics). It would be possible to describe each batter's plate appearance in just a few characters.

As is the case with so many other baseball topics, one can turn to Bill James for a creative idea about how to do this. In the 1983 Abstract, he introduced what he called the account-form box score. It was a box score that retained much of the data from a scoresheet, yet was compact enough to be printed in a space no larger than that devoted to traditional box scores.

I recently threw together a new scoresheet that records only the information necessary to create an account-form box score of a game. I made a few tweaks to James' coding system, but for the most part I just used it as is.

First, the ways to reach base:

S = single
D = double
T = triple
H = home run
W = walk
I = intentional walk
B = hit batter
E = error
f = fielder's choice (at end of code)

Three baserunning events are noted, by use of a lowercase letter after the on-base code:

s = steal
c = caught stealing
x = other out on base (That is charged to the runner, like a pickoff or out advancing when not forced. If the runner is retired on a force, no "x" is added).

Outs are recorded in typical scorekeeping notation--a ground out to short is "63", a fly to right is "9", a strikeout is "K". Double plays get a D prefix--"D643". A sacrifice hit gets a b suffix (for bunt); a sac fly needs no modifier because it will involve a RBI. If a fielder's choice results in a batter-runner not reaching base (i.e. it is the third out of the inning, so the batter-runner never actually gains first), then it is recorded with a "f" suffix (for example, 46f).

When a batter gets a RBI for driving in someone other than himself, James uses a ', so D' is a RBI double. D" would be a two-run double. When three other runners score, an exclamation point is used, so a bases-clearing double is D!. Thus, a solo homer is just H, a three-run homer is H", and a grand slam is H! A sac fly to left is 7'.

In James' system, batter-runners that reached base and do not end up scoring are given lower case symbols...s, d, t, etc. If you are keeping a paper scoresheet in this style, this is problematic, as you'll have to do a lot of erasing. So what I did instead was embolden the codes of batter-runners who scored. When paper scoring, this is easily accomplished by tracing a pencil marking with black ink. When keeping a paper scoresheet, it is also helpful to mark off the innings, which I did with a number in the far left portion of the box.

The biggest drawback to the account-form box as James envisioned it was that pitcher records are hard to read. James simply noted the pitchers by the number of batters they faced each time through the order. Suppose for instance that CC Sabathia faced 32 hitters, Phil Hughes faced one, and Mariano Rivera faced five. James notation would be:
Sabathia (9995), Hughes(xxx1), Rivera (xxx32)

Sabathia faced nine hitters the first three times through the order, than five the fourth. Hughes faced no batters the first three times through, and one on the fourth. Rivera faced no batters the first three times, the final three on the fourth, and two on the fifth.

This makes it a chore to read through the account and tally up a pitcher's statistics (the same issue exists for batters of course, but it is a lot easier to read across a row to take in the result of three to six plate appearances than it is to read up and down column to take in the result of potentially dozens of plate appearances). There's no easy solution--the best option might be to just give each pitcher a basic line, perhaps in the format of (IP H R ER W K Dec), like:

Sabathia (7.2 8 3 2 2 6 W) or Sabathia (7.2, 8, 3, 2, 2, 6, W)

Also, I tend to think that a better notation to indicate batters faced for the above example would be something like:

Sabathia (1-32), Hughes (33), Rivera (34-38)

This does not solve the problem with reading a pitcher's performance I described above, and it may force you to count off by nine (9, 18, 27, 36, etc.) in order to figure out where to start reading the scoresheet to find a particular pitcher, but I think it's more intuitive and it saves a little space that is being given back by listing out pitcher lines. Of course, counting off by nine shouldn't be a problem for those versed in systems (such as Project Scoresheet) that use numbered boxes.

Another issue is how to list substitutions. When printing a box score, it is easy to insert an extra line for substitutes and indent it. But with a pre-printed scoresheet, it's trickier. What I've done is use letters (a, b, c, d, etc.) to show substitutions. The timing of the substitutions is noted by using the first time through the batting order that the player appeared. For example:

[a Giambi 4] indicates that Giambi pinch hit, with his first PA coming the fourth time through the order in the batting order slot where a is listed.
[d PR Guzman 5] indicates that Guzman pinch-ran the fifth time through the order.
[c 6 Everett 3] indicates that Everett was a defensive replacement at shortstop, and that his first PA (if in fact he got one) would come the third-time through the order. The exact time he entered the game defensively is not recorded, as it isn't on a typical basic box score.

This allows the scoresheet to be more compact, but it does make it a little tougher to quickly read batting lines for lineup slots in which multiple players appeared.

Honestly, I don't have a lot of use for an account-form scoresheet myself; it's not detailed enough for my tastes as a scorekeeper. It's the box score that I want--I want to be able to get on the internet in the morning and read a mini-scoresheet rather than a box score. Which line tells you more, and more importantly IMO tells you a better story:

63, K, H", 9, W

or

4 1 1 3 1 1
with a note below saying HR: Player X

or

4 1 1 3 1 1 HR
as the more helpful Baseball-Reference box score does?

It's true that the account-form style requires you to do a little counting in your head if you want to know H-AB, or runs scored, or the other box score categories. However, I'd argue that it's very easy to do that sort of addition, and that it also is a cosmetic improvement--there are an awful lot of zeros in box scores.

Incidentally, this is the same reason that all of the scoresheets I design myself omit columns in which to record batting lines. I'd rather devote that space to larger scoreboxes and do some math in my head than waste a bunch of space on columns I'll have to fill in to create a box score that I could just as easily read on the internet. Pitchers are a different story, as it is much harder to get a quick read on their performance as the number of scoreboxes under consideration is much larger (and more spread out) than for a batter.

Any site that implemented an account-form box score would instantly get all of my morning box score hits. I'm a little surprised that someone hasn't implemented it or something similar on online just to stand out, as there are so many places that you can go to get box scores. As it stands now, only Baseball-Reference offers something truly unique.

Below is an account-form scoresheet for Cincinnati @ Milwaukee, 4/10/2008, copied from a scoresheet I kept in a different format. I simply typed the account into Excel, although ideally I would have paper-scored it and scanned. It will be easier to read this way though. A drawback is that since I am using the cells in the spreadsheet, if a leadoff hitter scored, the inning number is in bold as well. I did not include pitcher lines, just the batters they faced:

Let me go through batter-by-batter and write what they did in English:

Patterson: grounded to second, grounded to second, grounded to second, grounded to second, flied to center
Keppinger: grounded to short, popped to short, grounded to pitcher, intentional walk, grounded to pitcher
Griffey: struck out, flied to center, walk, hit into double play
Phillips: flied to right, flied to right, struck out, grounded to second
Dunn: grounded to first (a pop to first would have been marked 3^; although James used 3 for a pop to first and 3- for a grounder to first), grounded to first (pitcher covering), single (scored), grounded to short
Encarnacion: grounded to third, walk (scored), two-run home run, single
Hatteberg: walk, single (put out on base), double (scored), grounded to second
Bako: struck out, RBI double, RBI single, walk
Harang: sac bunt to first, struck out, single
Votto: flied to center

Weeks: popped to third, flied to center, hit into fielder's choice at second, popped to second
Gross: double, flied to center, grounded to second
Counsell: hit into fielder's choice at short
Fielder: grounded to short, flied to left, flied to right, popped to third
Braun: grounded to short, popped to third, grounded to third, struck out
Hall: double (scored), grounded to short, grounded to short, grounded to second (via pitcher)
Hart: sac bunt to first, grounded to short, struck out
Villanueva: sac bunt to pitcher, struck out
Kapler: grounded to first (pitcher covering)
Kendall: grounded to short, single, single

Here is a link to download the account-form scoresheet in Excel format (scroll down to Account-Form scoresheet).

Great Moments in Yahoo! Box Scores

Hide your women and children...the R. Ortiz's are multiplying.

Great Moments in Yahoo! Box Scores

At first I thought they might be thinking of the Dodgers' other washed-up R. Ortiz, but he has a loss too.  And don't ask me which is which; Yahoo! box scores don't bother to elaborate.

2010 Predictions

As always, these are offered in the spirit of fun and not serious analysis. If you want the latter, there are plenty of smart folks in the saber-sphere that do W-L projections based on their projection systems. Even with fancy tools, the margin of error inherent in this exercise is enormous.

So I don't do anything special. I put together a very rudimentary spreadsheet and than I just make my best guess as to how the teams will finish and put it out on the internet for the world to see.

One thing to always keep in mind about predictions: picking a team first doesn't mean that I think they'll win. It just means that I think they have the highest probability of winning. If you don't understand what I mean, don't waste your time by reading on.

I'll keep the rambling introduction short this year and get to the point.

AL EAST

1. New York
2. Tampa Bay (wildcard)
3. Boston
4. Baltimore
5. Toronto

On paper, it's extremely hard to pick against the Yankees. They have the most potent projected offense and strong pitching to boot. That being said, any notion that the Yankees are some sort of unbeatable juggernaut is silly, as such notions usually are. It's an old team, there are multiple players that have noteworthy lingering injury concerns, and one wonders how sure even their brass is about the structure of their pitching staff. They are an easy choice as favorite for me, but not a super-team.

The Rays would probably be my pick to win any other division save the NL East and perhaps the NL Central, but we all know they're in a tough spot in the East. It would be foolish to write them off. The Red Sox have retooled around pitching and defense, but it seems as if the mainstream take is underestimating the offensive prowess of a healthy Beltre and Cameron. Picking them third is just a means to deviate a little bit from the conventional wisdom; they could easily end up with the best record in MLB. The Orioles are improving, and I expect big things from Matt Wieters, but it's not their year to get back in the hunt and even if it were, they'd be buried in this division. The Blue Jays may be a challenger...for the Royals, that is.

AL CENTRAL

1. Minnesota
2. Chicago
3. Detroit
4. Cleveland
5. Kansas City

My track record at predicting AL Central results is horrible. Some of it has to do with the fact that I'm a Cleveland fan, and am probably a little biased towards them in my picks (although when I've picked the Indians I've hardly been the only one doing so, I've admittedly tended to pick them whenever it is reasonable to do so).

Joe Nathan's elbow injury is obviously a blow to the Twins, and it does cut into the gap between them and the rest of the division, but I still think they stand out as the clear on-paper favorite. Chicago figures to have excellent pitching, but their offense is among the worst in the AL. The additions of JJ Putz and Juan Pierre and the subtraction of Jim Thome just serve to increase my (somewhat irrational) antipathy towards the team. The Tigers didn't seem to have much of a plan; apparently they really like Max Scherzer, because otherwise the Granderson trade is very puzzling given that they acted like contenders by signing Damon and Valverde (Just to make it clear, I don't have a problem with the trade for the Tigers, except that it doesn't make a whole lot of sense for winning in 2010). I've written about the Tribe in detail; suffice it to say, the pitching is a major weakness but they should score some runs and aren't as bad as many seem to think. The Royals...I don't even know where to begin.

AL WEST

1. Los Angeles
2. Texas
3. Seattle
4. Oakland

On paper, this should be the best pennant race of the year. It's easy to see any of the first three winning it, and Oakland can't be dismissed out of hand. I had a hard time deciding which team to pick, as the Angels and Mariners are just about even in my crude spreadsheet. On one hand, it's boring to pick LA; on the other hand, Seattle has gotten so much love from certain segments of the saber-sphere that it almost feels like groupthink to pick them.

I'm thoroughly unimpressed by the Angels off-season maneuvers and like just about everyone else, I love what Seattle did. However, the Mariners are still short on offense, and carrying Ken Griffey and Mike Sweeney does not help. Their spring training moves have not impressed as much as their winter deals, to say the least. The Rangers have high hopes, but I'd be surprised if they matched the 87 games they won last year; less could be enough, though, and while I would have pegged them third when spring training started, I've moved them up a notch. I'm probably a little behind the curve in admitting this, but Oakland does seem to be floundering about without a long-term plan. Whether they're in a holding pattern until their stadium situation is resolved, changing their minds on the strength of the rest of the division, or just plain confused is a different question. There's some talent here, but the offense also leaves a lot to be desired.

NL EAST

1. Atlanta
2. Philadelphia (wildcard)
3. Florida
4. New York
5. Washington

Just after the turn of the century, I was always picking against the Braves--it had gotten so boring to pick them that anytime I thought they had a halfway-serious challenger, I'd pick the challenger. However, in the last three seasons I've gotten in the habit of picking Atlanta when I probably should know better.

This season, my crude analysis indicates that the Phillies are the best team in the NL. But picking them would mean picking a World Series rematch, which is beyond boring. If my goal here was to be as painstaking as possible and strive for the highest possible chance of being right, I'd do it. But this whole exercise is for kicks and nothing else, so I'm going with Atlanta.

I hope it doesn't sound as if I think Atlanta really doesn't have a chance--of course they do, and they'd be my wildcard pick regardless. Their offense is still shaky (in Jason Heyward I trust, apparently), the back end of the bullpen is a ticking time bomb of age and injury, and trading one of the NL's top starters of 2009 may make business sense but it certainly doesn't improve their 2010 hopes. Meanwhile, the Phillies have gotten too much of a halo in my opinion from the mainstream based on their playoff successes over the past two seasons. They have been remarkably consistent since 2003 (86 wins, 86, 88, 85, 89, 92, 93), but 93 wins doesn't make you a super-team and I see no real reason to expect improvement. It wouldn't be shocking to see them on the outside looking in during October.

The Marlins are interesting in their own right; they could very well slip in as the wildcard or challenge for the division with the right set of circumstances. What's kind of funny about a team that's consistently had good young talent is they've never had that one magical year in which everything comes together and they win 95+ games--their highwater mark is still 92 in 1997. The Rays, Twins, A's, and Indians have all managed to put together at least one season of that type in the same time period. The Mets' offense should bounce back to some extent, but the starting pitching is awfully thin to expect a strong contender. The Nationals' 2010 should vaguely resemble the Orioles' 2009--hoping for mediocrity while the primary drama is the debut of a much-hyped prospect.

NL CENTRAL

1. St. Louis
2. Milwaukee
3. Chicago
4. Cincinnati
5. Houston
6. Pittsburgh

The Cardinals look like a clear favorite amongst a pretty motley group. Of course the proper candles must be lit for the health of Pujols and the twin aces. The Brewers look like a thoroughly average team, which makes them the safest pick for second here, but I think the Cubs might have a better chance of winning it. On the other hand, when your rotation includes Carlos Silva and your outfield is filled with two albatross contracts and a third that may end up as one, it's hard to be too excited. Reds fans seem to think it's their year; contention is certainly possible, particularly if St. Louis comes back to the pack, but this team is still a year or two away from being obvious contenders. The Astros are spinning their wheels as they have since they won the pennant, which may have been the worst thing that ever happened to them. They have the potential to be the worst team in the NL, but there's enough reliable-if-healthy veteran talent to make me conservatively pick them ahead of the Pirates.

NL WEST

1. Colorado
2. Los Angeles
3. Arizona
4. San Francisco
5. San Diego

This is another division where any of the top three winning would not surprise me in the least. The Rockies get the nod just to shake things up a little bit; I actually have Los Angeles ahead on paper. Colorado probably can't expect on repeat performances from Tulo and de la Rosa among others, and I'd like to take this opportunity to again advance my Free Chris Iannetta! campaign. The Dodgers are being written off all too quickly by a lot of mainstream writers; no, they didn't improve themselves in the off-season, but I don't see why their starting pitching is invoking hand-wringing. The Diamondbacks starting pitching looks shaky, but they have a solid offense and bullpen and with any luck should stay in the race. The Giants are seemingly an injury to one of the big two away from disaster; the incompetence in assembling an offense shouldn't be surprising any more, but it makes me shake my head. The Padres should challenge the Pirates, Astros, and Nationals for league cellar position--I'm looking forward to the potential of a Cory Luebke sighting later in the year.

WORLD SERIES

New York over Atlanta

It's boring to pick the Yankees, and of course I think the odds against any particular team winning the pennant are fairly formidable.

AL Rookie of the Year: 1B Chris Carter, OAK
I realize he's been optioned; these picks are anything but scientific and are more or less the first name that pops into my head.
AL Cy Young: Jon Lester, BOS
AL MVP: 3B Evan Longoria, TB
NL Rookie of the Year: OF Jason Heyward, ATL
It's way too easy to pick Heyward, but he'll have the hype machine on his side from the start if nothing else.
NY Cy Young: Yovani Gallardo, MIL
I hate making the obvious Cy Young picks, so I usually look like a moron.
NL MVP: 2B Chase Utley, PHI
The World Series seems to have finally convinced a lot of media-types how good Utley really is. Unfortunately, it may have come too late to earn him any hardware.

First manager fired: Ron Washington, TEX
Worst pennant race: NL Central
Best pennant race: AL West
Worst teams in each league: KC, PIT
Most likely to go .500 in each league: SEA, MIL
Team in each league most likely to disappoint mainstream consensus: DET, SF
Team in each league most likely to surprise mainstream consensus: OAK, ARI
Most obnoxious stories of the year: This is no longer fun to predict, as it's always and forever steroids. This year Mark McGwire as Cards hitting coach seems to be the leading cause for hand-wringing by the insufferable steroid crusaders. Others: Joba and Phil as relievers or starters, Jeter's contract, the Twins' closer situation, the various Twitter-fueled controversies that are bound to pop up (Ozzie and Oney could just be the beginning)
How stupid am I likely to look if one reviews these predictions after the season and ignores the disclaimers: Very

Esoteric Ramblings About +

Recently, Sean Forman changed the formula for ERA+ used at Baseball-Reference. It turned out to be a fleeting alteration, as for now the old ERA+ is back. There's been a lot written about the properties of the two ERA+ formulas, and I have nothing new to add, but that's never stopped me before.

Everything that follows takes ERA and ERA+ for what they are. I'm not going to get into why I prefer total runs to earned runs, or the virtues of DIPS-based stats, or any other topic. Just how best to compare a pitcher's ERA to the league average, implicitly assuming that is something one is interested in doing.

A Brief, Trivial History of ERA+

As far as I can find, the earliest reference to an adjusted ERA in the literature occurs in Pete Palmer and John Thorn's seminal Hidden Game of Baseball, circa 1984. In The Hidden Game, "Normalized ERA" is calculated as LgERA/ERA*100 (there is a park-adjusted version as well, but I'll sidestep the issue of park effects for the remainder of the post as they just serve to muddy the water even more, and the concept of comparing ERA to LgERA stands whether park is accounted for or not. I'll also ignore the scalar of 100 for the rest of the post because it is cosmetic and does nothing to alter the properties of the method, provided that it is corrected for when appropriate).

Interestingly, at no point in the book (at least that I could find through skimming) does Palmer explicitly explain why he chose to calculate LgERA/ERA rather than ERA/LgERA. There is a general explanation that normalized statistics set the average at 100 and above-average performances receive marks greater than 100, but the case of ERA is not given special attention.

In their later work, Total Baseball, the statistic was called Adjusted ERA but the ERA+ abbreviation was used (at least as of Total Baseball III, published in 1994). In the ESPN Baseball Encyclopedia, a collaboration between Palmer and Gary Gillette, it remains Adjusted ERA but with a new abbreviation (AERA); the names of most of the derived stats changed, presumably to avoid any sort of issue between the two works.

In his Historical Baseball Abstract (1985), Bill James uses what he calls "Percentage of League" as his main rate stat for pitchers. Percentage of League is RA/LgRA. Obviously, it differs from ERA+ by using total runs allowed, but more relevant for the purpose of this discussion is that it puts the individual's average in the numerator. James does not discuss the properties of one form against the other.

Properties of ERA+

Treating LgERA as a constant (and it is from the backwards-looking perspective of the historical record), the formula for ERA+ is:

LgERA/ERA = LgERA/(ER*9/IP) = (LgERA*IP)/(ER*9)

The denominator of ERA is innings pitched; the denominator of ERA+ is earned runs. This is a point that many people have overlooked (embarrassingly, I made this mistake myself in 2007, although I had come to my senses by early 2008). The practical consequence is that if you are averaging ERA+ across samples, you cannot weight it by innings pitched--you must weight it by earned runs.

An example is hardly necessary given that this point has been repeated ad nauseam, but consider a pitcher with a ERA of 3.25 in one season and 4.00 in a second season, both achieved in leagues with LgERA = 4.50 and both in 200 innings. His ERA+ is 1.385 in year one and 1.125 in year two. The natural inclination is to simply average the two and produce an ERA+ of 1.255.

However, his ERA for the two seasons combined is 3.625 against a league average of 4.50, producing an ERA+ of 1.241. The discrepancy is because the denominator of ERA+ is not innings.

In order to accurately figure ERA+ for the two seasons combined from the single-season ERA+s, one has to weight them by earned runs. In that case (we can use ERA as a stand-in for earned runs since we've specified that innings = 200 in both cases):

(3.25*1.385 + 4.00*1.125)/(3.25 + 4.00) = 1.241

Admittedly, the discrepancy in a case like this is quite small. Others might argue that weighting by earned runs is not that big of a hassle. However, once park factors are introduced into the mix, the task becomes trickier, as it's necessary to park-adjust the earned run totals being used to weight ERA+.

A common way to explain ERA+ figures in English is to say that an ERA+ of 1.07 means a pitcher is "7% better than average", or an ERA+ of .84 means that he is "16% worse than average". This usage doesn't really hold up mathematically, though. In a 4.5 LgERA context, an ERA+ of 1.07 corresponds to a 4.206 ERA. But 4.206 is 94.37% of 4.5, or 6.53% less. What ERA+ actually says, in English, is that the League ERA was 7% higher than the pitcher's ERA.

Using "better" and "worse" rather than the mathematically definable "less" and "more" is one of the causes of all the confusion. What does it mean to be 50% better with a 1.50 ERA+? 50% fewer runs (no, that's not it--that would be an ERA+ of 2.00)? 50% more wins than average? And if one pitcher is 50% better (1.50 ERA+) and another is 50% worse (.50 ERA+), then they should add up to be average, right? As was already demonstrated, they don't.

I'm repeating the takeaway from the second-to-last paragraph now, but I think it's worthwhile. ERA+ does not tell you that a pitcher's ERA was X% less or more than the league's ERA. It tells you that the league's ERA was X% less or more than the pitcher's ERA. If you want to claim that the league's ERA being X% more than an individual's ERA means that the pitcher was X% better than average, you can try, but it will require a very odd and very convoluted definition of "better" to hold up as logically consistent.

The end result is that while ERA+ ratios are meaningful, ERA+ differentials are not. Let's return to the two pitchers above, one with a 4.206 ERA and 1.07 ERA+ and the other with a 5.357 ERA and .84 ERA+ in a 4.50 LgERA context. The ratio between their ERAs is 5.357/4.206 = 1.274. The ratio between their ERA+ is also 1.274 (although we have to take Pitcher A/Pitcher B in this case because of the inversion of ERA+)--1.07/.84 = 1.274.

However, there's nothing you can do with (1.07-.84). Simple mathematical operations to compare the differences between ERA+s are impossible because there are no common denominators. (One might object at this point and say that because each batter has a different number of PA, OBA doesn't have a common denominator. That's not true, though, since anything per PA is a rate of batter performance per unit of playing time. While the PA for any two batters may not be equal, OBA remains the proportion of PA in which the batter gets on base. Using ER as a denominator, on the other hand, leaves one with a ratio rather than a rate--a ratio that, as Colin Wyers points out, is outs to runs rather than the much more useful runs per out that ERA itself returns).

There is one positive quality of ERA+ worth noting. If one wishes to convert ERA to an estimated W% for an individual pitcher (while ignoring factors like IP/G that will make team W% when the pitcher works in a game different), ERA+ provides a run ratio that can be plugged into a Pythagorean estimator. An individual W% can be easily figures as (ERA+)^2/((ERA+)^2 + 1). (ERA+)^2, therefore, is an estimate of individual win ratio.

Proponents of ERA+ have pointed out that ERA+ has a wider spread of observed values than aERA does. This is true, at least for above-average pitchers, but I fail to see why that's a positive if the numbers themselves don't mean what people often interpret them to mean. We could increase the spread even more by, say, cubing ERA+, but the results would have no mathematical or baseball meanings attached. ERA+ also depresses differences for pitchers with very bad ERAs, but of course this is brushed off because most discussion of ranking players focuses on the greats rather than the scrubs.

Runs allowed has a floor at zero; a pitcher can't possibly prevent more than 100% fewer runs than average.

Moving into the realm of pure opinion at this point:

Pros of ERA+ construction:

1. Inverted scale makes larger numbers better, which is aesthetically pleasing to some
2. Maintains ratio comparability
3. Corresponds to Pythagorean win ratio
4. People have become familiar with the scale

Cons of ERA+ construction:

1. Inverted scale wrecks havoc--makes earned runs the denominator quantity making averaging confusing, and makes differentials between ERA+s useless
2. Doesn't offer a coherent definition of "better" or "worse" to be put into English, and "less" or "more" cannot be used due to mathematical properties

I did ERA+ some favors there by splitting pros up as much as possible, while the inverted scale drawback could have been expanded into several associated complaints.

Properties of ERA/LgERA

Personally, I use RA/LgRA when I need a statistic in this family, and I just call it Adjusted RA. "Fancy Pants Handle" on BTF coined the name "ERA-" for ERA/LgERA, which is clever, but I'll just call it aERA for this post.

One of the most basic properties of aERA is that the lower it is, the better. The convention with adjusted statistics dating back to Palmer has been to make higher better, but I don't see why that needs to be the case. Lower is better for ERA and many other pitching statistics in their unadjusted form, and even the most casual of fans understands this.

An aERA of .75 means that the pitcher allowed runs at 75% of the league average, or that he allowed 25% less runs per inning than the league average. Unlike similar statements made with ERA+, these kinds of statements are mathematically accurate.

Ratio comparisons between pitchers remain workable. Going back to pitcher A with a 4.206 ERA and pitcher B with a 5.357 ERA in a 4.50 LgERA context, pitcher A's aERA is .9347 and pitcher B's is 1.1904. The ratio, .9347/1.1904 = .785, is the same as the ratio between their ERAs (4.206/5.357 = .785).

Where aERA distinguishes itself from ERA+ is the differential comparison, which is possible with the former but not the latter. The difference between the two pitcher's ERAs is 1.151 runs; the difference between their aERAs multiplied by the LgERA by definition is also 1.151 runs ((1.1904-.9347)*4.5 = 1.151).

aERA keeps innings pitched as the denominator, since aERA = ER*9/(IP*LgERA), so weighted averaging is achieved by using innings, which is far more natural than using earned runs.

The only mathematical property on which aERA arguably loses to ERA+ by my reckoning is conversion to estimated W%. ERA+ can be seen as a stand-in for run ratio, which when squared equals win ratio (using Pythagorean). aERA on the other hand serves as a stand-in for runs allowed ratio, and so W% = 1/(1 + aERA^2). Squaring aERA on its own produces not an approximate win ratio but the approximate win ratio of one's opponents, or the approximate loss ratio of a pitcher (in other words, an aERA of .5 means that a pitchers opponents are expected to have a win ratio of .25, or that he will have a .25:1 ratio of losses:opponents losses).

Properties of "ERA#"

A number of sabermetricians have taken issue with some or all of the questionable properties of ERA+ described above. In response to one of Tango Tiger's posts, Guy suggested an alternative to aERA that would retain the "higher is better" property of ERA+ but also retain some of the good properties of aERA. His formula was 2 - ERA/LgERA, which of course is equal to 2 - aERA.

This is the version that Forman briefly implemented on Baseball-Reference. Following the lead of BTF poster Greg Pope, some folks in the discussion there have taken to calling Guy's version ERA# to distinguish it from ERA+, and I'll use that naming convention here.

By subtracting aERA from two, ERA# caps itself on the high end at two. It also allows for negative numbers in cases where a pitcher's ERA is more than double the league average (ERA+ of .5 or less).

If you write it with a common denominator, ERA# = (2*LgERA - ERA)/LgERA. It could also be written as ERA# = 2 - ER*9/(IP*LgERA), which demonstrates that innings are the denominator as in aERA.

Like ERA+, ERA# is not able to sustain both ratio and differential comparisons between ERA like aERA. ERA+ sacrifices differential comparisons; ERA# sacrifices ratio comparisons. An ERA# of 1.15 means that the pitcher allowed 15% fewer runs per inning than average; and ERA# of -1 tells us that a pitcher allowed 200% more runs per inning than average. This property holds for all values of ERA#.

A natural reaction might be to wonder how that can be when ERA# is capped at two, but the cap at two is natural--one cannot possibly allow more than 100% fewer runs per inning than the league average. Negative runs may exist in our run estimators, but they do not exist in reality.

However, ERA# ratios are not comparable across pitchers. To illustrate, allow me to return to pitcher A with a 4.206 ERA and pitcher B with a 5.357 ERA, both in a 4.5 LgERA context. Pitcher A's ERA# is 1.0653, and Pitcher B's ERA# is .8096. The difference between their ERAs is 1.15 runs, which can be found by taking (1.0653-.8096)*4.5 = 1.15. But the ratio between their ERA#s is 1.0653/.8096 = 1.316, which does not correspond to the ratio between their ERAs.

Since ERA# does not retain ratio comparability, it cannot be plugged directly into Pythagorean to produce a W% estimate, which could be considered a drawback. However, as its proponents have pointed out, it does have a fairly solid relationship with 2*W% without making any adjustments at all.

Why does it work out this way? One way to demonstrate it is to use Bill Kross' W% estimator, which mathematically is equivalent to a linearization of Pythagorean with exponent 2 at win ratio = run ratio = 1 (in other words, a perfectly average team). I have written about his equation at (considerable) length before. It is:

W% = R/(2*RA) if R < RA, W% = 1 - RA/(2*R) if R > RA

The first equation, for R < RA, is the one that can be directly tied to ERA#. If you think about ERA as a stand-in for runs allowed and LgERA as a stand-in for runs scored, then 2*W% = ERA# using Kross' method (*). Of course, this is only true if R > RA, which for our purposes here is equivalent to ERA < LgERA. So for above-average pitchers, ERA# can be viewed as an estimate of 2*W%. For below-average pitchers, it's still pretty close--Kross' two equation solution ensures that W% estimates for a team and its opponents sum to one, but for normal situations, using one equation or the other does not cause massive distortions, so ERA# still approximates 2*W%. Of course, if someone wanted their adjusted ERA to track 2*W% for some reason, they could use two equations just like Kross does. For ERA < LgERA, you would use ERA# as discussed. For ERA > LgERA, one would mimic 2*(R/(2*RA)), which would simplify to R/RA, which using our stand-ins would simplify to...ERA+.

100% Opinion (as opposed to ~ 75% opinion in the rest of the post)

The major objections to the ERA+ change, as best as I can summarize them, are:

1. Familiarity with the existing ERA+

People have grown familiar with ERA+ as it is. They have a feel for how the all-time leaders stack up, perhaps they have explained it to their friends, etc. As a hard-core sabermetric sort, I can't really relate to this.

2. ERA+ distribution is comparable to OPS+ distribution

I'm not going to take the time to get into this right now, as this post is long enough as is, but these two statistics are measuring different things as is. I'm not sure why their scales should be expected to match or what it means if they do, without more context, and when you start adding that context you really shouldn't be using OPS+ anyway.

3. The new formula is less intuitive

I will admit that there is some validity in this objection, as it is marginally easier to figure LgERA/ERA than it is to figure 2 - ERA/LgERA, and the former is more intuitive. The problem is that once many people start trying to explain what ERA+ means, they demonstrate that they either don't really understand what it means or that they don't have a solid mathematical foundation for the claims they make. People are in the habit of saying that a 200 ERA+ means "twice as good as average", which is logically ambiguous and even worse mathematically. Very few people say that what it means is that the League ERA was twice the pitcher's ERA (or that the pitcher's ERA was half the league's ERA). What good is ease of computation if it comes at the price of easy explanation?

My own personal preference is for aERA, since I'm not bothered by lower being better and because, when possible, I prefer statistics that offer both ratio and differential comparability. However, if the adjusted ERA standard must display above-average as >100, then I prefer ERA#.

What bothers me most about the debate I've seen is not that people can't express what ERA+ really means or that fuzzy terminology is used, but the impression I get that some people have placed certain statistics on a pedestal and never want to see them change, even if the changes are sound. When sabermetric formulas are treated as inviolable constants, the entire purpose of sabermetrics goes out the window.

I don't mean to suggest that any alteration to a sabermetric statistic, no matter how marginal the benefit, should be implemented immediately and without discussion. That too would go against the spirit of the field. But it's important to remember that sabermetrics is always moving forward (or at least striving to move forward), and from time-to-time new and better methods will be developed. If and when there is a backlash to those newer methods simply because they are new, then ERA+ and OPS+ and the other statistics have simply become a new-age version of BA, HR, RBI, Wins--numbers celebrated for their own sake, rather than for what they tell us about the game.

Mathematical Miscellany

ERA+ = 1/aERA and aERA = 1/ERA+
ERA# = 2 - aERA and aERA = 2 - ERA#
ERA+ = 1/(2 - ERA#) and ERA# = 2 - 1/ERA+

(*) 2*W% = 2*(1 - RA/(2*R)) = 2 - 2*RA/(2*R) = 2 - RA/R ~= ERA#

2010 Indians: Bad (but not as bad as some think)

Even by the meager standards of my own writing, I am not good at writing micro- or trend analysis of players and teams (and even by the meager standards of my own post titles, this is a bad title). There will be dozens of better previews of the Indians' 2010 season that you can read elsewhere on the internet. What I hope to do here is express some broad thoughts on the team's outlook from the perspective of a fan without a tremendous amount of emotional investment.

Cleveland fans suffered through a second consecutive season of underperforming expectations in 2009, as a team that was expected to contend got off to a slow start, stumbled around for a while, and then fell all the way to sharing the cellar with Kansas City. This (along with the salary dump/rebuilding trades of Cliff Lee and Victor Martinez) has caused fans to take a pessimistic outlook towards the 2010 season and the organization in general.

Some of the frustration is certainly warranted--it is frustrating when a team trades away two homegrown Cy Young winners in as many years, and it's even worse when the team came within a game of the pennant just two and a half years ago. However, much of the pessimism is over the top.

To listen to the fans who call talk shows, the team will be lucky to match last season's win total of 65. Of course, such a finish is well within the confidence interval I'd set for the team, but I don't think it's a particularly good median projection.

Looking at the personnel, it's best to start with the team's biggest weakness: starting pitching. The good news about Cleveland's starting pitching is that there should be enough warm bodies to run out to the mound. The bad news is that the back-of-the-rotation options are nearly indistinguishable from those that would replace them.

The #1 starter will be Jake Westbrook, making his return after one and a half years on the mend from Tommy John. Behind him comes Fausto Carmona, still seeking to recapture the low-strikeout hard sinker magic of 2007. He's been pounded in every other major league stint of his career and in the last two has lost all semblance of command. The third starter is Justin Masterson, yet another sinkerballer, and one who has seen most of his major league appearances out of the Boston bullpen.

Westbrook, Carmona, and Masterson all have the potential to be valuable members of a rotation. But when they are the front three, there's cause for serious concern.

The fourth and fifth spots will likely go to Aaron Laffey and Mitch Talbot, besting David Huff. Jeremy Sowers' March injury left him out of the running and will likely provide an avenue to forestall a decision on his future (he's out of options), while OSU product Scott Lewis remains hampered by shoulder trouble. Carlos Carrasco should be second in line to get the call from Columbus, while the team will seek to get top prospect Hector Rondon another season on the farm prior to a possible September callup.

Laffey, Lewis, Sowers, and Huff (along with prospect TJ House) are all emblematic of what's become an organizational hallmark--the finesse left-handed starter. I certainly don't want to overstate their similarities lest the PitchF/x-ers correctly rebuff me with detailed analysis of the differing repertories of the group, but it's quite a departure from the teams of the 90s which had next to no left-handed starters (Brian Anderson led 1994-99 Indian southpaws with 17 starts).

The bullpen will be anchored by Kerry Wood, at least until he gets injured or is traded. (Editor's note: I wrote this about a week before Wood was officially pronounced out for eight weeks, but he was already behind schedule. Plus, seriously, it's Kerry Wood.) Chris Perez is penciled in as the top righty setting him up; his stuff continues to outrank his performance. If Rafael Perez can bounce back, he should combine with Tony Sipp to give the team a pair of lefties that can hold their own against right-handed batters as well.

Joe Smith and Jensen Lewis would appear to have the upper hand to gain middle relief jobs, but both had disappointing '09s and have not impressed to this point in the spring. Rule V pick Hector Ambriz is another option, and if Mitch Talbot fails to win a starting role he will surely be in the pen. Jess Todd will likely be sent to AAA, while Saul Rivera is a longshot despite his history with Acta in Washington. Jamey Wright may well win a spot as a veteran presence, even if a mediocre one; Jason Grilli was a prime contender for the spot but was injured and is out for the season. Josh Judy may be the nearest to ready among a group of young relief possibilities, but he's not pitched above AA and would it would be a surprise if he made the team.

My best guess (and it's just that) on the bullpen makeup is: Chris Perez, Rafael Perez, Tony Sipp, Jamey Wright, Joe Smith, Jensen Lewis, and Hector Ambriz.

The offense was essentially average last year, scoring 4.9 runs/game versus the 4.8 league average. I expect that the 2010 offense will perform similarly. Lou Marson should best Wyatt Toregas for the starting catching job, holding it down for a few months while Carlos Santana gets some AAA seasoning. Marson projects as an average-ish offensive catcher with the ability to draw some walks. Mike Redmond will serve as the veteran backup, and hopefully will not weasel his way into too much playing time at Marson or Santana's expense.

Russell Branyan, a late signing, will play first when healthy and drive fans crazy with his strikeouts. He also could spell Travis Hafner at DH if his shoulder does not allow him to resume everyday status. However, as of this writing Branyan has yet to make his spring debut and could open the season on the DL.

Second baseman Luis Valbuena showed surprising power spurts last year, but his fielding metrics and OBA leave lots of room for growth if he is justify his status as the centerpiece of the Franklin Gutierrez trade. At third, Jhonny Peralta should rebound from a dreadful offensive season to get back into the hunt for "most average player in the league" status. If he has any success, look for the Indians to move him at the deadline as they are unlikely to exercise his $7 million option for 2011. The team appears to be high on Lonnie Chisenhall and is positioning him as the third baseman of the future.

Asdrubal Cabrera will be under the microscope this year as he assumes leadoff duties. He has emerged as one of the better young shortstops in the game thanks to a good fielding and solid offensive contributions. His walk rate is not great, so if he has a BA dropoff he might become shaky as a leadoff option.

Thanks to the presence of Valbuena and Peralta (each of whom played shortstop last season) on the roster, it is not imperative that Cleveland carries a utility infielder that can play short, which could open the door for Mark Grudizelanek as utility-man and Valbuena insurance. Jason Donald, acquired in the Cliff Lee trade, will likely play everyday in Columbus, leaving former Met Anderson Hernandez as the top utility candidate that can play short. Brian Bixler is also under consideration but his inclusion would be a surprise.

Many fans want to see Michael Brantley in left field and leading off everyday, but Brantley projects to an ISO in the neighborhood of .100 and would probably be better served with another season at AAA. Depending on Branyan's status, he could open the season in Cleveland, but it's more likely that the team would go with a veteran in that case. Matt LaPorta originally appeared to be ticketed for first, but with the Branyan signing he figures to remain in left for another season.

Grady Sizemore looks to bounce back from an injury-plagued campaign, and this observer is still waiting for him to luck his way into a high-BA season that will make the average fan take note of what a great player he is when healthy. Sizemore will bat second this year, somewhat placating fans who still think he should be batting third. Shin-Soo Choo combines with Sizemore to give the Indians two of the better outfielders in the AL.

When Travis Hafner played last year, he rebounded to average production for a 1B/DH. Of course, it is not average production that the team is paying for, and 379 PA is far from full-time. He may be an albatross to the bottom line, but he wasn't an albatross to the lineup, at least in 2010. His largest cost to the team on the field was leaving them a man short every third day and a fielder short every day (the latter being expected for a DH-exclusive type of player like Pronk).

Trevor Crowe has a good shot at a reserve outfield spot as he can play center and could serve as a pinch-runner. Austin Kearns also has a decent shot to make it as a 1B/LF/DH and a right-handed bat in a lineup whose power threats are predominantly left-handed (Choo, Hafner, Sizemore). If I had to guess, I'd say that the bench is Mike Redmond, Anderson Hernandez, Trevor Crowe, and Austin Kearns. If Branyan is on the DL to start the season, add Andy Marte to that prognostication as I assume Brantley is headed for Columbus regardless. Marte had a superficially impressive season as a Clipper in '09 (.327/.369/.593), but didn't hit in the majors (85 OPS+ in 175 PA) and is now 26.

I'm not big on "best-case" scenario forecasts; they'd probably be better termed "90th percentile" or something along those lines. In any case, the best-case scenario I see for this team as the term is used by these types of previews is: Carmona, Westbrook, and Masterson are effective enough to drag the rotation close to average, while the bullpen is above-average and the offense pretty sold--enough to earn 86 wins or so with some magic Pythagorean dust the Indians have lacked lately and thus a team right in the thick of the AL Central race.

Worst-case? Disastrous pitching, continuing injuries for Sizemore, Hafner, and Branyan, and an epic anti-pennant race with the Royals.

Median? Solid offense, below average bullpen, and bad starting pitching leave the Indians at 74-88 or something in that neighborhood and a fourth-place finish several games ahead of Kansas City and a few games behind Detroit.

I actually happen to be fairly optimistic about the organization's long-term prospects--I generally think that Mark Shapiro's regime makes sound decisions (which is not to say that they are perfect, or to excuse the poor drafts the team has had). Chris Antonetti will become GM later this year as Shapiro is promoted to President (of the team, not the USA, although I doubt he'd be significantly worse in that capacity than the current occupant of the office), ensuring organizational stability for the next few years. Manny Acta got my tacit endorsement as a managerial hire. There remain legitimate questions about ownership's commitment to payroll, but I don't disagree with the decisions not to pay market value for Sabathia, Lee, and Martinez, so it's hard for me to criticize too much on that front, at least at this time.

Runs Anything and the Hall of Fame

I don't actually want to discuss specific Hall of Fame candidacies, but I will present career Runs Anything figures for predominantly post-1900 Hall of Famers (not including those selected in 2010, as I wrote this a week before the results were announced) and a number of players who often come up in Hall of Fame debates. While I don't care for many of the Hall's selections, using membership as a standard is a convenient way to keep the player pool to a reasonable size.

First, here is a list of the top ten Hall of Famers in R+. They are mostly the usual suspects, but I think there is one name that will surprise you--it certainly surprised me:



Frank Chance scored 60% more runs per out than the average player in his day. He did have a short career and he was on a historically great team, but his is still a surprising name to have pop up on a list of this type.

Here is a similar list for RBI+:



Willie Stargell is probably the most surprising name on the list (and perhaps Cobb if one fails to account for context in setting their expectations), but for the most part these are exactly the players you would expect to see: middle-of-the-order studs.

There are five players who crack the top ten in both R+ and RBI+ (Cobb, Gehrig, Hornsby, Ruth, and Williams)--it is thus no surprise that they make up the top five in Runs Anything:



A natural question to ask is "what is the degree of agreement between Runs Anything and a sabermetrically-approved context-neutral measure of the same thing, on the career level?" With the release of wRC+ on Fangraphs, it is very easy to compare ANY to such a metric. I did so for the Hall of Famers.

Of course, this is a very biased sample--it is only those players selected as the greatest of all-time, and there may well be a tendency for players who excel in R and RBI to a greater extent than context-neutral statistics to be voted into the Hall. The players in the sample all have long careers, there is a higher representation of players that excel in both runs and RBI than one would observe among the general player population, and there also bound to be more middle-of-the-order hitters, who are placed in a much better position to excel in both categories than leadoff hitters, for instance. wRC+ includes a park adjustment; ANY does not.

With all of the obvious problems with restricting the comparison to this group, why am I bothering? Because players of this group are the only ones for whom mainstream fans really examine career statistics closely. Ordinary players are hardly ever placed in historical perspective; the Hall of Fame debates are really the only time careers are given any sort of through examination. Ordinary players are talked about each season, but when their useful playing careers end, they are forgotten by and large. Few people are particularly concerned about Jeff King's career batting performance and how it compares to that of Al Martin. Those that are probably aren't using R and RBI anyway.

In fact, as a brief digression, the fact that HOF debates are the only time that players are really placed into historical context is what might make it seem as if I care about them. If you read this blog, you'll note a few posts tagged "Hall of Fame"--despite the fact that I do not think the current Hall of Fame structure is capable of being salvaged and don't care which players are chosen. But if someone like me (interested in both statistics and the history of the game) chooses to ignore the discussion surrounding the Hall altogether, they are basically shutting himself out from comparing players' careers. It's really the only time that the topic gets any sort of mainstream play (one could talk about top 100 lists, and comparisons between Aaron and Bonds when the home run record fell, or what have you, but those are in a certain sense subsets of the Hall debate--examining the records of no-brainer Hall of Famers).

If you want to skip the last two paragraphs, the takeaway is that I am only looking at the relationship between ANY and wRC+ for great players because that is the only situation in which ANY would be used in practice (in fact, it won't be at all, but that's besides the point).

So, for the Hall of Famers, the correlation between ANY and wRC+ is +.94. The average absolute value of the difference between the two is 7, and the average difference (ANY - wRC+) is -1 (in other words, the average player has a very slightly lower ANY). The players that ANY favors the most:



It's pretty interesting how close both rates are for the three catchers (Bench, Berra, Campanella); they all have ANY of 141-143 and wRC+ of 130. Traynor, Medwick, Klein, and Wilson can be tied into their own group as thirties hitters (although the others, as corner outfielders, form a tighter group than is made if Traynor is included).

The players for that ANY doesn't like:



A lot of high-OBA, top of the order types in this group, highlighted by Henderson and Morgan. It's interesting that two Hall of Famers who would probably be considered in the bottom half of the Hall show up here (Carey and Combs), and that a player often put forth as one of the very worst (Ferrell) is here as well.

The correlation between R+ and wRC+ is +.87; between RBI+ and wRC+ it is +.78. At least with this very biased sample, runs are more closely related to estimated context-neutral production than RBI. The next logical step is to run a regression to estimate wRC+ from R+ and RBI+ (then to never actually use it for a myriad of reasons, as this the type of mathematical gyration that would drive me crazy if it was being used for serious purposes):

wRC+ ~ = .616(R+) + .287(RBI+) + 14.7

This equation estimates that a player with average R+ and RBI+ would actually have a 105 wRC+, which is not quite satisfying; it also puts a double-weight on runs scored. We could try a regression without an intercept:

wRC+ ~ = .710(R+) + .302(RBI+)

This equation puts a R+ = RBI+ = 100 player at 101, which is more in line with what we'd expect, but produces less accurate estimates for this group.

To get back to the point of these posts, one can use runs scored and RBI to derive a decent estimate of a player's productivity, particularly over a career. I'd never use it and even when dressed up it retains the drawbacks present in looking at R and RBI in any event, but I think that it could have some limited persuasive value with mainstream fans if you could get them to accept the key premise (productivity must be expressed in relation to outs). Of course, if you can get one to accept that premise, they are probably open to further sabermetric indoctrination, and that would obviously be preferable.

This spreadsheet includes career totals in these categories for all of the predominantly post-1900 Hall of Famers as well as a few other selected players (the others are not in any way intended to be a comprehensive list of the most productive non-HOF players).

Runs Anything

In The Politics of Glory (later Whatever Happened to the Hall of Fame, although I still think the original title was much better), Bill James examines the history of the Hall of Fame and discusses some of the general arguments advanced on behalf of candidates.

After quoting a letter to the editor questioning Reggie Jackson's Hall of Fame worthiness due to his low batting average, James responds: "If you have to focus on one number, don't focus on batting average. Focus on runs--runs scored, runs batted in, runs created, runs produced, runs anything. Focus on what wins the ballgame."

For some reason the last phrase in his list--"runs anything"--has stuck with me since I first read the book. It seems like a great name to slap on a metric based on some tally of runs, but one the person writing about the metric doesn't actually believe in--and for that very reason I've used it here.

Let me accept the premise which is often either explicitly or implicitly put forth by non-sabermetricians, namely that estimates of runs (such as runs created) are unreliable and not based in reality. Ergo, if I want a metric expressed in runs for an individual batter, I will be limiting myself to those based on runs scored and RBI.

Let me also add an arbitrary constraint that I can only use R and RBI; I can't consider any other piece of statistical information (like home runs) in constructing the numerator of my metric. This allows us to sidestep the lively discussion about the merits or R+RBI versus R+RBI-HR.

Of course it goes without saying that I don't accept this premise, and you and I both know the myriad weaknesses of using runs scored and RBI to evaluate individual batters. I won't even insult your intelligence by listing them. Given these constraints, what is a reasonable (note that I did not say "the best", "ideal", or any of a number of possible phrases that would indicate I am taking this more seriously than I am) metric we can construct while still infusing it with some sabermetric-suggested properties?

There are three such properties that I think would be imperative in constructing such a metric. These are based on principles that are really not negotiable from my point of view; any analysis (even one constrained to using R and RBI) that does not consider them is one that is fatally flawed, IMO. Of course the points that follow are all obvious to you (although you are more than welcome to not care for how I express them):

1. You must consider runs scored AND runs batted in. Too often in mainstream analysis, runs are pushed aside while RBI are given all the attention. Of course driving in runs is just one side of the coin; someone must be on base to score them. The bias towards driving runs in seems to manifest itself most plainly in modern MVP voting, in which cleanup hitters with gaudy RBI totals are rewarded handsomely (see Juan Gonzalez, Ryan Howard, and Justin Morneau) while their teammates who get on base in front of them (Chase Utley and Joe Mauer in 2006) are ignored.

2. The proper opportunity factor for a rate of run production is outs. Certainly, we can complicate things by talking about the merits of R+/PA, RAA/PA, R+/O+, and the like, but that's beyond the scope of an exercise limiting one to only considering R and RBI, and really is a topic that has more theoretical than practical implications in any event.

It is imperative to consider the batter's role in using up his team's opportunities by making out and creating additional opportunities for his teammates by avoiding outs. Any attempt to quantify offensive performance that doesn't account for this is woefully incomplete.

3. When comparing players across eras, it is necessary to consider the win impact of a player's run contribution. Runs are not equally valuable in all contexts; the more runs are scored per game, the less win impact each run has.

I have not considered park factor here; after all, a metric of this type is designed for non-statheads, the effects are of a lesser magnitude than league scoring level, and it's way too much precision for such a clumsy metric.

So, here are the metrics I'm going to use:

R+ = (R/Out)/Lg(R/O)*100
RBI+ = (RBI/Out)/Lg(RBI/O)*100

Where outs are simply AB-H; again, we really should consider caught stealing at the least, and could consider the ancillary outs as well. I'm going to error on the side of simplicity given the inputs.

To combine R+ and RBI+ into one number, I simply average them. League RBI usually are around 95% of league runs scored in recent years, as of course RBI are not credited on every run. Averaging R+ and RBI+ counts them both equally, and I'll call this figure Runs Anything:

ANY = [(R+) + (RBI+)]/2

The result is a figure similar in scale to OPS or wRC+ and similar metrics--efficiency of run production expressed as a ratio to the league average, with the decimal point discarded.

I have also figured a runs above average figure, which I call ANYA. It is figured as the average of runs above average and RBI above average, which admittedly gives a little more weight to runs scored for an average player. I have not bothered to convert to win value; that is partially accounted for by figuring runs above average, but not completely, which is why sabermetricians are so fond of converting to wins when comparing players across divergent contexts. However, that is too much precision for me to worry about in such an inherently imprecise metric:

ANYA = average(R - Lg(R/O)*O, RBI - Lg(RBI/O)*O)

Let's look at an example that I mentioned in passing earlier through the Runs Anything framework: Ryan Howard and Chase Utley in 2009. Howard finished third in the MVP voting, while Utley languished back in eighth place. (Context-neutral) sabermetric measures are unanimous in evaluating Utley's performance as being more valuable to the Phillies than Howard, but Howard drove in 141 runs to tie for the NL lead while Utley drove in 93. I obviously can't identify exactly what the thought process was for the NL MVP voters, but it seems like a safe bet that the 48 RBI gap between the two was a key factor in their placement.

Utley scored more runs than Howard, although the gap is only seven (112-105). So by any kind of "runs produced" metric, regardless of whether you take out homers or not, Howard ranks well ahead of Utley.

However, what that analysis does not consider is that Howard made 34 more outs than Utley (444-410). The NL averages were .176 runs and .168 RBI per out, which means that Howard's R+, RBI+, ANY is 134, 189, 162 and Utley came in at 155, 135, 145.

This is a useful illustration, as we still get the "wrong" result even when using ANY. ANY does not adjust for the way batting order position affects R and RBI production, and the other pitfalls of using actual R and RBI counts are still in play. Still, the gap between 162-145 is a lot less than a cursory look at R and RBI totals for Howard and Utley would indicate. And of course the analysis ignores the difference in fielding value between the two. Even if ANY captures the true offensive value of the two players, the slick-fielding second baseman is going to come out on top when defense is taken into account.

Believe it or not, I'm going to wring another post out of this topic.