Musings on Positional Adjustments

This is an old post that I never published. It is sort of an attempt to justify why I use offensive positional adjustments, which is an even more dated position today than it was when I wrote it. In re-reading it, though, I thought my comments about zero-level defense were at least somewhat pertinent (if not particularly insightful) given Bill James' current effort at developing "Runs Saved Against Zero".

This post is not intended to be a comprehensive discussion of the issue of position adjustments; it will just quickly sketch out a system to classify adjustments and then I’ll offer a few of my opinions on them. There is a lot more that could be said and most of it could and has been said more eloquently by others.

The most important technical distinction between position adjustments (which I’ll shorten to PADJ sometimes) is which type of metric is used to set them--offensive or defensive. This distinction is well-known and gets a lot of attention. One that is talked about less is the difference between explicit and implicit position adjustments, and while people who get their hands dirty with various rating systems are well aware of implicit position adjustments, the average reader presented with a metric might gloss over them.

Explicit position adjustments are obvious and are acknowledged as being position adjustments. The first well-known example of their usage was in Pete Palmer’s linear weights system. They have also been used in VORP, just about every implementation of WAR, and many other metrics.

Implicit position adjustments usually crop up in the work of Bill James, although there are other metrics out there that utilize them. An implicit position adjustment is not really implicit in the truest sense of the word--they are obviously position adjustments if you look at them and consider what their function is. James likes to hide them in his fielding systems.

James’ metrics have always attempted to measure absolute wins and losses. I’ve always maintained that this is a fool’s errand, and that absolute wins and losses only make sense on the team level, not the player level. Most sabermetricians are in general agreement on this, and construct systems that are built to yield results against some baseline.

This is especially true for defensive metrics, whether for pitching for fielding. Absolute metrics (such as runs created) are tempting to apply to individual batters because there is a theoretical minimum on the number of runs a player can create (zero, of course), and such a performance represents the worst possible performance. There is no such cap on the poor performance of a defense; a team could allow an infinite number of runs. The only real cap on the poor performance of an individual fielder is the number of balls that are hit to the locations on the field that fall under his responsibility.

As such, it is impossible to develop a true zero baseline metric to evaluate pitchers or fielders (one can certainly argue that it’s impossible for batters as well, but the existence of the theoretical floor makes it undeniably more tempting). You have to start by comparing to a non-zero baseline (average being the most straightforward), but the problem is compounded for fielders by the fact that it’s also impossible to directly compare fielders at different positions. The fielding standards, be it in fielding average, range factor, or more complex methods vary wildly from one position to another. While all fielders have the same objective (record outs and prevent the opponent from scoring), the primary ways in which fielders at different positions contribute to the common goal are very different.

That pretty much leaves comparing a player to the average fielder at his position as the only viable starting point for the developer of a fielding metric. As is, the results are not satisfactory for inclusion in a total value metric, because they implicitly assume that an average fielder at any position is equal in value to an average fielder at any other position.

There is no one with any degree of baseball knowledge that believes this to be true. Everyone agrees that an average shortstop is harder to find than an average first baseman--that is, the pool of available players that can adequately field shortstop is much smaller than the pool of adequate available first basemen. This basic truth is sometimes obfuscated by silly hypotheticals (i.e. “if you didn’t have a catcher, every pitch with a runner on base would be a passed ball” and “without a first baseman, it would be nearly impossible to convert a groundball into an out”), but serious people agree on this.

So what is one to do about this problem? You have to do something--you cannot have a functioning estimate of total value that pretends first basemen and shortstops are equal in fielding value. The easiest answer is a position adjustment.

While James attempts to express all of his value metrics relative to an absolute baseline, he of course can’t pull off a clean implementation. His solution, in both his early 1980s Defensive Winning Percentage and his more recent Win Shares, is to develop a fielding winning percentage for each position and convert this to wins and losses (the terminology and procedure is a little different in Win Shares but that’s a long story).

To make the conversion to from a rate to a total of wins and losses, James multiplies by a number of games for which each position is assumed to be responsibility. Positions on the left side of the defensive spectrum are assigned less responsibility than those on the right side…and thus this is an implicit position adjustment.

In pointing this out, I don’t mean to suggest that James is in any way dishonest in describing his systems--the assigned games are clearly defined in the system and aren’t hidden. The characterization I’ve offered of these adjustments as “implicit” is therefore not really accurate. The real difference between James-style position adjustments and the ones I’ve defined as “explicit” is that explicit adjustments either add or subtract a set number of runs dependent upon a player’s position or apply a different baseline to their rate in converting to a value stat.

The other major characteristic that defines a position adjustment’s type is whether it is an offensive PADJ or a defensive PADJ. The categories are not black and white--many positional adjustments incorporate subjective weighting of various factors, which could include offensive performance by players at a position, the range of offensive performance, the performance of fielders at multiple positions, comparisons of average salary as a stand-in for how teams value players, subjective corrections that the developer feels better matches the results of the system to common sense--but usually the primary basis can be identified as either offensive or defensive.

Offensive position adjustments have fallen out of favor recently, although there are still some people using them (including me). The offensive PADJ originated with Pete Palmer, who used it as part of his linear weights system. The other most prominent use came in Keith Woolner’s VORP.

Defensive positional adjustments are a more recent phenomenon, but are key to both the Fangraphs and Chone WAR methodology. Tango Tiger was the driving force behind their development, and Chone has also done his own research to establish the adjustments for his WAR.

Before deciding how to construct a position adjustment, it’s a good idea to take a step back and figure out why you need a position adjustment at all. Taking the reason behind your metric for granted is a path to just slapping numbers around indiscriminately and failing to model baseball reality. From my perspective, the only real reason that a PADJ is necessary is that it is essentially impossible to measure a player’s fielding value independent of his position. Therefore, one has to have a way of comparing the value of fielding performances across positions--a position adjustment.

A common misperception regarding all position adjustments among people not that well-versed in sabermetrics is that they provide a bonus “just for playing the position”. While I suppose that might be technically true in the sense of calculation, the underlying need for such an adjustment is discussed above. If one does not believe in applying a positional adjustment, and accepts the use of defensive metrics baselined to an average fielder at the position, then they must conclude that, as a group, the most valuable players are those at left-side of the spectrum positions. Or, in other words, that the overall average value of players at a given position is strictly a function of their aggregate offensive production.

It is possible to complicate the question of position adjustments by talking about baselines (particularly replacement level) and other considerations, but at the heart of the issue is the need to compare the value of a shortstop -5 fielding runs relative to an average shortstop to a third baseman +10 runs relative to an average third baseman to an average first baseman.

Such a viewpoint suggests that a defensive PADJ is the way to go, since the sole reason for needing the adjustment is consideration of defense. So while the overwhelming positive of a defensive PADJ is that it is defined in the terms that necessitate the entire endeavor, it also carries a few negatives.

One is the difficulty of accurately evaluating fielding value, even within the constraints of one’s own position. While it is quite possible that any biases or methodological errors will balance out when aggregated over a large population of players, it would nonetheless be more comforting to begin from metrics in which one had a great deal of confidence.

Another key issue is that the pool of players who log times at multiple positions, while relatively large when comparing similar position groups (particularly outfielders, but also middle infielders, corner infielders, etc.), there is a much smaller available sample of players who play very different positions, at least in the same or adjacent seasons. And catcher? Forget about it--Bill James even left catcher off of the defensive spectrum due to the difficulty of comparing it directly to the positions whose occupants stand in fair territory.

Players that move positions introduce all kinds of selective sampling issues as well. Consider the problem comparing positions where left-handers are de facto ineligible, and the fact that players moved off a position are more likely to have been stopgaps. For a more complete discussion of these issues (and an all-around good discussion of PADJ issues), see Colin Wyers’ article at Baseball Prospectus.

Thus, to avoid strange conclusions, defensive position adjustments are always going to require a little subjective massaging. That’s not necessarily a bad thing--the construction of any metric requires subjective decisions made on the part of the developer--but it makes them inherently high maintenance.

Of course, offensive position adjustments are best employed with a measure of caution as well. The pros of offensive adjustments are that they are very easy to work with. Offensive statistics are more reliable than fielding statistics, require much more basic data to calculate, and are available throughout the entire history of the game. Rather than having to compare performance of players across positions, one can at least start by simply looking at the average performance of all players at a particular position.

An offensive PADJ implicitly assumes that teams allocate their talent in such a manner that the average player at any position is equal to the average player at any other position--alternatively stated, that the offensive value gap between positional averages is equal to the defensive value gap. This is certainly never 100% truly the case for any sample period, particularly for single years. Offensive PADJs based on one year of data or other short stretches should be viewed with a great deal of skepticism.

Another problem lurking is what Wyers, in the linked article, refers to as the “Mays problem”--the existence of supremely talented players that excel at both hitting and fielding. Such players might be superstars at any position (ignoring handedness and other impediments), even first base, thanks to their hitting alone but are able to handle the defensive rigors of right-side defensive spectrum positions. While more ordinary players offer a package of offensive and defensive skills that limits their possible fielding positions commensurate to their offensive production, these players are playable anywhere. There are also potential issues with the very worst players at a position.

The Mays problem skews offensive positional averages, so Wyers proposes using an alternative offensive PADJ that adjusts the overall positional average for the gap between the upper and lower median of observed performance at the position. This approach (and other similar algorithms that could be offered) is novel but involves subjective choices similar to those necessitated by defensive PADJs.

The offensive PADJ will surely fail at lower levels of baseball thanks to the Mays problem--the best high school players, for instance, are often the cleanup hitter, ace pitcher, and center fielder or shortstop when not pitching. Such all-around stars are also more common in college ball or in earlier, less developed major leagues than they are in the modern major leagues with their high overall quality of play and relatively strong competitive balance. An offensive PADJ approach will surely break down at those low levels without serious alterations.

There are other relevant issues to discuss with respect to position adjustments, such as their relationship to replacement level and the manner in which they are applied (That is, if they should be used to change the baseline to which performance is compared or if they should be assigned as a lump sum based on playing time. The possible answers to this question is also closely tied to one’s choice of offensive or defensive adjustment), but those will have to wait for some other time.

Tripod: Baselines

See the first paragraph of this post for an explanation of this series.

This essay will touch on the topics of various baselines and which are appropriate(in my opinion) for what you are trying to measure. In other words, it discusses things like replacement level. This is a topic that creates a lot of debate and acrimony among sabermetricians. A lot of this has to do with semantics, so all that follows is my opinion, some of it backed by facts and some of it just opinion.

Again, I cannot stress this enough; different baselines for different questions. When you want to know what baseline you want to use, first ask the question: what am I trying to measure?

Anyway, this discussion is kind of disjointed, so I'll just put up a heading for a topic and write on it.

Individual Winning Percentage

Usually the baseline is discussed in terms of a winning percentage. This unfortunate practice stems from Bill James' Offensive Winning Percentage. What is OW%? For instance, if Jason Giambi creates 11 runs per game in a context where the average team scores 5 runs per game, than Giambi's OW% is the W% you would expect when a team scores 11 runs and allows 5(.829 when using Pyth ex 2). It is important to note that OW% assumes that the team has average defense.

So people will refer to a replacement level of say .333, and what they mean is that the player's net value should be calculated as the number of runs or wins he created above what a .333 player would have done. This gets very confusing when people try to frame the discussion of what the replacement level should be in terms of actual team W%s. They'll say something like, "the bottom 1% of teams have an average W% of .300, so let's make .300 the replacement level". That's fine, but the .300 team got its record from both its offense and defense. If the team had an OW% of .300 and a corresponding DW% of .300, their record would be about .155.

Confusing, eh? And part of that comes from the silly idea of putting a player's individual performance in the form of a team's W%. So, I prefer to define replacement level in terms of percentage of the league average the player performed at. It is much easier to deal with, and it just makes more sense. But I may use both interchangeably here since most people discuss this in terms of W%.

ADDED 12/04: You can safely skip this part and understand the rest of the article; it's really about a different subject anyway. I should note the weakness of the % of league approach. The impact of performing at 120% of the league is different at different levels of run scoring. The reason for this is that the % of league for a particular player is essentially a run ratio(like runs scored/runs allowed for a team). We are saying that said player creates 20% more runs than his counterpart, which we then translate into a W% by the Pythagorean by 1.2^2/(1.2^2+1)=.590. But as you can read in the "W% Estimator" article, the ideal exponent varies based on RPG. In a 10 RPG context(fairly normal), the ideal exponent is around 1.95. But in a 8 RPG context, it is around 1.83. So in the first case a 1.2 run ratio gives a .588 W%, but in the other it gives a .583. Now this is a fairly minor factor in most cases, but we want to be as precise as possible.

So from this you might determine that indeed the W% display method is ideal, but the W% approach serves to ruin the proportional relationship between various Run Ratios(with a Pyth exponent of 2, a 2 RR gives an .800 W%, while a 1 RR gives .500, but 2 is 2 times as high as 1, not .8/.5). So the ideal thing as far as I'm concerned is to use the % of league, but translate it into a win ratio by raising it to the proper pythagorean exponent for the context(which can be figured approximately as RPG^.28). But this shouldn't have too big of an impact on the replacement level front. If you like the win ratio idea but want to convert it back into a run ratio, you can pick a "standard" league that you want to translate everybody back into(ala Clay Davenport). So if you want a league with a pyth exponent of 2, take the square root of the win ratio to get the run ratio. Generally (W/L) = (R/RA)^x or (R/RA) = (W/L)^(1/x).

Absolute Value

This is a good place to start. Why do we need a baseline in the first place? Why can't we just look at a player's Runs Created, and be done with it? Sabermetricians, I apologize, this will be quite patronizing for you.

Well, let's start by looking at a couple of players:

H D T HR W

145 17 1 19 13

128 32 2 19 25

The first guy has 68 RC, the second guy has 69. But when you discover that Player A made 338 outs and Player B made 284 outs, the choice becomes pretty clear, no? BTW, player A is Randall Simon and player B is Ivan Rodriguez(2003).

But you could say that we should have known player B was better, because we could just look at his runs/out. But of course I could give you an example of 2 guys with .2 runs/out, but one made 100 outs and had 20 RC and another made 500 outs and had 100 RC. And so you see that there must be some kind of balance between the total and the rate.

The common sense way to do this with a baseline. Some people, like a certain infamous SABR-L poster, will go to extreme lengths to attempt to combine the total and the rate in one number, using all sorts of illogical devices. A baseline is logical. It kills two or three birds with one stone. For one thing, we can incorporate both the total production and the rate of production. For another, we eventually want to evaluate the player against some sort of standard, and that standard can be the baseline that we use. And using a baseline automatically inserts an adjustment for league context.

There is value in every positive act done on a major league field. There is no way that you can provide negative absolute value. If you bat 500 times, make 499 outs and draw 1 walk, you have still contributed SOMETHING to your team. You have provided some value to that team.

But on the other hand, the team could have easily, for the minimum salary, found someone who could contribute far, far more than you could. So you have no value to the team in an economic sense. The team has no reason to pay you a cent, because they can find someone you can put up a .000/.002/.000 line panhandling on the street. This extreme example just goes to show why evaluating a major league player by the total amount of production he has put up is silly. That leads into the question of what is level at which a team can easily find a player who can play that well?

Minimum Level

This is where a lot of analysts like to draw the baseline. They will find the level at which there are dozens of available AAA players who perform that well, and that is the line against which they evaluate players. Those players are numerous and therefore have no real value to a team. They can call up another one from AAA, or find one on waivers, or sign one out of the Atlantic League. Whatever.

There are a number of different ways of describing this, though. One is the "Freely Available Talent" level. That's sort of the economic argument I spelled out. But is it really free? This might be nitpicking, but I think it is important to remember that all teams spend a great deal of money on player development. If you give your first round pick a $2 million bonus and he turns out to be a "FAT" player, he wasn't really free. Of course, he is freely available to whoever might want to take him off your hands. But I like the analogy of say, getting together with your friends, and throwing your car keys in a box, and then picking one randomly and taking that car. If you put your Chevy Metro keys in there and draw out somebody's Ford Festiva keys, you didn't get anywhere. And while you now have the Festiva, it wasn't free. This is exactly what major league teams do when they pick each other's junk up. They have all poured money into developing the talent and have given up something to acquire it(namely their junk). None of this changes the fact that it is freely available or really provides any evidence against the FAT position at all; I just think it is important to remember that the talent may be free now, but it wasn't free before. Someone on FanHome proposed replacing FAT with Readily Available Talent or something like that, which makes some sense.

Another way people define this is the level at which a player can stay on a major league 25 man roster. There are many similar ways to describe it, and while there might be slight differences, they all are getting at the same underlying principle.

The most extensive study to establish what this line is was undertaken by Keith Woolner in the 2002 Baseball Prospectus. He determined that the minimum level was about equal to 80% of the league average, or approximately a .390 player. He, however, looked at all non-starters, producing a mishmash of bench players and true FAT players.

The basic idea behind all of these is that if a player fell of the face of the earth, his team would have to replace him, and the player who would replace them would be one of these readily available players. So it makes sense to compare the player to the player who would replace him in case of injury or other calamity.

A W% figure that is often associated with this line of reasoning is .350, although obviously there is no true answer and various other figures might give a better representation. But .350 has been established as a standard by methods like Equivalent Runs and Extrapolated Wins, and it is doubtful that it will be going anywhere any time soon.

Sustenance Level

This is kind of similar to the above. This is the idea that there is some level of minimum performance at which the team will no longer tolerate the player, and will replace him. This could be other from his status on the roster or as a starting player(obviously, the second will produce a higher baseline in theory). You could also call this "minimum sustainable performance" level.

Cliff Blau attempted a study to see when regular players lost their jobs based on their RG, at each position. While I have some issues with Blau's study, such as that it did not include league adjustments while covering some fairly different offensive contexts, his results are interesting none the less. He found no black line, no one level where teams threw in the towel. This really isn't that surprising, as there are a number of factors involved in whether or not a player keeps his job other than his offensive production(such as salary, previous production, potential, defensive contribution, nepotism, etc). But Bill James wrote in the 1985 Abstract that he expected there would be such a point. He was wrong, but we're all allowed to be sometimes.

Anyway, this idea makes sense. But a problem with it is that it is hard to pin down exactly where this line is-or for that matter, where the FAT line is. We don't have knowledge of a player's true ability, just a sample of varying size. The team might make decisions on who to replace based on a non-representative sample, or the sabermetrician might misjudge the talent of players in his study and thus misjudge the talent level. There are all sorts of selective sampling issues here. We also know that the major leagues are not comprised of the 750 best players in professional baseball. Maybe Ricky Weeks could hit better right now then the Brewers' utility infielder, but they want him to play every day in the minors. The point is, it is impossible to draw a firm baseline here. All of the approaches involve guesswork, as they must.

Some people have said we should define replacement level as the W% of the worst team in the league. Others have said it should be based on the worst teams in baseball over a period of years. Or maybe we should take out all of the starting players from the league and see what the performance level of the rest of them is. Anyway you do it, there's uncertainty, large potential for error, and a need to remember there's no firm line.

Average

But the uncertainty of the FAT or RAT or whatever baseline does leave people looking for something that is defined, and that is constant. And average fits that bill. The average player in the league always performs at a .500 level. The average team always has a .500 W%. So why not evaluate players based on their performance above what an average player would have done?

There are some points that can be made in favor of this approach. For one thing, the opponent that you play is on average a .500 opponent. If you are above .500, you will win more often then you lose. If you are below .500, you will lose more often that you win. The argument that a .500+ player is doing more to help his team win then his opponent is, while the .500- player is doing less to help his team win then his opponent is, makes for a very natural demarcation: win v loss.

Furthermore, the .500 approach is inherently built into any method of evaluating players that relies on Run Expectancy or Win Expectancy, such as empirical Linear Weights formulas. If you calculate the run value of each event as the final RE value minus the initial RE value plus runs scored on the play(which is what empirical LW methods are doing, or the value added approach as well), the average player will wind up at zero. Now the comparison to zero is not inevitable; you can fudge the formula or the results to compare to a non-.500 baseline, but initially the method is comparing to average.

An argument that I have made on behalf of the average baseline is that, when looking back in hindsight on the season, the only thing that ultimately matters is whether or not the player helped you to win more than your opponent. An opponent of the average baseline might look to a .510 player with 50 PA and say that he is less valuable than a .490 player with 500 PA, since the team still had 450 additional PA with the first player. This is related to the "replacement paradox" which I will discuss later, but ignoring that issue for now, my argument back would be that it is really irrelevant, because the 450 PA were filled by someone, and there's no use crying over spilled milk. The .490 player still did less to help his team win than his opponent did to help his team win. It seems as if the minimum level is more of a forward looking thing, saying "If a team could choose between two players with these profiles, they would take the second one", which is surely true. But the fact remains that the first player contributed to wins more than his opponent. From a value perspective, I don't necessarily have to care about what might have happened, I can just focus on what did happen. It is similar to the debate about whether to use clutch hitting stats, or actual pitcher $H data, even when we know that these traits are not strongly repetitive from season to season. Many people, arguing for a literal value approach, will say that we should use actual hits allowed or a player's actual value added runs, but will insist on comparing the player to his hypothetical replacement. This is not a cut and dry issue, but it reminds us of why it is so important to clearly define what we are trying to measure and let the definition lead us to the methodology.

Average is also a comfortable baseline for some people to use because it is a very natural one. Everybody knows what an average is, and it is easy to determine what an average player's Batting Average or walk rate should be. Using a non-.500 baseline, some of this inherent sense is lost and it is not so easy to determine how a .350 player for instance should perform.

Finally, the most readily accessible player evaluation method, at least until recently, was Pete Palmer's Linear Weights system. In the catch-all stat of the system, Total Player Rating, he used an average baseline. I have heard some people say that he justified because if you didn't go .500, you couldn't make the playoffs in the later editions of Total Baseball. However, in the final published edition, on page 540, he lays out a case for average. I will quote it extensively here since not many people have access to the book:

The translation from the various performance statistics into the wins or losses of TPR is accomplished by comparing each player to an average player at his position for that season in that league. While the use of the average player as the baseline in computing TPR may not seem intuitive to everyone, it is the best way to tell who is helping his team win games and who is costing his team wins. If a player is no better than his average counterparts on other teams, he is by definition not conferring any advantage on his team. Thus, while he may help his team win some individual games during the season--just as he will also help lose some individual games--over the course of a season or of a career, he isn't helping as much as his opponents are. Ultimately, a team full of worse-than-average players will lose more games than it wins.

The reason for using average performance as the standard is that it gives a truer picture of whether a player is helping or hurting his team. After all, almost every regular player is better than his replacement, and the members of the pool of replacement players available to a team are generally a lot worse than average regulars, for obvious reasons.

If Barry Bonds or Pedro Martinez is out of the lineup, the Giants or the Red Sox clearly don't have their equal waiting to substitute. The same is typically true for lesser mortals: when an average ballplayer cannot play, his team is not likely to have an average big-league regular sitting on the bench, ready to take his place.

Choosing replacement-level performance as the baseline for measuring TPR would not be unreasonable, but it wouldn't give a clear picture of how the contributions of each player translate into wins or losses. Compared to replacement-level performance, all regulars would look like winners. Similarly, when compared to a group of their peers, many reserve players would have positive values, even though they would still be losing games for their teams. Only the worst reserves would have negative values if replacement level were chosen as the baseline.

The crux of the problem is that a team composed of replacement-level players(which would be definition be neither plus nor minus in the aggregate if replacement-level is the baseline) would lose the great majority of its games! A team of players who were somewhat better than replacement level--but still worse than their corresponding average regulars--would lose more games than it won, even though the player values(compared to a replacement-level baseline) would all be positive.

Median

This is sort of related to the average school of thought. But these people will say that since the talent distribution in baseball is something like the far right hand portion of a bell curve, there are more below average players than above average players, but the superior performance of the above average players skew the mean. The average player may perform at .500, but if you were given the opportunity to take the #15 or #16 first baseman in baseball, they would actually be slightly below .500. So they would suggest that you cannot fault a player for being in below average if he is in the top half of players in the game.

It makes some sense, but for one thing, the median in Major League baseball is really not that dissimilar to the mean. A small study I did suggested that the median player performs at about 96% of the league mean in terms of run creation(approx. .480 in W% terms). It almost a negligible difference. Maybe it is farther from the mean than that(as other studies have suggested), but either way, it just does not seem to me to be a worthwhile distinction, and most sabermetricians are sympathetic to the minimum baseline anyway, so few of them would be interested in a median baseline that really is not much different from the mean.

Progressive Minimum

The progressive minimum school of thought was first expressed by Rob Wood, while trying to reconcile the average position and the minimum position, and was later suggested independently by Tango Tiger and Nate Silver as well. This camp holds that while if a player is injured and the team must scramble to find a .350 replacement, that does not bind them to using the .350 replacement forever. A true minimal level supporter wants us to compare Pete Rose, over his whole 20+ year career, to the player that would have replaced him had he been hurt at some point during that career. But if Pete Rose had been abducted by aliens in 1965, would the Reds have still been forced to have a .350 player in 1977? No. A team would either make a trade or free agent signing to improve, or the .350 player would become better and save his job, or a prospect would eventually come along to replace him.

Now the minimum level backer might object, saying that if you have to use resources to acquire a replacement, you are sacrificing potential improvement in other areas. This may be true to some extent, but every team at some point must sacrifice resources to improve themselves. It is not as if you can run a franchise solely on other people's trash. A team that tried to do this would eventually have no fans and would probably be repossessed by MLB. Even the Expos do not do this; they put money into their farm system, and it produced players like Guerrero and Vidro. They produced DeShields who they turned into Pedro Martinez. Every team has to make some moves to improve, so advocates of the progressive or time dependent baseline will say that it is silly to value a player based on an unrealistic representation of the way teams actually operate.

So how do we know how fast a team will improve from the original .350 replacement? Rob Wood and Tango looked at it from the perspective of an expansion teams. Expansion teams start pretty much with freely available talent, but on average, they reach .500 in 8 years. So Tango developed a model to estimate the W% of such a team in year 1, 2, 3, etc. A player who plays for one year would be compared to .350, but his second year might be compared to .365, etc. The theory goes that the longer a player is around, the more chances his team has had to replace him with a better player. Eventually, a team will come up with a .500 player. After all, the average team, expending an average amount of resources, puts out a .500 team.

Another area you could go to from here is whether or not the baseline should ever rise above .500. This is something that I personally am very uneasy with, since I feel that any player who contributes more to winning than his opponent does should be given a positive number. But you could make the case that if a player plays for 15 years in the show, at some point he should have provided above average performance. This approach would lead to a curve for a player's career that would rise from .350, up over .500 maybe to say .550 at its peak, and then tailing back down to .350. Certainly an intriguing concept.

Silver went at it differently, by looking at player's offensive performance charted against their career PA. It made a logarithmic curve and he fitted a line to it. As PA increase, offensive production rapidly increases, but then the curve flattens out. Comparing Silver's work to Tango's work, the baselines at various years were similar. This was encouraging to see similar results coming from two totally different and independent approaches.

A common argument against the progressive baseline is that even if you can eventually develop a .500 replacement, the presence of your current player does not inhibit the development of the replacement, so if your player does not get hurt or disappear, you could peddle the replacement to shore up another area, or use him as a backup, or something else. This is a good argument, but my counter might be that it is not just at one position where you will eventually develop average players; it is all over the diamond. The entire team is trending toward the mean(.500) at any given time, be it from .600 or from .320. Another potential counter to that argument is that some players can be acquired as free agent signings. Of course, these use up resources as well, just not human resources.

The best argument that I have seen against the progressive level is that if a team had a new .540 first baseman every year for 20 years, each would be evaluated against .350 first baseman. But if a team had the same .540 first baseman for 20 years, he would be evaluated against a .350, then a .365, then a .385, etc, and would be rated as having less value then the total of the other team's 20 players, even though each team got the exact same amount of production out of their first base position. However, this just shows that the progressive approach might not make sense from a team perspective, but does makes sense from the perspective of an individual player's career. Depending on what we want to measure, we can use different baselines.

Chaining

This is the faction that I am most at home in, possibly because I published this idea on FanHome. I borrowed the term "chaining" from Brock Hanke. Writing on the topic of replacement level in the 1998 BBBA, he said something to the effect that win you lose your first baseman, you don't just use him. You lose your best pinch hitter, who now has to man first base, and then he is replaced by some bum.

But this got me to thinking: if the team is replacing the first baseman with its top pinch hitter, who must be a better than minimum player or else he could easily be replaced, why must we compare the first baseman to the .350 player who now pinch hits? The pinch hitter might get 100 PA, but the first baseman gets 500 PA. So the actual effect on the team when the first baseman is lost is not that it gives 500 PA to a .350 player; no, instead it gives 500 PA to the .430 pinch hitter and 100 PA to the .350 player. And all of that dynamic is directly attributable to the first baseman himself. The actual baseline in that case should be something like .415.

The fundamental argument to back this up is that the player should be evaluated against the full scenario that would occur if he has to be replaced, not just the guy who takes his roster spot. Let's run through an example of chaining, with some numbers. Let's say that we have our starting first baseman who we'll call Ryan Klesko. We'll say Klesko has 550 PA, making 330 outs, and creates 110 runs. His backup racks up 100 PA, makes 65 outs, and creates 11 runs. Then we have a AAA player who will post a .310 OBA and create .135 runs/out, all in a league where the average is .18 runs/out. That makes Klesko a .775 player, his backup a .470 player, and the AAA guy a .360 player(ignoring defensive value and the fact that these guys are first baseman for the sake of example; we're also ignoring the effect of the individual's OBAs on their PAs below-the effect might be slight but it is real and would serve to decrease the performance of the non-Klesko team below). Now in this league, the FAT level is .135 R/O. So a minimalist would say that Klesko's value is (110/330-.135)*330 = +65.5 RAR. Or, alternatively, if the AAA player had taken Kleko's 550 PA(and it is the same thing as doing the RAR calculation), he would have 380 outs and 51 runs created.

Anyway, when Klesko and his backup are healthy, the team's first baseman have 650 PA, 395 outs, and 121 RC. But what happens if Klesko misses the season? His 550 PA will not go directly to the bum. The backup will assume Klesko's role and the bum will assume his. So the backup will now make 550/100*65=358 outs and create 11/65*358=61 runs. The bum will now bat 100 times, make 69 outs, and create .135*69=9 runs. So the team total now has 427 outs and 70 RC from its first baseman. We lose 51 runs and gain 32 outs. But in the first scenario, with the bum replacing Klesko directly(which is what a calculation against the FAT line implicitly assumes), the team total would be 445 outs and 62 runs created. So the chaining subtracts 18 outs and adds 8 runs. Klesko's real replacement is the 70/427 scenario. That is .164 runs/out, or 91% of the league average, or a .450 player. That is Klesko's true replacement. A .450 player. A big difference from the .360 player the minimalists would assume.

But what happens if the backup goes down? Well, he is just replaced directly by the bum, and so his true replacement level is a .360 player. Now people will say that it is unfair for Klesko to be compared to .450 and his backup to be compared to .360. But from the value perspective, that is just the way it is. The replacement for a starting player is simply a higher level than the replacement for a backup. This seems unfair, and it is a legitamite objection to chaining. But I suggest that it isn't that outlandish. For one thing, it seems to be the law of diminishing returns. Take the example of a team's run to win converter. The RPW suggested by Pythagorean is:

RPW = RD:G/(RR^x/(RR^x+1)-.5)

Where RD:G is run differential per game, RR is run ratio, and x is the exponent. We know that the exponent is approximately equal to RPG^.29. So a team that scores 5 runs per game and allows 4 runs per game has an RPW of 9.62. But what about a team that scores 5.5 and allows 4? Their RPW is 10.11.

So a team that scores .5 runs more than another is buying their wins at the cost of an additional .49 runs. This is somewhat similar to a starting player deriving value by being better than .450, and a backup deriving value by being better than .360. Diminishing returns. Now obviously, if your starter is .450, your backup must be less than that. So maybe the chained alternative should be tied to quality in the first place. Seems unfair again? Same principle. It's not something that we are used to considering in a player evaluation method, so it seems very weird, but the principle comes into play in other places(such as RPW) and we don't think of it as such because we are used to it.

Now an alternative way of addressing this is to point out the concept of different baselines for different purposes. A starting player, to keep his starting job, has a higher sustenance level than does a backup. Now since backups max out at say 200 PA, we could evaluate everyone's first 200 PA against the .360 level and their remaining PA against the .450 level. This may seem unfair, but I feel that it conforms to reality. A .400 player can help a team, but not if he gets 500 PA.

Some other objections to chaining will invariably come up. One objection is that not all teams have a backup to plug in at every position. Every team will invariably have a backup catcher, and somebody who can play some infield positions and some outfield positions, but maybe not on an everyday basis. And this is true. One solution might be to study the issue and find that say 65% of teams have a bench player capable of playing center field. So then the baseline for centerfield would be based 65% on chaining and 35% on just plugging the FAT player into the line. Or sometimes, more than one player will be hurt at once and the team will need a FAT player at one position. Another is that a player's position on the chain should not count against him. They will say that it is not the starter's fault that he has more room to be replaced under him. But really, it's not counting against him. This is the diminishing returns principle again. If he was not a starter, he would have less playing time, and would be able to accrue less value. And if you want to give "Klesko" credit for the value that his backup has that is greater than his backup, fine. You are giving him credit for a .360 player, but only over 100 PA, rather than the minimalist, who will extend him that value over all 550 of his PA. That is simply IMO not a realistic assessment of the scenario. All of these things just demonstrate that the baseline will not in the end be based solely on chaining; it would incorporate some of the FAT level as well.

When chaining first came up on FanHome, Tango did some studies of various things and determined that in fact, chaining coupled with adjusting for selective sampling could drive the baseline as high as 90%. I am not quite as gung ho, and I'm not sure that he still is, but I am certainly not convinced that he was wrong either.

Ultimately, it comes down to whether or not we are trying to model reality as best as possible or if we have an idealized idea of value. It is my opinion that chaining, incorporated at least somewhat in the baseline setting process, best models the reality of how major league teams adjust to loss in playing time. And without loss in playing time(actually variance in playing time), everyone would have equal opportunity and we wouldn't be having this darn discussion. Now I will be the first to admit that I do not have a firm handle on all the considerations and complexities that would go into designing a total evaluation around chaining. There are a lot of studies we would need to do to determine certain things. But I do feel that it must be incorporated into any effort to settle the baseline question for general player value.

Plus-.500 Baselines

If a .500 advocate can claim that the goal of baseball is to win games and sub-.500 players contribute less to winning then do their opponent, couldn't someone argue that the real goal is to make the playoffs, and that requires say a .560 W%, so shouldn't players be evaluated against .560?

I suppose you could make that argument. But to me at least, if a player does more to help his team win than his opponent does to help his team win, he should be given a positive number of a rating. My opinion, however, will not do much to convince people of this.

A better argument is that the idea of winning pennants or making the playoffs is a separate question than just winning. Let's take a player who performs at .100 one season and at .900 in the other. The player will rate, by the .560 standard, as a negative. He has hurt his team in its quest to win pennants.

But winning a pennant is a seasonal activity. In the season in which our first player performed at .900, one of the very best seasons in the history of the game, he probably added 12 wins above average to his team. That would take an 81 win team up to 93 and put them right in the pennant hunt. He has had an ENORMOUS individual impact on his team's playoff hopes, similar to what Barry Bonds has done in recent years for the Giants.

So his team wins the pennant in the .900 season, and he hurts their chances in the second season. But is their a penalty in baseball for not winning the pennant? No, there is not. Finishing 1 game out of the wildcard chase is no better, from the playoff perspective, than finishing 30 games out. So if in his .100 season he drags an 81 win team down to 69 wins, so what? They probably weren't going to make the playoffs anyway.

As Bill James said in the Politics of Glory, "A pennant is a real thing, an object in itself; if you win it, it's forever." The .100 performance does not in any way detract from the pennant that the player provided by playing .900 in a different season.

And so pennant value is a totally different animal. To properly evaluate pennant value, an approach such as the one proposed by Michael Wolverton in the 2002 Baseball Prospectus is necessary. Using a baseline in the traditional sense simply will not work.

Negative Value/Replacement Paradox

This is a common area of misunderstanding. If we use the FAT baseline, and a player rates negatively, we can safely assume that he really does have negative value. Not negative ABSOLUTE value--nobody can have negative absolute value. But he does have negative value to a major league team, because EVERYBODY from the minimalist to the progressivists to the averagists to the chainists would agree that they could find, for nothing, a better player.

But if we use a different baseline(average in particular is used this way), a negative runs or wins above baseline figure does not mean that the player has negative value. It simply means that he has less value then the baseline he is being compared to. It does not mean that he should not be employed by a major league team.

People will say something like, ".500 proponents would have us believe that if all of the sub-.500 players in baseball retired today, there would be no change in the quality of play tomorrow". Absolute hogwash! An average baseline does not in anyway mean that its proponents feel that a .490 player has no value or that there is an infinite supply of .500 players as there are of .350 players. It simply means that they choose to compare players to their opponent. It is a relative scale.

Even Bill James does not understand this or pretends not to understand this to promote his own method and discredit TPR(which uses a .500 baseline). For instance, in Win Shares, he writes: "Total Baseball tells us that Billy Herman was three times the player that Buddy Myer was." No, that's not what it's telling you. It's telling you that Herman had three times more value above his actual .500 opponent than did Myer. He writes "In a plus/minus system, below average players have no value." No, it tells you that below average players are less valuable than their opponent, and if you had a whole team of them you would lose more than you would win.

These same arguments could be turned against a .350 based system too. You could say that I rate at 0 WAR, since I never played in the majors, and that the system is saying that I am more valuable than Alvaro Espinoza. It's the exact same argument, and it's just as wrong going the other way as it is going this way.

And this naturally leads into something called the "replacement paradox". The replacement paradox is essentially that, using a .500 baseline, a .510 player with 10 PA will rate higher than a .499 player with 500 PA. And that is true. But the same is just as true at lower baselines. Advocates of the minimal baseline will often use the replacement paradox to attack a higher baseline. But it the sword can be turned against them. They will say that nobody really cares about the relative ratings of .345 and .355 players. But hasn't a .345 player with 500 PA shown themselves to have more ability than a .355 player with 10 PA. Yes, they have. Of course, on the other hand, they have also provided more evidence that they are a below average player as well. That kind of naturally leads in to the idea of using the baseline to estimate a player's true ability. Some have suggested a close to .500 baseline for this purpose. Of course, the replacement paradox holds wherever you go from 0 to 1 on the scale. I digress; back to the replacement paradox as it pertains to the minimal level. While we may not care that much about how .345 players rate against .355 players, it is also true that we're not sure exactly where that line really is as we are with the .500 line. How confident are we that it is .350 and not .330 or .370? And that uncertainty can wreck havoc with the ratings of players who for all we know could be above replacement level.

And now back to player ability; really, ability implies a rate stat. If there is an ability to stay healthy(some injuries undoubtedly occur primarily because of luck--what could Geoff Jenkins have done differently to avoid destroying his ankle for instance), and there almost certainly is, than that is separate from a player's ability to perform when he is on the field. And performance when you are on the field is a rate, almost by definition. Now a player who has performed at a .600 level for 50 PA is certainly not equal to someone with a .600 performance over 5000 PA. What we need is some kind of confidence interval. Maybe we are 95% confident that the first player's true ability lies between .400 and .800, and we are 95% confident that the second player's true ability lies between .575 and .625. Which is a safer bet? The second. Which is more likely to be a bum? The first, by a huge margin. Which is the more likely to be Babe Ruth? The first as well. Anyway, ability is a rate and does not need a baseline.

Win Shares and the Baseline

A whole new can of worms has been opened up by Bill James' Win Shares. Win Shares is very deceptive, because the hook is that a win share is 1/3 of an absolute win. Except it's not. It can't be.

Win Shares are divided between offense and defense based on marginal runs, which for the purpose of this debate, are runs above 50% of the league average. So the percentage of team marginal runs that come from offense is the percentage of team win shares that come from offense. Then each hitter gets "claim points" for their RC above a 50% player.

Anyway, if we figure these claim points for every player, some may come in below the 50% level. What does Bill James do? He zeroes them out. He is saying that the minimum level of value is 50%, but you do not have negative value against this standard, no matter how bad you are. Then, if you total up the positive claim points for the players on the team, the percentage of those belonging to our player is the percentage of offensive win shares they will get. The fundamental problem here is that he is divvying up absolute value based on marginal runs. The distortion may not be great, but it is there. Really, as David Smyth has pointed out, absolute wins are the product of absolute runs, and absolute losses are the product of absolute runs allowed. In other words, hitters don't make absolute losses, and pitchers don't make absolute wins. The proper measure for a hitter is therefore the number of absolute wins he contributes compare to some baseline. And this is exactly what every RAR or RAA formula does.

Now, to properly use Win Shares to measure value, you need to compare it to some baseline. Win Shares are incomplete without Loss Shares, or opportunity. Due to the convoluted nature of the method, I have no idea what the opportunity is. Win Shares are not really absolute wins. I think the best explanation is that they are wins above approximately .200, masqueraded as absolute wins. They're a mess.

My Opinions

The heading here is a misnomer, because my opinions are littered all throughout this article. Anyway, I think that the use of "replacement level" to mean the FAT, or RAT, or .350 level, has become so pervasive in the sabermetric community that some people have decided they don't need to even defend it anymore, and can simply shoot down other baselines because they do not match. This being an issue of semantics and theory, and something you can't prove like a Runs Created method, you should always be open to new ideas and be able to make a rational defense of your position. Note that I am not accusing all of the proponents of the .350 replacement rate as closed-minded; there are however some who don't wish to even consider other baselines.

A huge problem with comparing to the bottom barrel baseline is intelligibility. Let's just grant for the sake of discussion that comparing to .350 or .400 is the correct, proper way to go. Even so, what do I do if I have a player who is rated as +2 WAR? Is this a guy who I want to sign to a long term contract? Is he a guy who I should attempt to improve upon through trade, or is he a guy to build my team around?

Let's say he's a full time player with +2 WAR, making 400 outs. At 25 outs/game, he has 16 individual offensive games(following the IMO faulty OW% methodology as discussed at the beginning), and replacement level is .350, so his personal W% is .475. Everyone, even someone who wants to set the baseline at .500, would agree that this guy is a worthwhile player who can help a team, even in a starting job. He may contribute less to our team than his opponent does to his, but players who perform better than him are not that easy to find and would be costly to acquire.

So he's fine. But what if my whole lineup and starting rotation was made up of +2 WAR players? I would be a below average team at .475. So while every player in my lineup would have a positive rating, I wouldn't win anything with these guys. If I want to know if my team will be better than the team of my opponent, I'm going to need to figure out how many WAR an average player would have in 400 outs. He'd have +2.4, so I'm short by .4. Now I have two baselines. Except what is my new baseline? It's simply WAA. So to be able to know how I stack up against the competition, I need the average baseline as well.

That is not meant to be a refutation of value against the minimum; as I said, that theory of value could be completely true and the above would still hold. But to interpret the results, I'm going to need to introduce, whether intentionally or not, a higher baseline. Thus, I am making the case here that even if the minimum baseline properly describes value, other baselines have relevance and are useful as well.

As for me, I'm not convinced that it does properly define value. Why must I compare everybody to the worst player on the team? The worst player on the team is only there to suck up 100 PA or 50 mop up IP. A starting player who performs like the worst player is killing my team. And chaining shows that the true effect on my team if I was to lose my starter is not to simply insert this bum in his place in many cases. So why is my starter's value based on how much better he is than that guy?

Isn't value supposed to measure the player's actual effect on the team? So if the effect on my team losing a starter is that a .450 composite would take his place, why must I be compelled to compare to .350? It is ironic that as sabermetricians move more and more towards literal value methods like Win Expectancy Added and Value Added Runs or pseudo-literal value methods like Win Shares, which stress measuring what actually happened and crediting to the player, whether we can prove he has an ability to repeat this performance or not, they insist on baselining a player's value against what a hypothetical bottom of the barrel player would do and not by the baseline implied by the dynamics of a baseball team.

I did a little study of the 1990-1993 AL, classifying 9 starters and 4 backups for each team by their primary defensive position. Anyone who was not a starter or backup was classified as a FAT player. Let me point out now that there are numerous potential problems with this study, coming from selective sampling, and the fact that if a player gets hurt, his backup may wind up classified as the starter and the true starter as the backup, etc. There are many such issues with this study but I will use it as a basis for discussion and not as proof of anything.

The first interesting thing is to look at the total offensive performance for each group. Starters performed at 106.9 in terms of adjusted RG, or about a .530 player. Backups came in at 87.0, or about .430. And the leftovers, the "FAT" players, were at 73.8, or about .350.

So you can see where the .350 comes from. The players with the least playing time have an aggregate performance level of about .350. If you combine the totals for bench players and FAT players, properly weighted, you get 81.7, or about .400. And there you can see where someone like Woolner got 80%--the aggregate performance of non-starters.

Now, let's look at this in terms of a chaining sense. There are 4 bench players per team, and let's assume that each of them can play one position each on an everyday basis. We'll ignore DH, since anyone can be a DH. So 50% of the positions have a bench player, who performs at the 87% level, who can replace the starter, and the other half must be replaced by a 73.8%. The average there is 80.4%, or about .390. So even using these somewhat conservative assumptions, and even if the true FAT level is .350, the comparison point for a starter is about .390, in terms of the player who can actually replace him.

Just to address the problems in this study again, one is that if the starter goes down, the backup becomes the starter and the FAT guy becomes the backup. Or the backup goes downit doesn't adjust for the changes that teams are forced to make, that skew the results. Another is that if a player does not perform, he may be discarded. So there is selective sampling going on. A player might have the ability to perform at a .500 level, but if he plays at a .300 level for 75 PA, they're going to ship him out of their. This could especially effect the FAT players; they could easily be just as talented as the bench players, but hit poorly and that's why they didn't get enough PA to qualify as a bench player.

The point was not to draw conclusions from the study. Anyway, why can't we rate players against three standards? We could compare(using the study numbers even though we know they are not truly correct) to the .530 level for the player's value as a starter; to .430 for the player's value as a bench player; and to .350 for a player's value to play in the major leagues at any time. Call this the "muti-tiered" approach. And that's another point. The .350 players don't really have jobs! They get jobs in emergencies, and don't get many PA anyway. The real level of who can keep a job, on an ideal 25 man roster, is that bench player. Now if the average bench player is .430, maybe the minimum level for a bench player is .400.

Anyway, what is wrong with having three different measures for three different areas of player worth? People want one standard on which to evaluate players, for the "general" question of value. Well why can't we measure the first 50 PAs against FAT, the next 150 against the bench, and the others against a starter? And you could say, "Well, if a .400 player is a starter, that's not his fault; he shouldn't have PAs 200-550 measured against a starter." Maybe, maybe not. If you want to know what this player has actually done for his team, you want to take out the negative value. But if you're not after literal value, you could just zero out negative performances.

A player who gets 150 PA and plays at a .475 level has in a way helped his team, relative to his opponent. Because while the opponent as a whole is .500, the comparable piece on the other team is .430. So he has value to his team versus his opponent; the opponent doesn't have a backup capable of a .475 performance. But if the .475 player gets 600 PA, he is hurting you relative to your opponent.

And finally, let's tie my chaining argument back in with the progressive argument. Why should a player's career value be compared to a player who is barely good enough to play in the majors for two months? If a player is around for ten years, he better at some point during his career at least perform at the level of an average bench player.

Now the funny thing is that I am about to end this article, I am not going to tell you what baseline I want to use, if I was forced to choose one baseline. I know for sure it wouldn't be .350. It would be something between .400 and .500. .400 on the conservative side, evaluating the raw data without taking its biases into account. .500 if I say "screw it all, I want literal value." But I think I will use .350, and publish RAR, because even though I don't think it is right, it seems to be the consensus choice among sabermetricians, many of whom are smarter than me. Is that a sell out? No. I'll gladly argue with them about it any time they want. But since I don't really know what it is, and I don't really know how to go about studying it to find it and control all the biases and pitfalls, I choose to error on the side of caution. EXTREME caution(got to get one last dig in at the .350 level).

I have added a spreadsheet that allows you to experiment with the "multi-tiered" approach. Remember, I am not endorsing it, but I am presenting it as an option and one that I personally think has some merit.

Let me first explain the multi-tiered approach as I have sketched it out in the spreadsheet. There are two "PA Thresholds". The first PA threshold is the "FAT" threshold; anyone with less PA than this is considered a FAT player. The second is the backup threshold; anyone with more PA than that is a starter. Anyone with a number of PA in between is the two thresholds is a backup.

So, these are represented as PA Level 1 and PA Level 2 in the spreadsheet, and are set at 50 and 200. Play around with those if you like.

Next to the PA Levels, there are percentages for FAT, backup, and starter. These are the levels at which value is considered to start for a player in each class, expressed as a percentage of league r/o. Experiment with these too; I have set the FAT at .73, the backup at .87, and the starter at 1.

Below there, you can enter the PA, O, and RC for various real or imagined players. "N" is the league average runs/game. The peach colored cells are the ones that do the calculations, so don't edit them.

RG is simply the players' runs per game figure. V1 is the player's value compared to the FAT baseline. V2 is the player's value compared to the backup baseline. V3 is the player's value compared to the starter baseline. v1 is the player's value against the FAT baseline, with a twist; only 50 PA count(or whatever you have set the first PA threshold to be). Say you have a player with 600 PA. The multi-tiered theory holds that he has value in being an above FAT player, but only until he reaches the first PA threshold. Past that, you should not have to play a FAT player, and he no longer has value. If the player has less than 50 PA, his actual PA are used.

v2 does the same for the backup. Only the number of PA between the two thresholds count, so with the default, there are a maximum of 150 PA evaluated against this level. If the player has less than the first threshold, he doesn't get evaluated here at all; he gets a 0.

v3 applies the same concept to the starter level. The number of PA that the player has over the second threshold are evaluated against the starter level. If he has less than the second threshold, he gets a 0.

SUM is the sum of v1, v2, and v3. This is one of the possible end results of the multi-tiered approach. comp is the baseline that the SUM is comparing against. A player's value in runs above baseline can be written as:

(R-xN)*O/25 = L

Where x is the baseline, 25 is the o/g default, and L is the value. We can solve for x even if the equation we use to find L does not explicitly use a baseline(as is the case here):

x = (RG-25*L/O)/N

So the comp is the effective baseline used by the multi-tiered approach. As you will see, this will vary radically, which is the point of the multi-tiered approach.

The +only SUM column is the sum of v1, v2, and v3, but only counting positive values. If a player has a negative value for all 3, his +only SUM is 0. If a player has a v1 of +3, a v2 of +1, and a v3 of -10, his +only SUM is +4, while his SUM would have been -6. The +only SUM does nto penalize the player if he is used to an extent at which he no longer has value. It is another possible final value figure of the multi-tiered approach. The next column, comp, does the same thing as the other comp problem, except this time it is based on the +only SUM rather than the SUM.

So play around with this spreadsheet. See what the multi-tiered approach yields and whether you think it is worth the time of day.

One of the objections that would be raised to the multi-tiered approach is that there is a different baseline for each player. I think this is the strength of it, of course. Think of it like progressive tax brackets. It is an uncouth analogy for me to use, but it helps explain it, and we're not confiscating property from these baseball players. So, let's just say that the lowest bracket starts at 25K at 10%, and then at 35K it jumps to 15%. So would you rather make 34K or 36K?

It's an absurd question, isn't it? At 34K, you will pay $900 in taxes, while at 36K you will pay $1150, but your net income is $33,100 against $34,850. Of course you want to make 36K, even if that bumps you into the next bracket.

The same goes for the multi-tiered value. Sure, a player who performs at 120% of the league average in 190 PA is having his performance "taxed" at 83%, and one who has 300 PA is being "taxed" at 89%. But you'd still rather have the guy in 300 PA, and you'd still rather make 36 grand.

Now, just a brief word about the replacement level(s) used in the season statistics on this website. I have used the .350 standard although I personally think it is far too low. The reason I have done this is that I don't really have a good answer for what it should be(although I would think something in the .8-.9 region would be a reasonable compromise). Anyway, since I don't have a firm choice myself, I have decided to publish results based on the baseline chosen by plurality of the sabermetric community. This way they are useful to the most number of people for what they want to look at. Besides, you can go in on the spreadsheet and change the replacement levels to whatever the heck you want.

I am now using 73% (.350) for hitters, 125% (.390) for starters, and 111% (.450) for relievers. There is still a lot of room for discussion and debate, and I'm hardly confident that those are the best values to use.

Replacement Level Fielding

There are some systems, most notably Clay Davenport’s WARP, that include comparisons to replacement level fielders. I believe that these systems are incorrect, as are those that consider replacement level hitters; however, the distortions involved in the fielding case are much greater.

The premise of this belief is that players are chosen for their combined package of offense and defense, which shouldn’t be controversial. Teams also recognize that hitting is more important, even for a shortstop, than is fielding. Even a brilliant defensive shortstop like Mario Mendoza or John McDonald doesn’t get a lot of playing time when they create runs at 60% of the league average. And guys who hit worse than that just don’t wind up in the major leagues.

It also turns out at the major league level that hitting and fielding skill have a pretty small correlation, for those who play the same position. Obviously, in the population at large, people who are athletically gifted at hitting a baseball are going to carry over that talent to being gifted at fielding them. When you get to the major league level, though, you are dealing with elite athletes. My model (hardly groundbreaking) of the major league selection process is that you absolutely have to be able to hit at say 60% of the league average (excluding pitchers of course). If you can’t reach this level, no amount of fielding prowess is going to make up for your bat (see Mendoza example). Once you get over that hitting threshold, you are assigned to a position based on your fielding ability. If you can’t field, you play first base. Then, within that position, there is another hitting threshold that you have to meet (let’s just say it’s 88% of average for a first baseman).

The bottom line is that there is no such thing as a replacement level fielder or a replacement level fielder. There is only a replacement level player. A replacement level player might be a decent hitter (90 ARG) with limited defensive ability (think the 2006 version of Travis Lee), who is only able to fill in some times at first base, or he might be a dreadful hitter (65 ARG) who can catch, and is someone’s third catcher (any number of non-descript reserve catchers floating around baseball).

Thus, to compare a player to either a replacement level fielder or hitter is flawed; that’s not how baseball teams pick players. Your total package has to be good enough; if you are a “replacement level fielder” who can’t hit the ball out of the infielder, you probably never even get a professional contract. If you are a “replacement level hitter” who fields like Dick Stuart, well, you’d be a heck of a softball player.

However, if you do compare to a “replacement level hitter” at a given position, you can get away with it. Why? Because, as we agreed above, all players are chosen primarily for their hitting ability. It is the dominant skill, and by further narrowing things down by looking at only those who play the same position, you can end up with a pretty decent model. Ideally, one would be able to assign each player a total value (hitting and fielding) versus league average, but the nature of defense (how do you compare a first baseman to a center fielder defensively?) makes it harder. Not impossible, just harder, and since you can get away fairly well with doing it the other way, a lot of people (myself included) choose to do so.

Of course, there are others that just ignore it. I saw a NL MVP analysis for 2007 just yesterday on a well-respected analytical blog (I am not going to name it because I don’t want to pick on them) that simply gave each player a hitting Runs Above Average figure and added it to a fielding RAA figure which was relative to an average fielder at the position. The result is that Hanley Ramirez got -15 runs for being a bad shortstop, while Prince Fielder got -5 runs for being a bad first baseman. Who believes that Prince Fielder is a more valuable defensive asset to a team than Hanley Ramirez? Anyone?

Comparing to a replacement level fielder as Davenport does is bad too, but it is often not obvious to people. I hope that my logic above has convinced you why it is a bad idea; now let’s talk about the consequences of it. Davenport essentially says that a replacement level hitter is a .350 OW%, or 73 ARG hitter. This is uncontroversial and may be the most standard replacement level figure in use. But most people agreed upon this figure under the premise that it is an average defender who hits at that level. Davenport’s system gives positive value above replacement to anyone who can hit at this level, even if they are a first baseman. Then, comparing to a replacement level fielder serves as the position adjustment. Shortstops have a lower replacement level than first baseman (or, since the formula is not actually published, it seems like this is the case), and so even Hanley picks up more FRAR than Prince. However, now the overall replacement level is now much lower than .350.

So Davenport’s WARP cannot be directly compared to the RAR/WAR figures I publish, or even BP’s own VORP. If one wants to use a lower replacement level, they are free to do so, but since the point Davenport uses is so far out of line with the rest of the sabermetric community, it seems like some explanation and defense of his position would be helpful. Also, even if one accepts the need for a lower replacement level, it is not at all clear that the practice of splitting it into hitting and fielding is the best way to implement it.

LWR Component Deflators and Replacement Hitters’ Batting Lines

Last time, I explained how we could use Linear Weight Ratio, an offensive metric developed by Tango Tiger, as a shortcut in finding a variable which I call the “component deflator” and symbolized as “a”.

Last time I focused on its application to park adjustments, but that’s not necessary. In fact, the component deflator can be applied generally to any situation in which you’d like to know what across the board percentage change in mutually exclusive offensive events would you have to see in order to alter run scoring by some scalar (assuming you are willing to accept that the linear weight values stay constant, which is certainly an assumption that must be applied with care).

So there are any number of questions that this kind of approach can address. One that I will consider in this piece is “What would a replacement-level hitter’s batting line look like?" First a few caveats, though. As Tango has pointed out, there really is no such thing as a replacement-level hitter. A replacement player is a replacement player because his overall contribution, offense and defense, is at a level so as to have no marginal value. Thinking about it in terms of a replacement level hitter only confuses the issue.

However, the analytical structure of assuming that replacement level players are average in the field, and thus calculating their value as their offensive contribution above a “replacement” performer specific to their position plus their defensive contribution above an average performer at their position can be a useful approximation. It is the structure used by a number of approaches, including Pete Palmer’s TPR (which is above average, but the same principle holds), Keith Woolner’s VORP, and the RAR figures I post here at the end of each season. I am not claiming that this approach is optimal or superior to the others, only that if applied with caution it can be a useful model of player value.

For the sake of this post, let’s just assume that we are going to use a model where a replacement level player hits at some percentage of the league average. Then, if we’d like to know what his batting line might look like, we can use the component deflator approach. The good thing is that we don’t have to worry so much about the fact that we have static linear weights, since we are now applying the process to individuals for whom we’d like to hold the weights constant (ignoring the Theoretical Team arguments). So that caveat is loosened in this application.

Of course, this approach carries some of its own caveats with it: one is that we are again developing a model in which all events are equally deflated. It might actually be that replacement level hitters tend to not be as deficient in BA as one might expect. Or maybe teams are willing to trade BA for power in a replacement level hitter. This is a specific model with specific assumptions, and it is not necessarily reality.

Anyway, if we define R as the percentage of league average (or positional average or anything else if you’d like), then we can just plug it into one of the formulas from last time, and carry out the rest of the calculations as explained in that post:

New LWR = ((LWR/s' + x)*R - x)*s'

In my RAR estimates, I assume that a replacement player’s R/O is 73% of the positional average, where the positional average is figured by taking the overall league average times a long-term offensive positional adjustment. The positional adjustments I use are (note: you can tell how long ago I wrote this by the use of 2008 league totals):

C = .89, 1B/DH = 1.19, 2B = .93, 3B = 1.01, SS = .86, LF/RF = 1.12, CF = 1.02

Combining these adjustments, the LWR component deflator procedure, and the overall 2008 MLB offensive averages, here is the offensive output expected from a replacement player at each position:



How do these numbers look to you? My impression is that the batting averages are too low; teams may resort to replacement level players at 73% of league R/O, but they may be those that trade secondary skills for BA points of equivalent value. (assuming that players of this profile even exist in reality)

Anyway, you don’t have to take any of this too seriously, and I’ve already stated that the assumptions and admitted they may not model reality, so I’m not going to spend too much time justifying the results. Instead, I have another potentially amusing if not completely realistic application.

Namely, it is to take the initial statistics of a real hitter, and maintaining the proportional relationships between his positive events, projecting what his line would look like at a different level of productivity. For example, what would a replacement level hitter with Barry Bonds’ bizarre 2004 proportional relationships look like?

In this case, I’ll assume that a replacement player would have a 3.50 RG. Bonds’ 2004 line comes out to a 18.26 RG, so our “R” will be 3.5/18.26 = 19.2%. This line results:



The bizarro Bonds would hit just .140, but would still manage to put up a .416 secondary average. Of course, such a player would never really exist, but if he did, his offensive value would be about the same as the other replacement level guys above.

Let’s look at Tony Gwynn, 1994 to see what this would look like for a very good singles-type hitter:




And we could try going the other way. What would Mario Mendoza, superstar look like? Here’s the transformation from Mendoza’ career line to a 8 RG:



In order to turn Mendoza’s no-secondary skills profile into an all-time upper echelon great, you have to allow him to hit .400, and increase all of his positive rates by 81%.

This translation approach falls squarely under the category of "toy"; please don’t get the impression that I’m elevating it to any greater pedestal.

Baseline Matters

One of the common traps I fall into in attempting to come up with topics for this blog is latching on to a sentiment expressed by one or two random people on a message board somewhere, proceeding as if this viewpoint is widely held and needs to be debunked. Hopefully, the resulting missives have some general value as a discussion of the underlying topic rather than as simply a response to a strawman.

A sentiment of this type that has caught my eye a few times recently is the notion that the actual baseline or reference point used in a total value metric doesn’t really matter. This seems so self-evidently wrong that it’s not worth addressing, but perhaps it would be useful to take a step back from the question of what the baseline should be (i.e. average v. replacement, or where exactly the replacement level line is drawn), or the theoretical justification for the baseline of choice (such a concept of “freely available talent”, a way to peg the value of a minimum salary veteran, the level of an average bench player, etc.), and instead ask why so many are compelled to use a baseline when looking at baseball statistics at all.

The most basic reason to use a baseline is to strike a balance between quantity and quality. One can rank batters by total hits (quantity) or by batting average (quality), but both of these are incomplete on their own as they consider only one of the dimensions of the observed performance. Sometimes, you might only be interested in one or the other, and that’s okay--there's certainly nothing wrong with looking at a total or a rate only as the question under investigation demands. But for the majority of questions people ask that involve player comparisons, both quantity and quality must enter into the equation.

A baseline is a very simple way to accomplish this. Define some floor below which additional playing time is not deemed helpful, or possibly even should be penalized. Use this floor as a reference point and you have now incorporated both quality and quantity. Again, I’ll stress that this is the simplest rationale for using a baseline that I could offer--I've said nothing about ancillary benefits (such as theoretical purity or a reference point that models a meaningful real world concept). Such ancillary benefits are often the reason behind advocates’ choice of a particular baseline, but they are not the focus of this post.

That all should seem simple enough, but there are a few other points to make, which will work best in bullet format.

* People will sometimes object to the use of a baseline by stating that the use of a particular baseline makes the exact value used as a baseline the focal point of one’s player evaluation system. I disagree with this characterization, although it may be true of some extreme baseline choices (a baseline of 1.000, for instance, would have incredibly distortive effects). As long as a reasonable baseline is selected, the performance of the players in the segment of the game under consideration remains the focal point of the metric. That is not to say that the baseline is of no consequence (more on this below), but if sensible it does not greatly distort a comparison between two players--it facilitates it.

* Baselined metrics are a natural target for criticism since it admittedly can be difficult to explain exactly why the baseline has been set where it is (average is an exception to this since it is intuitive). Even if one can eloquently articulate the definition of replacement level (or alternative) that has been used, the process that led to a given value is not so easy to define. Usually there is a fair amount of judgment involved, and that means that the sabermetrician’s subjective judgment has come into play.

The good news is that the minute differences between baselines are rarely material--using a .340 baseline or a .360 baseline will result in very similar results, especially for single seasons. So if one can agree on the logic used to hone in on the baseline, the actual point at which the line is drawn is not as important.

* The notion that the choice of baseline doesn’t matter at all is another matter. It’s difficult to respond to this claim, since it’s obviously wrong on its face. It’s more interesting to try to figure out the perspective(s) from which it might be true. I have two suggestions, one well-developed and one a pseudo-psychological mess:

a) Baselines don’t matter if you have two players with equal playing time. For example, let’s suppose that the league average BA is .260 and “replacement level” is .230. Player A hits .315 and Player B hits .260, each in 600 AB. Relative to average, Player A is +33 hits and Player B is +0 hits. Relative to replacement level, Player A is +51 hits and Player B is +18 hits. In each case, the difference between the two players is 33 hits relative to baseline (including the zero baseline, raw hits, since Player A had 189 and Player B had 156).

But as soon as you change playing time, the baseline comes into play. If Player A had 400 AB, and Player B had 600 AB, then relative to average Player A is +22 hits and Player B is +0 hits. Relative to replacement, Player A is +34 hits and Player B remains +18 hits. We’ve gone from a difference of 22 baselined hits to a difference of 16. This example is not extreme enough to change the rank order, which is precisely why I chose it--to illustrate subtle differences even in cases in which the conclusion appears obvious before applying a baseline at all.

b) A disproportionate number of assertions that baseline doesn’t matter seem to come from people who consider themselves the smartest guy in the room. Perhaps these people don’t need a baseline to balance quantity and quality. But the rest of us need some help, a systematic manner in which to compress two dimensions of information onto one scale for comparison. Doing so does not require one to discard the rate and playing time components, nor does it require one to follow in lockstep with the conclusions implied by the chosen baseline.

A Brief, Incomplete History of Replacement Level

Baseball Prospectus was kind enough to run a meta-sabermetric history piece written by me on the history of replacement level in sabermetric analysis. As you can imagine, it's not a topic that has generated a lot of responses, but the one person who has commented on the article so far seems to like it, and perhaps you will too.

End of Season Statistics 2012

The spreadsheets are published as Google Spreadsheets, which you can download in Excel format by changing the extension in the address from "=html" to "=xls". That way you can download them and manipulate things however you see fit. The player spreadsheets are not ready yet, but I want to get the team stuff posted.

The data comes from a number of different sources. Most of the basic data comes from Doug's Stats, which is a very handy site, or Baseball-Reference. KJOK's park database provided some of the data used in the park factors, but for recent seasons park data comes from B-R. Data on pitcher's batted ball types allowed, doubles/triples allowed, and inherited/bequeathed runners comes from Baseball Prospectus.

The basic philosophy behind these stats is to use the simplest methods that have acceptable accuracy. Of course, "acceptable" is in the eye of the beholder, namely me. I use Pythagenpat not because other run/win converters, like a constant RPW or a fixed exponent are not accurate enough for this purpose, but because it's mine and it would be kind of odd if I didn't use it.

If I seem to be a stickler for purity in my critiques of others' methods, I'd contend it is usually in a theoretical sense, not an input sense. So when I exclude hit batters, I'm not saying that hit batters are worthless or that they *should* be ignored; it's just easier not to mess with them and not that much less accurate.

I also don't really have a problem with people using sub-standard methods (say, Basic RC) as long as they acknowledge that they are sub-standard. If someone pretends that Basic RC doesn't undervalue walks or cause problems when applied to extreme individuals, I'll call them on it; if they explain its shortcomings but use it regardless, I accept that. Take these last three paragraphs as my acknowledgment that some of the statistics displayed here have shortcomings as well.

The League spreadsheet is pretty straightforward--it includes league totals and averages for a number of categories, most or all of which are explained at appropriate junctures throughout this piece. The advent of interleague play has created two different sets of league totals--one for the offense of league teams and one for the defense of league teams. Before interleague play, these two were identical. I do not present both sets of totals (you can figure the defensive ones yourself from the team spreadsheet, if you desire), just those for the offenses. The exception is for the defense-specific statistics, like innings pitched and quality starts. The figures for those categories in the league report are for the defenses of the league's teams. However, I do include each league's breakdown of basic pitching stats between starters and relievers (denoted by "s" or "r" prefixes), and so summing those will yield the totals from the pitching side. The one abbreviation you might not recognize is "N"--this is the league average of runs/game for one team, and it will pop up again.

The Team spreadsheet focuses on overall team performance--wins, losses, runs scored, runs allowed. The columns included are: Park Factor (PF), Home Run Park Factor (PFhr), Winning Percentage (W%), Expected W% (EW%), Predicted W% (PW%), wins, losses, runs, runs allowed, Runs Created (RC), Runs Created Allowed (RCA), Home Winning Percentage (HW%), Road Winning Percentage (RW%) [exactly what they sound like--W% at home and on the road], Runs/Game (R/G), Runs Allowed/Game (RA/G), Runs Created/Game (RCG), Runs Created Allowed/Game (RCAG), and Runs Per Game (the average number of runs scored an allowed per game). Ideally, I would use outs as the denominator, but for teams, outs and games are so closely related that I don’t think it’s worth the extra effort.

The runs and Runs Created figures are unadjusted, but the per-game averages are park-adjusted, except for RPG which is also raw. Runs Created and Runs Created Allowed are both based on a simple Base Runs formula. The formula is:

A = H + W - HR - CS
B = (2TB - H - 4HR + .05W + 1.5SB)*.76
C = AB - H
D = HR
Naturally, A*B/(B + C) + D.

I have explained the methodology used to figure the PFs before, but the cliff’s notes version is that they are based on five years of data when applicable, include both runs scored and allowed, and they are regressed towards average (PF = 1), with the amount of regression varying based on the number of years of data used. There are factors for both runs and home runs. The initial PF (not shown) is:

iPF = (H*T/(R*(T - 1) + H) + 1)/2
where H = RPG in home games, R = RPG in road games, T = # teams in league (14 for AL and 16 for NL). Then the iPF is converted to the PF by taking x*iPF + (1-x), where x = .6 if one year of data is used, .7 for 2, .8 for 3, and .9 for 4+.

It is important to note, since there always seems to be confusion about this, that these park factors already incorporate the fact that the average player plays 50% on the road and 50% at home. That is what the adding one and dividing by 2 in the iPF is all about. So if I list Fenway Park with a 1.02 PF, that means that it actually increases RPG by 4%.

In the calculation of the PFs, I did not get picky and take out “home” games that were actually at neutral sites, like the Astros/Cubs series that was moved to Milwaukee in 2008. I have also reset the NYN park factor due to park modifications; only 2012 data for the Mets is being considered.

There are also Team Offense and Defense spreadsheets. These include the following categories:

Team offense: Plate Appearances, Batting Average (BA), On Base Average (OBA), Slugging Average (SLG), Secondary Average (SEC), Walks per At Bat (WAB), Isolated Power (SLG - BA), R/G at home (hR/G), and R/G on the road (rR/G) BA, OBA, SLG, WAB, and ISO are park-adjusted by dividing by the square root of park factor (or the equivalent; WAB = (OBA - BA)/(1 - OBA) and ISO = SLG - BA).

Team defense: Innings Pitched, BA, OBA, SLG, Innings per Start (IP/S), Starter's eRA (seRA), Reliever's eRA (reRA), Quality Start Percentage (QS%), RA/G at home (hRA/G), RA/G on the road (rRA/G), Battery Mishap Rate (BMR), Modified Fielding Average (mFA), and Defensive Efficiency Record (DER). BA, OBA, and SLG are park-adjusted by dividing by the square root of PF; seRA and reRA are divided by PF.

The three fielding metrics I've included are limited it only to metrics that a) I can calculate myself and b) are based on the basic available data, not specialized PBP data. The three metrics are explained in this post, but here are quick descriptions of each:

1) BMR--wild pitches and passed balls per 100 baserunners = (WP + PB)/(H + W - HR)*100

2) mFA--fielding average removing strikeouts and assists = (PO - K)/(PO - K + E)

3) DER--the Bill James classic, using only the PA-based estimate of plays made. Based on a suggestion by Terpsfan101, I've tweaked the error coefficient. Plays Made = PA - K - H - W - HR - HB - .64E and DER = PM/(PM + H - HR + .64E)

Next are the individual player reports. I defined a starting pitcher as one with 15 or more starts. All other pitchers are eligible to be included as a reliever. If a pitcher has 40 appearances, then they are included. Additionally, if a pitcher has 50 innings and less than 50% of his appearances are starts, he is also included as a reliever (this allows some swingmen type pitchers who wouldn’t meet either the minimum start or appearance standards to get in).

For all of the player reports, ages are based on simply subtracting their year of birth from 2011. I realize that this is not compatible with how ages are usually listed and so “Age 27” doesn’t necessarily correspond to age 27 as I list it, but it makes everything a heckuva lot easier, and I am more interested in comparing the ages of the players to their contemporaries, for which case it makes very little difference. The "R" category records rookie status with a "R" for rookies and a blank for everyone else; I've trusted Baseball Prospectus on this. Also, all players are counted as being on the team with whom they played/pitched (IP or PA as appropriate) the most.

For relievers, the categories listed are: Games, Innings Pitched, estimated Plate Appearances (PA), Run Average (RA), Relief Run Average (RRA), Earned Run Average (ERA), Estimated Run Average (eRA), DIPS Run Average (dRA), Strikeouts per Game (KG), Walks per Game (WG), Guess-Future (G-F), Inherited Runners per Game (IR/G), Batting Average on Balls in Play (%H), Runs Above Average (RAA), and Runs Above Replacement (RAR).

IR/G is per relief appearance (G - GS); it is an interesting thing to look at, I think, in lieu of actual leverage data. You can see which closers come in with runners on base, and which are used nearly exclusively to start innings. Of course, you can’t infer too much; there are bad relievers who come in with a lot of people on base, not because they are being used in high leverage situations, but because they are long men being used in low-leverage situations already out of hand.

For starting pitchers, the columns are: Wins, Losses, Innings Pitched, Estimated Plate Appearances (PA), RA, RRA, ERA, eRA, dRA, KG, WG, G-F, %H, Pitches/Start (P/S), Quality Start Percentage (QS%), RAA, and RAR. RA and ERA you know--R*9/IP or ER*9/IP, park-adjusted by dividing by PF. The formulas for eRA and dRA are based on the same Base Runs equation and they estimate RA, not ERA.

* eRA is based on the actual results allowed by the pitcher (hits, doubles, home runs, walks, strikeouts, etc.). It is park-adjusted by dividing by PF.

* dRA is the classic DIPS-style RA, assuming that the pitcher allows a league average %H, and that his hits in play have a league-average S/D/T split. It is park-adjusted by dividing by PF.

The formula for eRA is:

A = H + W - HR
B = (2*TB - H - 4*HR + .05*W)*.78
C = AB - H = K + (3*IP - K)*x (where x is figured as described below for PA estimation and is typically around .93) = PA (from below) - H - W
eRA = (A*B/(B + C) + HR)*9/IP

To figure dRA, you first need the estimate of PA described below. Then you calculate W, K, and HR per PA (call these %W, %K, and %HR). Percentage of balls in play (BIP%) = 1 - %W - %K - %HR. This is used to calculate the DIPS-friendly estimate of %H (H per PA) as e%H = Lg%H*BIP%.

Now everything has a common denominator of PA, so we can plug into Base Runs:

A = e%H + %W
B = (2*(z*e%H + 4*%HR) - e%H - 5*%HR + .05*%W)*.78
C = 1 - e%H - %W - %HR
cRA = (A*B/(B + C) + %HR)/C*a

z is the league average of total bases per non-HR hit (TB - 4*HR)/(H - HR), and a is the league average of (AB - H) per game.

In the past couple years I’ve presented a couple of batted ball RA estimates. I’ve removed these this year, not just because batted ball data exhibits questionable reliability but because these metrics were complicated to figure, required me to collate the batted ball data, and were not personally useful to me. I figure these stats for my own enjoyment and have in some form or another going back to 1997. I share them here only because I would do it anyway, so if I’m not interested in certain categories, there’s no reason to keep presenting them.

Instead, I’m showing strikeout and walk rate, both expressed as per game. By game I mean not 9 innings but rather the league average of PA/G. I have always been a proponent of using PA and not IP as the denominator for non-run pitching rates, and now the use of per PA rates is widespread. Usually these are expressed as K/PA and W/PA, or equivalently, percentage of PA with a strikeout or walk. I don’t believe that any site publishes these as K and W per equivalent game as I am here. This is not better than K%--it’s simply applying a scalar multiplier. I like it because it generally follows the same scale as the familiar K/9.

To facilitate this, I’ve finally corrected a flaw in the formula I use to estimate plate appearances for pitchers. Previously, I’ve done it the lazy way by not splitting strikeouts out from other outs. I am now using this formula to estimate PA (where PA = AB + W):

PA = K + (3*IP - K)*x + H + W
Where x = league average of (AB - H - K)/(3*IP - K)

Then KG = K*Lg(PA/G) and WG = W*Lg(PA/G).

G-F is a junk stat, included here out of habit because I've been including it for years. It was intended to give a quick read of a pitcher's expected performance in the next season, based on eRA and strikeout rate. Although the numbers vaguely resemble RAs, it's actually unitless. As a rule of thumb, anything under four is pretty good for a starter. G-F = 4.46 + .095(eRA) - .113(K*9/IP). It is a junk stat. JUNK STAT JUNK STAT JUNK STAT. Got it?

%H is BABIP, more or less--%H = (H - HR)/(PA - HR - K - W), where PA was estimated above. Pitches/Start includes all appearances, so I've counted relief appearances as one-half of a start (P/S = Pitches/(.5*G + .5*GS). QS% is just QS/(G - GS); I don't think it's particularly useful, but Doug's Stats include QS so I include it.

I've used a stat called Relief Run Average (RRA) in the past, based on Sky Andrecheck's article in the August 1999 By the Numbers; that one only used inherited runners, but I've revised it to include bequeathed runners as well, making it equally applicable to starters and relievers. I am using RRA as the building block for baselined value estimates for all pitchers this year. I explained RRA in this article , but the bottom line formulas are:

BRSV = BRS - BR*i*sqrt(PF)
IRSV = IR*i*sqrt(PF) - IRS
RRA = ((R - (BRSV + IRSV))*9/IP)/PF

The two baselined stats are Runs Above Average (RAA) and Runs Above Replacement (RAR). RAA uses the league average runs/game (N) for both starters and relievers, while RAR uses separate replacement levels for starters and relievers. Thus, RAA and RAR will be pretty close for relievers:

RAA = (N - RRA)*IP/9
RAR (relievers) = (1.11*N - RRA)*IP/9
RAR (starters) = (1.28*N - RRA)*IP/9

All players with 300 or more plate appearances are included in the Hitters spreadsheets (along with some players close to the cutoff point who I was interested in). Each is assigned one position, the one at which they appeared in the most games. The statistics presented are: Games played (G), Plate Appearances (PA), Outs (O), Batting Average (BA), On Base Average (OBA), Slugging Average (SLG), Secondary Average (SEC), Runs Created (RC), Runs Created per Game (RG), Speed Score (SS), Hitting Runs Above Average (HRAA), Runs Above Average (RAA), Hitting Runs Above Replacement (HRAR), and Runs Above Replacement (RAR).

I do not bother to include hit batters, so take note of that for players who do get plunked a lot. Therefore, PA are simply AB + W. Outs are AB - H + CS. BA and SLG you know, but remember that without HB and SF, OBA is just (H + W)/(AB + W). Secondary Average = (TB - H + W)/AB = SLG - BA + (OBA - BA)/(1 - OBA). I have not included net steals as many people (and Bill James himself) do--it is solely hitting events.

BA, OBA, and SLG are park-adjusted by dividing by the square root of PF. This is an approximation, of course, but I'm satisfied that it works well. The goal here is to adjust for the win value of offensive events, not to quantify the exact park effect on the given rate. I use the BA/OBA/SLG-based formula to figure SEC, so it is park-adjusted as well.

Runs Created is actually Paul Johnson's ERP, more or less. Ideally, I would use a custom linear weights formula for the given league, but ERP is just so darn simple and close to the mark that it’s hard to pass up. I still use the term “RC” partially as a homage to Bill James (seriously, I really like and respect him even if I’ve said negative things about RC and Win Shares), and also because it is just a good term. I like the thought put in your head when you hear “creating” a run better than “producing”, “manufacturing”, “generating”, etc. to say nothing of names like “equivalent” or “extrapolated” runs. None of that is said to put down the creators of those methods--there just aren’t a lot of good, unique names available. Anyway, RC = (TB + .8H + W + .7SB - CS - .3AB)*.322.

RC is park adjusted by dividing by PF, making all of the value stats that follow park adjusted as well. RG, the rate, is RC/O*25.5. I do not believe that outs are the proper denominator for an individual rate stat, but I also do not believe that the distortions caused are that bad. (I still intend to finish my rate stat series and discuss all of the options in excruciating detail, but alas you’ll have to take my word for it now).

I have decided to switch to a watered-down version of Bill James' Speed Score this year; I only use four of his categories. Previously I used my own knockoff version called Speed Unit, but trying to keep it from breaking down every few years was a wasted effort.

Speed Score is the average of four components, which I'll call a, b, c, and d:

a = ((SB + 3)/(SB + CS + 7) - .4)*20
b = sqrt((SB + CS)/(S + W))*14.3
c = ((R - HR)/(H + W - HR) - .1)*25
d = T/(AB - HR - K)*450

James actually uses a sliding scale for the triples component, but it strikes me as needlessly complex and so I've streamlined it. I also changed some of his division to mathematically equivalent multiplications.

There are a whopping four categories that compare to a baseline; two for average, two for replacement. Hitting RAA compares to a league average hitter; it is in the vein of Pete Palmer’s Batting Runs. RAA compares to an average hitter at the player’s primary position. Hitting RAR compares to a “replacement level” hitter; RAR compares to a replacement level hitter at the player’s primary position. The formulas are:

HRAA = (RG - N)*O/25.5
RAA = (RG - N*PADJ)*O/25.5
HRAR = (RG - .73*N)*O/25.5
RAR = (RG - .73*N*PADJ)*O/25.5

PADJ is the position adjustment, and it is based on 2002-2011 offensive data. For catchers it is .89; for 1B/DH, 1.17; for 2B, .97; for 3B, 1.03; for SS, .93; for LF/RF, 1.13; and for CF, 1.02. I had been using the 1992-2001 data as a basis for the last ten years, but finally have done an update. I’m a little hesitant about this update, as the middle infield positions are the biggest movers (higher positional adjustments, meaning less positional credit). I have no qualms for second base, but the shortstop PADJ is out of line with the other position adjustments widely in use and feels a bit high to me. But there are some decent points to be made in favor of offensive adjustments, and I’ll have a bit more on this topic in general below.

That was the mechanics of the calculations; now I'll twist myself into knots trying to justify them. If you only care about the how and not the why, stop reading now.

The first thing that should be covered is the philosophical position behind the statistics posted here. They fall on the continuum of ability and value in what I have called "performance". Performance is a technical-sounding way of saying "Whatever arbitrary combination of ability and value I prefer".

With respect to park adjustments, I am not interested in how any particular player is affected, so there is no separate adjustment for lefties and righties for instance. The park factor is an attempt to determine how the park affects run scoring rates, and thus the win value of runs.

I apply the park factor directly to the player's statistics, but it could also be applied to the league context. The advantage to doing it my way is that it allows you to compare the component statistics (like Runs Created or OBA) on a park-adjusted basis. The drawback is that it creates a new theoretical universe, one in which all parks are equal, rather than leaving the player grounded in the actual context in which he played and evaluating how that context (and not the player's statistics) was altered by the park.

The good news is that the two approaches are essentially equivalent; in fact, they are equivalent if you assume that the Runs Per Win factor is equal to the RPG. Suppose that we have a player in an extreme park (PF = 1.15, approximately like Coors Field pre-humidor) who has an 8 RG before adjusting for park, while making 350 outs in a 4.5 N league. The first method of park adjustment, the one I use, converts his value into a neutral park, so his RG is now 8/1.15 = 6.957. We can now compare him directly to the league average:

RAA = (6.957 - 4.5)*350/25.5 = +33.72

The second method would be to adjust the league context. If N = 4.5, then the average player in this park will create 4.5*1.15 = 5.175 runs. Now, to figure RAA, we can use the unadjusted RG of 8:

RAA = (8 - 5.175)*350/25.5 = +38.77

These are not the same, as you can obviously see. The reason for this is that they take place in two different contexts. The first figure is in a 9 RPG (2*4.5) context; the second figure is in a 10.35 RPG (2*4.5*1.15) context. Runs have different values in different contexts; that is why we have RPW converters in the first place. If we convert to WAA (using RPW = RPG, which is only an approximation, so it's usually not as tidy as it appears below), then we have:

WAA = 33.72/9 = +3.75
WAA = 38.77/10.35 = +3.75

Once you convert to wins, the two approaches are equivalent. The other nice thing about the first approach is that once you park-adjust, everyone in the league is in the same context, and you can dispense with the need for converting to wins at all. You still might want to convert to wins, and you'll need to do so if you are comparing the 2010 players to players from other league-seasons (including between the AL and NL in the same year), but if you are only looking to compare Jose Bautista to Miguel Cabrera, it's not necessary. WAR is somewhat ubiquitous now, but personally I prefer runs when possible--why mess with decimal points if you don't have to?

The park factors used to adjust player stats here are run-based. Thus, they make no effort to project what a player "would have done" in a neutral park, or account for the difference effects parks have on specific events (walks, home runs, BA) or types of players. They simply account for the difference in run environment that is caused by the park (as best I can measure it). As such, they don't evaluate a player within the actual run context of his team's games; they attempt to restate the player's performance as an equivalent performance in a neutral park.

I suppose I should also justify the use of sqrt(PF) for adjusting component statistics. The classic defense given for this approach relies on basic Runs Created--runs are proportional to OBA*SLG, and OBA*SLG/PF = OBA/sqrt(PF)*SLG/sqrt(PF). While RC may be an antiquated tool, you will find that the square root adjustment is fairly compatible with linear weights or Base Runs as well. I am not going to take the space to demonstrate this claim here, but I will some time in the future.

Many value figures published around the sabersphere adjust for the difference in quality level between the AL and NL. I don't, but this is a thorny area where there is no right or wrong answer as far as I'm concerned. I also do not make an adjustment in the league averages for the fact that the overall NL averages include pitcher batting and the AL does not (not quite true in the era of interleague play, but you get my drift).

The difference between the leagues may not be precisely calculable, and it certainly is not constant, but it is real. If the average player in the AL is better than the average player in the NL, it is perfectly reasonable to expect the average AL player to have more RAR than the average NL player, and that will not happen without some type of adjustment. On the other hand, if you are only interested in evaluating a player relative to his own league, such an adjustment is not necessarily welcome.

The league argument only applies cleanly to metrics baselined to average. Since replacement level compares the given player to a theoretical player that can be acquired on the cheap, the same pool of potential replacement players should by definition be available to the teams of each league. One could argue that if the two leagues don't have equal talent at the major league level, they might not have equal access to replacement level talent--except such an argument is at odds with the notion that replacement level represents talent that is truly "freely available".

So it's hard to justify the approach I take, which is to set replacement level relative to the average runs scored in each league, with no adjustment for the difference in the leagues. The best justification is that it's simple and it treats each league as its own universe, even if in reality they are connected.

The replacement levels I have used here are very much in line with the values used by other sabermetricians. This is based both on my own "research", my interpretation of other's people research, and a desire to not stray from consensus and make the values unhelpful to the majority of people who may encounter them.

Replacement level is certainly not settled science. There is always going to be room to disagree on what the baseline should be. Even if you agree it should be "replacement level", any estimate of where it should be set is just that--an estimate. Average is clean and fairly straightforward, even if its utility is questionable; replacement level is inherently messy. So I offer the average baseline as well.

For position players, replacement level is set at 73% of the positional average RG (since there's a history of discussing replacement level in terms of winning percentages, this is roughly equivalent to .350). For starting pitchers, it is set at 128% of the league average RA (.380), and for relievers it is set at 111% (.450).

I am still using an analytical structure that makes the comparison to replacement level for a position player by applying it to his hitting statistics. This is the approach taken by Keith Woolner in VORP (and some other earlier replacement level implementations), but the newer metrics (among them Rally and Fangraphs' WAR) handle replacement level by subtracting a set number of runs from the player's total runs above average in a number of different areas (batting, fielding, baserunning, positional value, etc.), which for lack of a better term I will call the subtraction approach.

The offensive positional adjustment makes the inherent assumption that the average player at each position is equally valuable. I think that this is close to being true, but it is not quite true. The ideal approach would be to use a defensive positional adjustment, since the real difference between a first baseman and a shortstop is their defensive value. When you bat, all runs count the same, whether you create them as a first baseman or as a shortstop.

That being said, using "replacement hitter at position" does not cause too many distortions. It is not theoretically correct, but it is practically powerful. For one thing, most players, even those at key defensive positions, are chosen first and foremost for their offense. Empirical work by Keith Woolner has shown that the replacement level hitting performance is about the same for every position, relative to the positional average.

Figuring what the defensive positional adjustment should be, though, is easier said than done. Therefore, I use the offensive positional adjustment. So if you want to criticize that choice, or criticize the numbers that result, be my guest. But do not claim that I am holding this up as the correct analytical structure. I am holding it up as the most simple and straightforward structure that conforms to reality reasonably well, and because while the numbers may be flawed, they are at least based on an objective formula that I can figure myself. If you feel comfortable with some other assumptions, please feel free to ignore mine.

That still does not justify the use of HRAR--hitting runs above replacement--which compares each hitter, regardless of position, to 73% of the league average. Basically, this is just a way to give an overall measure of offensive production without regard for position with a low baseline. It doesn't have any real baseball meaning.

A player who creates runs at 90% of the league average could be above-average (if he's a shortstop or catcher, or a great fielder at a less important fielding position), or sub-replacement level (DHs that create 4 runs a game are not valuable properties). Every player is chosen because his total value, both hitting and fielding, is sufficient to justify his inclusion on the team. HRAR fails even if you try to justify it with a thought experiment about a world in which defense doesn't matter, because in that case the absolute replacement level (in terms of RG, without accounting for the league average) would be much higher than it is currently.

The specific positional adjustments I use are based on 1992-2001 data. There's no particular reason for not updating them; at the time I started using them, they represented the ten most recent years. I have stuck with them because I have not seen compelling evidence of a change in the degree of difficulty or scarcity between the positions between now and then, and because I think they are fairly reasonable. The positions for which they diverge the most from the defensive position adjustments in common use are 2B, 3B, and CF. Second base is considered a premium position by the offensive PADJ (.94), while third base and center field are both neutral (1.01 and 1.02).

Another flaw is that the PADJ is applied to the overall league average RG, which is artificially low for the NL because of pitcher's batting. When using the actual league average runs/game, it's tough to just remove pitchers--any adjustment would be an estimate. If you use the league total of runs created instead, it is a much easier fix.

One other note on this topic is that since the offensive PADJ is a proxy for average defensive value by position, ideally it would be applied by tying it to defensive playing time. I have done it by outs, though.

The reason I have taken this flawed path is because 1) it ties the position adjustment directly into the RAR formula rather then leaving it as something to subtract on the outside and more importantly 2) there’s no straightforward way to do it. The best would be to use defensive innings--set the full-time player to X defensive innings, figure how Derek Jeter’s innings compared to X, and adjust his PADJ accordingly. Games in the field or games played are dicey because they can cause distortion for defensive replacements. Plate Appearances avoid the problem that outs have of being highly related to player quality, but they still carry the illogic of basing it on offensive playing time. And of course the differences here are going to be fairly small (a few runs). That is not to say that this way is preferable, but it’s not horrible either, at least as far as I can tell.

To compare this approach to the subtraction approach, start by assuming that a replacement level shortstop would create .86*.73*4.5 = 2.825 RG (or would perform at an overall level of equivalent value to being an average fielder at shortstop while creating 2.825 runs per game). Suppose that we are comparing two shortstops, each of whom compiled 600 PA and played an equal number of defensive games and innings (and thus would have the same positional adjustment using the subtraction approach). Alpha made 380 outs and Bravo made 410 outs, and each ranked as dead-on average in the field.

The difference in overall RAR between the two using the subtraction approach would be equal to the difference between their offensive RAA compared to the league average. Assuming the league average is 4.5 runs, and that both Alpha and Bravo created 75 runs, their offensive RAAs are:

Alpha = (75*25.5/380 - 4.5)*380/25.5 = +7.94

Similarly, Bravo is at +2.65, and so the difference between them will be 5.29 RAR.

Using the flawed approach, Alpha's RAR will be:

(75*25.5/380 - 4.5*.73*.86)*380/25.5 = +32.90

Bravo's RAR will be +29.58, a difference of 3.32 RAR, which is two runs off of the difference using the subtraction approach.

The downside to using PA is that you really need to consider park effects if you, whereas outs allow you to sidestep park effects. Outs are constant; plate appearances are linked to OBA. Thus, they not only depend on the offensive context (including park factor), but also on the quality of one's team. Of course, attempting to adjust for team PA differences opens a huge can of worms which is not really relevant; for now, the point is that using outs for individual players causes distortions, sometimes trivial and sometimes bothersome, but almost always makes one's life easier.

I do not include fielding (or baserunning outside of steals, although that is a trivial consideration in comparison) in the RAR figures--they cover offense and positional value only). This in no way means that I do not believe that fielding is an important consideration in player valuation. However, two of the key principles of these stat reports are 1) not incorporating any data that is not readily available and 2) not simply including other people's results (of course I borrow heavily from other people's methods, but only adapting methodology that I can apply myself).

Any fielding metric worth its salt will fail to meet either criterion--they use zone data or play-by-play data which I do not have easy access to. I do not have a fielding metric that I have stapled together myself, and so I would have to simply lift other analysts' figures.

Setting the practical reason for not including fielding aside, I do have some reservations about lumping fielding and hitting value together in one number because of the obvious differences in reliability between offensive and fielding metrics. In theory, they absolutely should be put together. But in practice, I believe it would be better to regress the fielding metric to a point at which it would be roughly equivalent in reliability to the offensive metric.

Offensive metrics have error bars associated with them, too, of course, and in evaluating a single season's value, I don't care about the vagaries that we often lump together as "luck". Still, there are errors in our assessment of linear weight values and players that collect an unusual proportion of infield hits or hits to the left side, errors in estimation of park factor, and any number of other factors that make their events more or less valuable than an average event of that type.

Fielding metrics offer up all of that and more, as we cannot be nearly as certain of true successes and failures as we are when analyzing offense. Recent investigations, particularly by Colin Wyers, have raised even more questions about the level of uncertainty. So, even if I was including a fielding value, my approach would be to assume that the offensive value was 100% reliable (which it isn't), and regress the fielding metric relative to that (so if the offensive metric was actually 70% reliable, and the fielding metric 40% reliable, I'd treat the fielding metric as .4/.7 = 57% reliable when tacking it on, to illustrate with a simplified and completely made up example presuming that one could have a precise estimate of nebulous "reliability").

Given the inherent assumption of the offensive PADJ that all positions are equally valuable, once RAR has been figured for a player, fielding value can be accounted for by adding on his runs above average relative to a player at his own position. If there is a shortstop that is -2 runs defensively versus an average shortstop, he is without a doubt a plus defensive player, and a more valuable defensive player than a first baseman who was +1 run better than an average first baseman. Regardless, since it was implicitly assumed that they are both average defensively for their position when RAR was calculated, the shortstop will see his value docked two runs. This DOES NOT MEAN that the shortstop has been penalized for his defense. The whole process of accounting for positional differences, going from hitting RAR to positional RAR, has benefited him.

I've found that there is often confusion about the treatment of first baseman and designated hitters in my PADJ methodology, since I consider DHs as in the same pool as first baseman. The fact of the matter is that first baseman outhit DH. There is any number of potential explanations for this; DHs are often old or injured, players hit worse when DHing than they do when playing the field, etc. This actually helps first baseman, since the DHs drag the average production of the pool down, thus resulting in a lower replacement level than I would get if I considered first baseman alone.

However, this method does assume that a 1B and a DH have equal defensive value. Obviously, a DH has no defensive value. What I advocate to correct this is to treat a DH as a bad defensive first baseman, and thus knock another five or ten runs off of his RAR for a full-time player. I do not incorporate this into the published numbers, but you should keep it in mind. However, there is no need to adjust the figures for first baseman upwards --the only necessary adjustment is to take the DHs down a notch.

Finally, I consider each player at his primary defensive position (defined as where he appears in the most games), and do not weight the PADJ by playing time. This does shortchange a player like Ben Zobrist (who saw significant time at a tougher position than his primary position), and unduly boost a player like Joe Mauer (who logged a lot of games at a much easier position than his primary position). For most players, though, it doesn't matter much. I find it preferable to make manual adjustments for the unusual cases rather than add another layer of complexity to the whole endeavor.

2012 Leagues

2012 Park Factors

2012 Teams

2012 Team Offense

2012 Team Defense

2012 AL Relievers

2012 NL Relievers

2012 AL Starters

2012 NL Starters

2012 AL Hitters

2012 NL Hitters