Extrapolated Wins

by G. Jay Walker and Jim Furtado

[ Webmaster's Note: The following article appears in The 1999 Big Bad Baseball Annual. ]

On November 19, 1998, Major League Baseball posted the following on its web site:

Was Sammy Sosa or Mark McGwire the best offensive player in 1998? Both players are decent but non-stellar fielders playing a lesser-skill position, so one would assume defense didn’t play much of a role in deciding the final vote. One would also like to assume that for a group as astute as the baseball writers, the "supporting cast" argument (Sosa’s team making the playoffs while McGwire’s didn’t) wasn’t the deciding criteria in this one-on-one matchup of history’s most prolific single-season home run hitters (actually a big if). So were the writers correct in their near unanimous preference for Sosa as MVP? Bonus question - how many more wins did the real winner add to his team compared to the runner-up? One? Two? Do I hear three? We’ll reveal the answer at the end of this article.

Many of you are familiar with the Offensive Wins Above Replacement (OWAR) methods used in previous editions of this book. The basic idea was to measure how many offensive wins a player was worth to his team over a replacement-level player. The general method to calculate OWAR would be to compute a player’s run production rate and subtract that from the run production rate of a replacement level player. You would then multiply the result by the number of offensive games a player was credited with (basically a measure of playing time) to get his OWAR.

Extrapolated Wins (XW) is the second pillar of our new XR/XW methodology and it replaces the OWAR methodology. The concept is the same – to measure value over a replacement player – but the entire methodology has been revamped.

In last year’s BBBA, we made our case that the way of measuring OWAR had some underlying problems. We’re not going to run through the whole argument again, but the gist of it centered around measuring the games a player was credited with in terms of the outs he recorded. So if a player is perfect and records no outs, then under that system, the player would be credited with zero offensive games and his OWAR would be zero. So if a perfect player is worth nothing, at what point did the system start breaking down and do something other than what it was intended to do?

We outlined some possible solutions, including using plate appearances as the way to measure how many offensive games to credit a player with. While plate appearances are a better than outs, it is still not the optimal solution. That's because 1) it doesn't take into account the different contexts players operate in, and 2) it ignores the indirect effect a player has on his team's run scoring.

Of course, these two limitations are intertwined. Context is not only driven by outside influences like park effects, but is also dependent on the interaction between players and the decisions the manager makes about whom to play and where to place those players in the lineup.

These two notions of placing a player in a team context and measuring his indirect effect on team run scoring are at the heart of the new Extrapolated Wins methodology.

How should run production be modeled?

Bill James and many others have long postulated that run scoring could be represented by:

With a small loss of accuracy, this can be re-written as OBA*TB (which, of course, is the foundation of James' Runs Created). Although this is close, we think it's slightly off the mark. A better representation would be:

Run Scoring = Opportunities * Run Production Ability

This is a very simple concept. The more opportunities you have and the better you are at taking advantage of those opportunities, the more runs you'll score. Of course the trick is how to best represent this basic concept.

Measuring Opportunity

The first concern in setting up a system to measure players is how could we properly represent the interaction between players on a team. We centered on the idea that "players consume plate appearances, teams consume outs". In particular, we wondered what would be the best way to properly account for the difference between the opportunity for the team (measured in outs) and the opportunity for the team's players (measured in plate appearances).

We start with the 27 outs that a team gets to work with each game. At this point of the process, the number of opportunities for the team (27) is the same as the opportunities for players (27). Of course, when the game starts, the clock starts ticking and things change.

Outs are the minutes on baseball's clock. When a team runs out of outs, it can't score any more runs. However, unlike a regular clock, baseball's timing mechanism nudges along sporadically. That's because whenever a player avoids making an out, time stands still.

We can account for that piece of the picture by defining opportunity as follows:

In the numerator are the outs a team gets in a season. (In a normal season this would equate to 4374 outs or 162 games times 27 outs.) This figure equates to the team's opportunity.

In the denominator is the rate that the players on the team consume outs. The faster the players consume outs, the faster the baseball clock ticks and as a result, the fewer opportunities (in plate appearances) the players on the team will get. This step represents the conversion process from the team's opportunities to the players' opportunities. The result is the number of opportunities the team's players have to generate runs—the team's total plate appearances.

The denominator, (Outs Consumed/Plate Appearances), we call the OUT% and is defined as:

OUT% = (AB-H+CS+GIDP+SF+SH) / PA

Then we have:

and

Measuring Run Production Ability

To generate the number of runs a team will score, we must come up with a measure of the ability of players to generate runs or, more specifically, the rate that players produce runs given a certain opportunity. Since we've already developed a run production method, Extrapolated Runs (XR), this process is easy. All we have to do is divide XR by the plate appearances. If you’ve been reading the other articles in this section, you already know that we refer to this as Extrapolated Average or XAVG.

The formula can now be written as:

1. (TeamOuts/OUT%) * XAVG
2. TeamOuts/(TeamOuts/TeamPA) * XR/TeamPA
3. TeamOuts*(TeamPA/TeamOuts)*XR/TeamPA
4. XR ]

Adding the player into the mix

Of course, if we only wanted to measure teams, we wouldn't have to go through all these steps, XR would suffice. However, to properly assess players we need these extra measures.

This leads us to the next few steps which measure how an individual player affects a team's run scoring, or more specifically, how a player affects the team's Opportunities and Run Production Ability.

To do that, we first place each player into a standard team context. This step is important. As Jim said in the historical piece on offensive production measures, "Baseball is a team game; therefore, players should be evaluated in a team context." That's what the XW methodology does. It places players into a standard team context. The standard context we chose is an average team. Now, you may be shaking your head because Jim questioned Pete Palmer's choice of an average baseline for his Linear Weights method. Don't worry; we're not being inconsistent, because we're talking about two different things.

Pete's baseline was the comparison of a player to an average player. We don't do that. This is what we do:

1. Place a player in an average team context
2. Compute his effect on the team's run production
3. Replace the player with a replacement player
4. Compute his effect on the team's run production
5. Compare the difference

In effect, we use an average team’s context and a replacement player’s baseline.

Returning to our formula from above, we have:

To evaluate a player's effect on the team, we must calculate the effect his play has on both the team's Opportunity (TeamOuts/OUT%) and its Run Production Ability (XAVG). To accomplish this task we factor in playing time.

Bill James used outs to represent the player's part of this process, but as Jay pointed out last year in his Beauty With A Blemisharticle, there are problems with using outs in this way. Instead, we factor this part of the puzzle by using something we call Play Percentage (PLAY%). PLAY% is simply the player's total plate appearances (AB+TBB+HP+SF+SH) divided by his team's total plate appearances.

For example, say a player had 600 PA, while his team had 6000 PA. That player would then have a PLAY% of 10.0%. This method takes into account not only a player's playing time, but also, his line-up placement. Additionally, it filters out the run scoring ability of his original team. Players who play on high scoring teams get to the plate more often because their team's offense is very efficient. That's one of the reasons Derek Jeter got to the plate 694 times in 1998 even though he missed 13 games. Of course in Jeter's case, he was helped because he also batted either first or second in the lineup all year.

The Big Hurt Example

We'll now illustrate how we incorporate these factors with an example. We'll use the 1998 American League and Frank Thomas.

Let’s start off with some league figures:

LXR = League Extrapolated Runs = 11426.3
LPA = League Plate Appearances = 88165
LOuts = League Outs = AB - H + CS + SH + SF + GDP = 60915

which are used to compute the three stats we’ll need:

LOUT%  =  League Out % = LOuts/LPA = 60915 / 88165 = .6909
LXAVG =  League XAVG = LXR/LPA = 11426.3/88165 = .1296
TeamOuts  =  # of Outs for an average team = 60915/14 = 4350.9

With these stats we can calculate the expected number of Extrapolated Runs an average team would score using the (TeamOuts/OUT%) * XAVG formula.

For the 1998 AL: (4350.9/.6909)*.1296 = 816.1 Extrapolated Runs

What would an average team score with Frank Thomas in the lineup? First, his basic numbers:

Extrapolated Runs = 113.51
Plate Appearances = 712
Outs = 455
White Sox Plate Appearances = Team PA = 6280

Important note - to keep this first example straight-forward, we’re going to use Thomas’ actual numbers instead of his park-adjusted ones. The results we derive from this example for Thomas will vary a bit from results you see elsewhere in the book, because those other results use park-adjusted figures.

The stats we’ll use are:

OUT% =  Outs/PA = 455 / 712 = .6390
XAVG  =  Extrapolated Average =  XR/PA =  113.51 / 712 = .1594
PLAY%  =  Playing Time Percent = PA/Team PA = 712 / 6280  =  .1134

Using these three stats, in addition to the league average stats, we can calculate how many runs an average team would score with Frank Thomas. To illustrate, we’ll break the formula into its two components: Opportunity and Run Production Ability.

Opportunities - Frank Thomas in the lineup

Plate appearances represent the number of opportunities for a team, and we saw above that this can be expressed as (TeamOuts/OUT%). To adjust this figure for Thomas, we use Thomas’ OUT% for the amount of playing time he had, and use the league OUT% (LOUT%) for the balance. Since Thomas’ OUT% is less than the league average, he will generate additional plate appearances for his team.

NewTmPA = (TeamOuts / ( (OUT%*PLAY%) + (LOUT%*(1-PLAY%) )

For Thomas the calculation is:

4350.9/( (.6390*.1134) + (.6909*(1-.1134)) ) = 6351.5 PA

Without Thomas, an average team would be estimated to have 6297.4 PA, so with the outs Thomas avoids over an average player, he gives his team another 54 plate appearances.

Before we move on, a brief digression. Note that to calculate the plate appearances gained by Thomas, we couldn’t use the formula:

(Thomas plate appearances)*(LOUT% - Thomas’ OUT%)=(712)*(.6909 - .6390) = 36.95.

While Thomas directly saves 37 outs, to assume the team gains only 37 plate appearances makes the assumption that all those additional plate appearances resulted in outs. In fact, on average about 25 or 26 will result in outs, while the remaining 11 or 12 will result in men getting on base, generating still more plate appearances. Once all is said and done, the team should have about 54 additional plate appearances resulting from the outs Thomas avoided.

Run Production Ability - Frank Thomas in the lineup

Next, we calculate how Thomas would affect the team's Run Production Ability. The basic idea is the same as in the previous step, we use Thomas’ Extrapolated Average (XAVG) for the amount of playing time he had, and use the LXAVG for the balance:

NewTmXAVG =  (XAVG*PLAY%) + (LXAVG*(1-PLAY%))

For Thomas the calculation is:

(.1594 * .1134) + (.1296 * (1-.1134)) = .1330

Putting it all together:

Run Scoring = Opportunities * Run Production Ability = NewTmPA * NewTmXAVG.

For Thomas this equals (6351.5) * (.1330) or 844.7 runs.

An average team with Thomas playing in place of an average player will score 844.7 - 816.1 or 28.6 additional runs.

The Replacement Example

Now we give a replacement player the same treatment Thomas got. We need three figures for the replacement player, the replacement PLAY% (RPLAY%), the replacement OUT% (ROUT%) and the replacement XAVG (RXAVG).

For the replacement:
RPLAY% = .1134
ROUT% = .7484
RXAVG = .1001

Since the replacement player is taking the place of Thomas, his PLAY% is the same as Thomas’. Appendix A and Appendix B show how the values for ROUT% and RXAVG are derived.

Appendix A is actually a discussion of what is meant by the replacement rate and replacement level. In it, we talk about the concept of the Runs Deflator. We strongly suggest you read it. Appendix B gets into the actual calculations for ROUT% and RXAVG. That one we leave up to you.

While there is general agreement as to what a good "ballpark" replacement rate should be, the actual replacement value is a subjective call. For Extrapolated Wins purposes, we define the replacement rate as a .350 offense (with an average defense).

Opportunities - Replacement Player in the lineup

We use the exact same idea here that we did for Frank Thomas. To adjust the team’s opportunities or plate appearances because of the replacement player, we use his OUT% for the amount of playing time he had, and use the LOUT% for the balance.

NewTmPA = (TeamOuts / ( (ROUT% * RPLAY%) + (LOUT% * (1-RPLAY%) )

For the replacement player this equals:

4350.9 / ( (.7484 * .1134) + (.6909 * (1-.1134)) ) = 6238.6 PA

It is estimated that the average team would have 6297.4 PA, so with the outs that the replacement player incurs over an average player (given Thomas’ playing time), an average team would lose 6297.4-6238.6 or about 59 plate appearances for the season.

Run Production Ability - Replacement Player in the lineup

This calculation also parallels the one for Frank Thomas. Again, the basic idea to use the replacement’s XAVG for the amount of playing time he had, and use the LXAVG for the balance:

NewTmXAVG = (RXAVG*PLAY%) + (LXAVG*(1-PLAY%))

For the replacement player this value is: (.1001*.1134) + (.1296*(1-.1134)) = .1263

Finally for the replacement player:

Run Scoring = Opportunities * Run Production Ability = NewTmPA * NewTmXAVG, which equals (6238.6) * (.1263) or 787.9 runs.

An average team with a replacement player in place of an average player will cost the team 816.1 – 787.9 or about 28 runs (given Thomas’ playing time).

Bringing It Home

Now that we know how many runs an average team would score with Frank Thomas in their lineup compared to a replacement player, the rest if just a mopping up operation using some familiar tools. We place these runs into the Pythagorean Projection formula to estimate the winning percentage for a team with Thomas versus a team with a replacement player:

Now with the actual numbers:

This is the estimated winning % for a team with Thomas against a team with a replacement player. If we were to compute winning % for an average team against the team with a replacement player, the result would be .5161. Computing the winning percent for a team with a replacement player against itself gives a trivial result of .500.

The final step in computing the Extrapolated Wins for Thomas is to plug this number into a variation of the old Offensive Wins Above Replacement (OWAR) formula. The original OWAR formula was of the form:

OWAR = (Player Games) x (Player Winning % - Replacement Player Winning %)

One problem with this formula is trying to define Player Games, especially when the measuring unit is in terms of outs. But now, everything is defined in a team context:

Extrapolated Wins = (Team Games) x (Thomas Team Winning % - Replacement Team Winning %)

= (162) x (.5318 - .5000) = 5.15 Extrapolated Wins

Frank Thomas has 5.15 Extrapolated Wins for 1998. This is the estimated number of games an average team would win with Thomas in their lineup in place of a replacement player. Keep in mind that this is the number of Extrapolated Wins (XW) for Thomas using non-park-adjusted numbers.

The number of XW that an average player would have, given Thomas 712 plate appearances, is 162 * (.5161 - .5000) = 2.61. At least for 1998, Thomas produced about two and a half more offensive wins than an average player.

Okay, now that the math is out of the way, who were the best offensive players in 1998?

You may be wondering how McGwire could possibly be 3.70 XW better than Sosa.

A comparison using basic statistics

To compare McGwire and Sosa, let's begin with a comparison of some of their raw numbers:

Sosa topped McGwire in Batting Average by nine points or 3%.

McGwire bested Sosa in OBP (25%), SLG (16%), and XR (16%).

Of course these are the "raw" numbers. Let's take a look at their park adjusted ones:

Notice that both players appear to have played in slight hitter's parks in 1998. Also notice that adjusting for park doesn't radically change either's production figures.

Taking these statistics in isolation doesn't paint a complete picture. Each individual stat can be thought of as an individual color on our player evaluation palette. To capture the subtle nuances of the game and to create a realistic depiction, we must capably utilize our statistical brush. That's what the XW methodology was designed to do—to better capture the interaction between player and team.

A comparison using some of the XW statistics

Let's take a look at how Sosa and McGwire compare using the XW methodology.

Here's the park-adjusted stats needed to calculate their XW:

Both McGwire's and Sosa's park-adjusted XAVG (aXAVG) are very impressive. They both rank at the top of the NL in 1998.

502 Plate Appearances needed to qualify.

McGwire's aXAVG, however, is .047 better than Sosa's. That means that, on average, McGwire produced .047 more runs each PA than Sosa did, or an additional run for about every 21 plate appearances.

Looking at the Top 15 in aOUT%, we see that McGwire again tops the list, while Sosa's name is conspicuously absent.

502 Plate Appearances needed to qualify.

As a matter of fact, we'd have to include twenty more names on the list before Sosa would appear. In 1998, his .668 OUT% ranked 35^th behind such notable players as Andy Fox (30^th with .662) and Chris Gomez (32nd with .664). This indicates that Mark McGwire used up far less outs in producing runs than Sammy Sosa did. This is the key reason why McGwire had so many more Extrapolated Wins than Sosa.

Regarding the last piece of the XW "stats pack", Sosa's PLAY% is slightly higher than McGwire's PLAY%. This means that over the course of the season, Sosa had more opportunities to affect his team's offense (not by a radical difference, though).

The stats translated to the XW methodology

If these numbers are inserted into the XW system, we can generate some other figures for comparison. We take a slightly different approach here by starting with the final XW result and working our way back through the numbers.

The final XW numbers indicate that McGwire (11.63) would be 3.70 wins better than Sosa (7.93) when both are compared to a replacement player.

If Mark McGwire and Sammy Sosa were placed on a team completely comprised of average players in 1998, and if each were given the same amount of their Plate Appearances as in real life, how many runs would their respective teams score?

McGwire's team would score 853.8 runs while Sosa's would score 809.9. That's a difference of 43.9 runs. Where does that stem from?

Step 1

Because of the high rate of runs per PA (XAVG) that McGwire and Sosa produced, each would generate more runs than an average player would. Given the same percentage of PA as each player had in 1998 , McGwire would generate 80.15 additional runs, while Sosa would generate 52.17 additional runs.

For this part of the equation, McGwire adds 27.98 more runs than Sosa because of his higher XAVG.

Step 2

Because McGwire and Sosa make outs at a lower rate than average player, they give their team more opportunities (more PA) to produce runs. McGwire presented 150.12 more PA to the Cards, while Sosa gave the Cubs an additional 31.96 PA. This is the part of run scoring that other methods ignore. That's why other methods don't properly evaluate a player's ability to get on base and avoid making outs.

Part of the additional plate appearances that McGwire and Sosa generate come full around to directly give them more plate appearances at their higher production rates. McGwire’s ability to avoid outs directly gave him the additional plate appearances to produce another 3.84 more runs for his team, while Sosa produced 0.70 more runs with his additional opportunities.

After Step 2, McGwire is now up to 83.99 runs. Sosa is up to 52.87. Big Mac now has a 31.12 run lead over Sosa.

Step 3

Most of McGwire's and Sosa's ability to avoid outs and generate additional opportunities goes to the benefit of their teammates in the form of additional plate appearances. Because of these additional opportunities, McGwire’s teammates would generate 16.16 more runs; Sosa's teammates would generate 3.42 more runs.

Adding these runs to the totals from Step 2 gives McGwire 100.15 runs produced and Sosa 56.29 runs produced.

Since McGwire has more runs produced, we'll subtract Sosa's totals to generate the difference in their run production ability: 100.15-56.29 = 43.86 or 43.9 runs.

McGwire was the better offensive player in 1998 because he was better in the two areas that are most important for run scoring: the ability to get on base and avoid making outs, and the ability to directly produce runs.

Wrapping It Up

Like the old OWAR method, the new Extrapolated Wins method attempts to measure offensive wins above replacement. The key differences is that XW places the player in a team context and measures not only a player’s direct contribution to the team but also his indirect contribution of (hopefully) avoiding outs and giving more opportunities to his teammates. Look for updates on the XR/XW methodology during the season at our web sites at www.backatcha.com and www.baseballstuff.com.

Appendix A – Replacement Rates, Replacement Levels

There’s been a lot of recent confusion about replacement rates, and I suppose BBBA isn’t totally innocent in this matter. In last year’s book when we were sketching out some proposed ideas and methods that evolved into the more rigorous XR/XW methodology, we inadvertently left off a replacement level-related term in one formula which caused our run estimates to be too conservative. Admittedly, our terminology was imprecise last year. Let’s see in we can clear up any problems we may have caused last year as well as some of the general confusion out there.

To get a better handle on this replacement rate business, this year we’ve introduced the use of the Runs Deflator (our thanks to Stephen Tomlinson for this recommendation). You can combine the Runs Deflator with the Pythagorean formula to answer questions like how many fewer wins can I expect my team to have if they score 10% fewer runs. Or how many fewer wins will they have if they allow 20% more runs. Or you can combine the two – if my batters score 10% less runs AND my pitchers allow 20% more runs how many wins will it cost me?

This leads us to a few other premises:

• There are separate offensive and defensive replacement rates
• The two are independent of each other (at least to the extent offense and defense are independent of each other in a real baseball game)
• If the offensive and defensive replacement rates are the same, the team replacement rate will be something different, except for the value of .500. (This isn’t so much of a premise but something that will be shown with the Pythagorean formula. We just thought we’d mention it here.)

Let’s illustrate an example of using the Pythagorean formula to compute the Runs Deflator: How many fewer runs would my offense have to score and how many more runs would my defense have to give up so that my team can expect to play at a .350 level? Let’s define terms first:

AvgRS – average runs scored (this would usually imply a league average)
AvgRA – average runs allowed (also usually a league average)
RD – Runs Deflator

Using the Pythagorean formula, we get an equation:

Make sure to note the differences in the two terms in the denominator. (AvgRS * RD) is the percent of runs you would expect to score compared to the average, and (AvgRA / RD) is the percent of runs you would give up compared to average.

If we solve for RD, we get RD=.8444. If my offense scores runs at 84.44% of an average offense and my defense gives up runs at (1/RD)=(1/.8444)=118.42% of an average defense, we can expect that team’s winning percentage to be around .350.

If you’re assuming this is what BBBA means by a replacement level of .350, you’re wrong. What we talk about is an offensive replacement level of .350. We don’t make any assumptions about the defense other than to assume the player is on an average defensive team. The equation to solve for the Runs Deflator using the BBBA definition of replacement rates is:

Make sure to note the difference in this equation and the one above it, namely the second part of the denominator is now (AvgRA) instead of (AvgRA/RD). We’re assuming a defense that gives up only an average number of runs. If we solve this equation for RD, we get RD=.713 as a Runs Deflator. In other words, if our offense scored only 71.3% of the runs of average offense, we would expect the team to play at the .350 level, given that the defense allowed an average number of runs. The .713 value is the amount of run reduction necessary to reach a replacement level as we define it.

The other point to note is that our .350 replacement rate is the offensive replacement rate given an average defense. What too many have been guilty of, including ourselves in the past, is to throw around the term "replacement rate", not clarifying as to whether it’s the replacement rate for the entire team or just for the offense, and not specifying if the assumption is that the offense plays with an average defense or with a replacement defense.

So if our .350 replacement rate represents a .350 offense with an average defense, what would be the expected winning percentage for a replacement team, that is one with both a replacement offense and defense? We know that at the .350 replacement level the Run Deflator value is .713, we just have to apply that now to both the offense and defense. Our Pythagorean equation would look like this:

which equals .225, the expected winning percentage for a replacement team, that is, replacement players on both offense and defense. Over a 162 game schedule, this replacement team would expect to go 36-126. In last year’s book, when we spoke of a replacement team going 57-105, we were imprecise. The 57-105 record is what would be expected of a replacement offense (with an average defense). A complete replacement team, both offense and defense, would be expected to have the 36-126 record.

Finally, why does BBBA set their replacement level at .350? There is no absolutely true replacement level, it is all a matter of conjecture. It’s set at whatever level you feel most comfortable with. Some argue that the replacement rate should be set at the value of a utility player, some say it should be at the value of a Triple-A level player, some think you should look at lousy teams to get a sense of what is a proper replacement rate. All these are interesting arguments, but they’re also theoretical arguments. If the replacement rate is part of a broader methodology like our XR/XW methodology, the replacement rate in practice becomes the point where you feel quantity and quality are in balance.

What do we mean by that? If you set you replacement rate too high, quality is going to trump quantity. Put it high enough and at some point guys with 50 Extrapolated Runs in 200 plate appearances are going to be rated higher than someone with 150 Extrapolated Runs in 700 plate appearances. Set the replacement rate too low, and quantity becomes foremost at the expense of quality. At some point guys with 90 Extrapolated Runs in 700 plate appearances will be rated ahead of players with 80 Extrapolated Runs in 500 plate appearances.

A general consensus has developed that a replacement level around .350 is the proper point where quantity and quality come into balance, and that is the value BBBA has used. In most seasons, only a handful of regular and semi-regular players have offensive seasons where they finish below this replacement level. The .350 level is a best guess, and theoretical arguments aside, replacement rates are always a best guess. There is no one right answer. It’s the place where you feel comfortable that quantity and quality are in balance.

Appendix B – Calculating the Out % (ROUT%) and the Extrapolated Average (RXAVG) for replacement players

If you’ve already read Appendix A, you’ve read about the Runs Deflator and how it’s equal to 71.3% or .713 as we define it. It represents the percent of runs a team would score compared to an average team if they were at the "replacement level", which we define the as having a .350 offense accompanied by an average defense.

To get the RXOUT% and RXAVG for the replacement player, we first need to compute the amount of direct runs lost by the team because of the replacement player. From that, we can then compute ROUT% and RXAVG. We then apply these figures to a team context in a manner similar to what we did for Frank Thomas to get the total runs the replacement player costs the team, both direct and indirect.

The amount of direct runs the replacement player costs his team is:

(PA) * (LXAVG) – (PA) * (LXAVG) * (Runs Deflator) = (712) * (.1296) – (712) *(.1296) * (.713) = 26.48 runs

How are these runs lost? Basically, the replacement player records outs in situations where the average player would get on base. If we had a way of measuring the expected runs gained for an offensive event where the batter reached base, and the expected runs lost for an offensive event where the batter made an out, we could estimate how much it would cost in runs each time a player made an out instead of getting on base. From there it would be a straightforward calculation to estimate the expected number of outs the replacement player makes over an average player to cost the average team the 26.48 runs.

How do we find the expected number of runs gained from each positive (on base) event and the number of runs lost from each negative (out) event? Actually, we already have this with our Extrapolated Runs formula. We only need to multiply the frequency of occurrence of the offensive events by the worth in runs for each of the events according to Extrapolated Runs coefficients.

If we take a weighted average of the coefficients, we get a value of .5499. That is, each on-base event represents on average of a little over half a run gained.

If we take a weighted average of the coefficients, we get a value of -.0964. That is, each negative event represents on average of about a tenth of a run lost.

So each time a Replacement player makes an out in a situation when an average player would get on base, he figures to cost his team .5499 + .0964 or .6463 runs. (For the NL in 1998, the value was .6411 runs).

Therefore, if a replacement player costs an average team 26.48 direct runs, it would be expected that he would record 26.48/.6463 or 40.97 additional outs more than an average player, given Thomas’ plate appearances.

We’ve already calculated the LOUT%, or the OUT% for an average team at .6909, and we estimate that a replacement player would record 40.97 additional outs over an average player (given Thomas’ plate appearances). Therefore ROUT%, the OUT% for a replacement player equals:

ROUT% = LOUT% + 40.97/PA=.6909 + 40.97/712 = .7484

To calculate our other value, RXAVG, we need to take the league Extrapolated Average, LXAVG, and multiply that by the Run Rate Deflator. Note that we are dealing with run rates here rather than runs themselves, so we need to adjust by the Run Rate Deflator as opposed to the Run Deflator.

What is the Run Rate Deflator? Well, we had earlier used this equation:

Run Scoring = (TeamOuts/OUT%) * Run Production Ability

We can use a similar type of structure to express the relationship between the production deflators for a replacement player:

Runs Deflator  = (Out Rate Deflator) * (Run Rate Deflator)

The Runs Deflator was discussed in Appendix A and equal to .713. For the Out Rate Deflator:

Out Rate Deflator = (League OUT% / Replacement OUT%)=.6909 / .7484=.9232

The Runs Deflator formula above can be re-expressed in terms of the Runs Rate Deflator:

Runs Rate Deflator = (Runs Deflator) / (Out Rate Deflator)=.713 / .9232=.7723

Finally, for the Replacement XAVG or RXAVG:

RXAVG = LXAVG * (Runs Rate Deflator) = (.1296) * (.7723)=.1001

With values now for both ROUT% and RXAVG, we can calculate how many runs, both direct and indirect, the replacement player costs his team.

From this point, you’d proceed to the formulas under the heading "Opportunities - Replacement Player in the lineup".

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