I have done two posts, the true value of a home run and the true value of a save. These posts sprung out of the question: which is worth more, 30 home runs or 30 saves?
We found that the true value of a home run was worth 1.406 runs, and that the true value of a save was 0.11415 WPA. So how do we compare these two variables in different units? Eventually, we want to set a dollar value to each event, but we must first translate each into a win value.
It has been estimated that a win is worth somewhere between 9.5 and 10 runs. There are many different explanations how that was calculated and why it is so, but for simplicity I am just going to accept the argument that 10 runs = 1 win. We can now change the run value of a home run into a win. One home run is worth 0.1406 wins, so 30 home runs would be worth 4.218 wins.
Although it is usually not helpful to sum up WPA, in this case, it is the best we can do to approximate the value of a save. We found that the average save is worth 0.114 wins, so 30 saves would be worth 3.425 wins, using WPA. If we use WPA/LI, the average save was worth 0.0614 wins, so 30 saves would now only be worth 1.841 wins.
We have found out that, mathematically, 30 home runs are clearly worth more than 30 saves. We can now figure out how much each are worth in dollars. It has been estimated each win is worth about $4.5 million on the open market (so each win above replacement will cost approximately $4.5 million to replace, obviously a player with 8 wins above replacement is not going to be paid $36 million per year). So the value of 30 home runs, on the open market, is $18.98 million. This seems to be an unrealistic number, but there are players such as Jayson Werth who hit 27 home runs last year and received a 7-year, $126 million contract (average of $18 million/year) this offseason from the Nationals.
30 saves measured by WPA are worth 3.425 wins, or $15.41 million, and 30 saves measured by WPA/LI are worth $8.28 million. This dollar amount for WPA/LI is much more realistic than the amount for home runs. One example is Bobby Jenks, who compiled 27 saves last year and got a 2-year, $12 million contract this offseason.
So, the answer to the question of 30 home runs or 30 saves has clearly been answered. Home runs are either only slightly more valuable, or much more valuable than saves, depending on your view of relief pitchers. I believe that the math agrees with intuition here, as it seems as though it would be much easier (and cheaper) to acquire a player that will get 30 saves as opposed to a player that will hit 30 home runs. The marginal difference between an average closer (like Frank Fransisco for the Jays) and another pitcher in the bullpen (say, Jason Frasor) is much smaller than the marginal difference between a player like Aaron Hill and a bench player, such as John MacDonald.
In conclusion, I want to show one more example. This is a list of the 18 players who hit at least 30 home runs last year. The average Wins Above Replacement for the players was 4.39. If we look only at Batting Wins (WAR with the defense and running statistics removed), the players still have an average of 3.58 wins. This is a list of the 14 pitchers who saved at least 30 games last year. They have an average WAR of 2.09 wins. I believe that this shows that the pitchers who save 30 games are less valuable to their teams than the players who hit 30 home runs, which we have seen over the past three posts.
Monday, January 31, 2011
Sunday, January 30, 2011
The True Value of a Save
This post will be a little different than the previous post, in that it is much more difficult to quantify the value of a save than the value of a home run. For home runs, we can fairly easily calculate the differences between each one, as there are only 24 different ways for a home run to occur (the base-out states). There are many, many more ways for a save to unfold. There are one and two inning saves; one, two, or three run leads; and a number of different base-out combinations throughout a save attempt.
To determine the true value of a save, we are going to look at all 1,204 saves in 2010, and determine the WPA of each save. A description on WPA can be found here, but put simply, it is the probability of a team winning after an event subtracted by the probability of a team winning before the event. It will show how much a player contributed to his team winning the game. Every save from last year can be found here, sorted by WPA. The true value of a save will then be the average WPA for all saves.
The most valuable save last year was recorded by Andy Sonnanstine of Tampa Bay, with a WPA of 0.662. The least valuable save was recorded by Matt Harrison of Texas, with a WPA of only 0.001 (he actually pitched 3 innings in a blow out game, which is one of the obscure ways a reliever can get a save). The average of all saves last year was a WPA of 0.114. What this means is that the average reliever recording a save will increase his team's expected win probability by about 11% (from 89.6% to 100%).
Unfortunately, there are many debates going on (such as here) as to whether or not WPA is an accurate measurement of a relief pitcher's value. The probability of a team winning when leading going into the 9th inning has not changed whatsoever from 1952 to 2010 (which is pretty amazing!). Naturally, this calls into question the value of the modern day closer. So instead of using WPA, many sabermetricians use WPA/LI, otherwise known as Context Neutral Wins, which is described here. LI is the leverage index of a certain play, as a tie game in the 9th inning will have much more pressure than a play in the 1st inning of a game. Simply using WPA will not account for the context of the situation, so the value of a reliever could be drastically overvalued merely because they pitch in higher-leverage situations.
WPA/LI takes care of this problem by neutralizing the leverage of the situation. As a result, a player's contribution will almost always be less, especially for relievers. If we look at the WPA/LI for all of the saves from 2010, the average is WPA/LI is now only 0.061 (almost half of the WPA value).
So the problem now becomes, which statistic do we use? WPA or WPA/LI? This really depends on your own beliefs. If you believe that closers are really good pitchers who can do things other relief pitchers cannot do, especially in high-pressure situations, then you would want to use WPA. However, personally I believe that closers are only marginally better pitchers than their bullpen counterparts, and as such, are getting some undue credit. So I believe that using WPA/LI is better, especially considering that many closers are failed starting pitchers. However, I will use both statistics in comparing home runs and saves. My next post will finally answer the question of whether 30 home runs or 30 saves are more valuable.
To determine the true value of a save, we are going to look at all 1,204 saves in 2010, and determine the WPA of each save. A description on WPA can be found here, but put simply, it is the probability of a team winning after an event subtracted by the probability of a team winning before the event. It will show how much a player contributed to his team winning the game. Every save from last year can be found here, sorted by WPA. The true value of a save will then be the average WPA for all saves.
The most valuable save last year was recorded by Andy Sonnanstine of Tampa Bay, with a WPA of 0.662. The least valuable save was recorded by Matt Harrison of Texas, with a WPA of only 0.001 (he actually pitched 3 innings in a blow out game, which is one of the obscure ways a reliever can get a save). The average of all saves last year was a WPA of 0.114. What this means is that the average reliever recording a save will increase his team's expected win probability by about 11% (from 89.6% to 100%).
Unfortunately, there are many debates going on (such as here) as to whether or not WPA is an accurate measurement of a relief pitcher's value. The probability of a team winning when leading going into the 9th inning has not changed whatsoever from 1952 to 2010 (which is pretty amazing!). Naturally, this calls into question the value of the modern day closer. So instead of using WPA, many sabermetricians use WPA/LI, otherwise known as Context Neutral Wins, which is described here. LI is the leverage index of a certain play, as a tie game in the 9th inning will have much more pressure than a play in the 1st inning of a game. Simply using WPA will not account for the context of the situation, so the value of a reliever could be drastically overvalued merely because they pitch in higher-leverage situations.
WPA/LI takes care of this problem by neutralizing the leverage of the situation. As a result, a player's contribution will almost always be less, especially for relievers. If we look at the WPA/LI for all of the saves from 2010, the average is WPA/LI is now only 0.061 (almost half of the WPA value).
So the problem now becomes, which statistic do we use? WPA or WPA/LI? This really depends on your own beliefs. If you believe that closers are really good pitchers who can do things other relief pitchers cannot do, especially in high-pressure situations, then you would want to use WPA. However, personally I believe that closers are only marginally better pitchers than their bullpen counterparts, and as such, are getting some undue credit. So I believe that using WPA/LI is better, especially considering that many closers are failed starting pitchers. However, I will use both statistics in comparing home runs and saves. My next post will finally answer the question of whether 30 home runs or 30 saves are more valuable.
Thursday, January 27, 2011
The True Value of a Home Run
I have been meaning to do a post of the true value of a home run for awhile, but unfortunately I put it on the back burner for awhile until I was asked this question: which is worth more, 30 home runs or 30 saves? In this post, I am going to examine the true value of a home run, and in the next post I will examine exactly how much a save is worth, so I can compare the two.
The data I am going to use is for all teams in the 2010 regular season. The first thing to do is to find the number of home runs hit in each base-out state, which can be found from baseball-reference:
The total number of home runs hit last year was 4613, and over half of those were solo home runs. It was very rare for players to hit home runs with no outs and runners on third, as it would usually require a triple, or a double and steal.
The next step is to find the expected runs matrix for 2010 (from Baseball Prospectus):
We can use these two matrices together to determine the true value of a home run. The equation we will use is: value of a home run = Expected runs at the end of the play - Expected runs at the beginning of the play + the number of runs scored during the play. What this means is that we are taking the expected runs after - before to determine the value of the play (e.g. a leadoff out would be calculated as 0.26151 - 0.49154 = -0.23003, meaning the expected runs for the team in that inning would decrease by 0.23 runs), and then adding the number of runs that were scored.
This matrix shows the true value of a home run for each base-out state. Obviously, when there are no runners on base, the value of a home run will be 1, as the beginning and end states will be the same.
The most valuable home runs, obviously, are grand slams, as they score 4 runs, while home runs hit with two outs are more valuable than those hit with 0 or 1 out as there will be fewer chances remaining in the inning to drive in the runners or base, thus making the home run more valuable.
Finally, we need to multiply the matrix containing the number of home runs hit by the matrix showing the true value of a home run for each base-out state to find the run values for each base-out state.
To find the true value of a home run, we simply add up all of the runs (6485) and divide by the total number of home runs hit (4613) to find the average value of a home run: 1.406 runs. What this means is that the average home run hit in 2010 was worth 1.406 runs for the player's team. We will use this number later to figure out exactly how much each home run is worth in a dollar amount, and whether or not it is worth more than a save.
The data I am going to use is for all teams in the 2010 regular season. The first thing to do is to find the number of home runs hit in each base-out state, which can be found from baseball-reference:
| RUNNERS | HR_OUTS_0 | HR_OUTS_1 | HR_OUTS_2 |
| None | 1220 | 811 | 617 |
| 1st | 248 | 318 | 312 |
| 2nd | 74 | 133 | 147 |
| 3rd | 9 | 39 | 52 |
| 1st and 2nd | 54 | 119 | 142 |
| 1st and 3rd | 27 | 49 | 47 |
| 2nd and 3rd | 9 | 30 | 30 |
| Bases Loaded | 23 | 43 | 60 |
The total number of home runs hit last year was 4613, and over half of those were solo home runs. It was very rare for players to hit home runs with no outs and runners on third, as it would usually require a triple, or a double and steal.
The next step is to find the expected runs matrix for 2010 (from Baseball Prospectus):
| RUNNERS | EXP_R_OUTS_0 | EXP_R_OUTS_1 | EXP_R_OUTS_2 |
| None | 0.49154 | 0.26151 | 0.10374 |
| 1st | 0.85877 | 0.50512 | 0.2282 |
| 2nd | 1.10113 | 0.67765 | 0.3215 |
| 3rd | 1.35798 | 0.93308 | 0.34192 |
| 1st and 2nd | 1.42099 | 0.88181 | 0.45503 |
| 1st and 3rd | 1.80042 | 1.0982 | 0.46571 |
| 2nd and 3rd | 1.96584 | 1.38849 | 0.58205 |
| Bases Loaded | 2.36061 | 1.51185 | 0.77712 |
We can use these two matrices together to determine the true value of a home run. The equation we will use is: value of a home run = Expected runs at the end of the play - Expected runs at the beginning of the play + the number of runs scored during the play. What this means is that we are taking the expected runs after - before to determine the value of the play (e.g. a leadoff out would be calculated as 0.26151 - 0.49154 = -0.23003, meaning the expected runs for the team in that inning would decrease by 0.23 runs), and then adding the number of runs that were scored.
This matrix shows the true value of a home run for each base-out state. Obviously, when there are no runners on base, the value of a home run will be 1, as the beginning and end states will be the same.
| RUNNERS | Value_OUTS_0 | Value_OUTS_1 | Value_OUTS_2 |
| None | 1 | 1 | 1 |
| 1st | 1.63277 | 1.75639 | 1.87554 |
| 2nd | 1.39041 | 1.58386 | 1.78224 |
| 3rd | 1.13356 | 1.32843 | 1.76182 |
| 1st and 2nd | 2.07055 | 2.3797 | 2.64871 |
| 1st and 3rd | 1.69112 | 2.16331 | 2.63803 |
| 2nd and 3rd | 1.5257 | 1.87302 | 2.52169 |
| Bases Loaded | 2.13093 | 2.74966 | 3.32662 |
The most valuable home runs, obviously, are grand slams, as they score 4 runs, while home runs hit with two outs are more valuable than those hit with 0 or 1 out as there will be fewer chances remaining in the inning to drive in the runners or base, thus making the home run more valuable.
Finally, we need to multiply the matrix containing the number of home runs hit by the matrix showing the true value of a home run for each base-out state to find the run values for each base-out state.
| RUNNERS | Value_OUTS_0 | Value_OUTS_1 | Value_OUTS_2 |
| None | 1220 | 811 | 617 |
| 1st | 404.92696 | 558.53202 | 585.16848 |
| 2nd | 102.89034 | 210.65338 | 261.98928 |
| 3rd | 10.20204 | 51.80877 | 91.61464 |
| 1st and 2nd | 111.8097 | 283.1843 | 376.11682 |
| 1st and 3rd | 45.66024 | 106.00219 | 123.98741 |
| 2nd and 3rd | 13.7313 | 56.1906 | 75.6507 |
| Bases Loaded | 49.01139 | 118.23538 | 199.5972 |
To find the true value of a home run, we simply add up all of the runs (6485) and divide by the total number of home runs hit (4613) to find the average value of a home run: 1.406 runs. What this means is that the average home run hit in 2010 was worth 1.406 runs for the player's team. We will use this number later to figure out exactly how much each home run is worth in a dollar amount, and whether or not it is worth more than a save.
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