30 HRs or 30 saves?

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.

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:

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.

Fact of the Week X: Multi Home Run Postseason Games

In game one of the NLCS last night, Cody Ross hit two home runs off of Roy Halladay. It is extremely hard to hit two home runs in the majors, let alone two against Roy Halladay in the postseason. Ross became only the 5th player to hit two home runs in the first game of a League Championship Series. As a comparison, there have been 10 players with at least 2 home runs in a LCS game two, 8 players in a game 3, 7 in a game 4, 2 in a game 5 or 7, and only 1 in a game 6. This is presumably based upon the starting pitcher on the opposing team, and the ace of the staff will usually pitch games 1, 4 or 5, and possibly 7. The lack of games in the later series games (5-7), is probably due to the lack of overall games (if a series ends in a sweep there will not be any game 5, 6, or 7), the quality of starting pitching, and the strategy of managers playing for one-run innings, so more sacrifice bunts and less at-bats.

Ross also became only the 4th player to hit 2 home runs in a postseason game while batting 8th in the lineup. More impressively, the other three players were all hitting with a position player behind them (a game involving the DH), so Ross became the first player to ever hit 2 home runs with the pitching batting behind him.

Finally, he became only the 19th player to hit at least 2 home runs in his first five postseason games in either the World Series or LCS (18 other players have hit at least two in their first five games in the LDS). All of these statistics are fairly impressive, and become more impressive when you consider that he did it against Roy Halladay, who threw the second no-hitter in postseason history in his last start.

Fact of the Week VIII: Home Run Records

The Blue Jays have accomplished a couple of notable feats in the past few days with home runs. The first happened on Wednesday night when John Buck homered, his 20th of the season, to become the 6th different Blue Jays to hit 20 home runs this year. They became only the 18th team in MLB history to have six hitters to hit at least 20 home runs each. If Edwin Encarnacion manages to hit two home runs in the last three games of the season (he currently has 18 HRs), the team will become only the 5th team to ever have seven players with at least 20 home runs, joining the 1996 Orioles, 2005 Rangers, 2009 Yankees, and the 2000 Blue Jays. No National League team has ever done it, possibly because they would need seven out of eight of their starting position players to do it as opposed to seven out of nine in the American League. One final note is that Alex Gonzalez, traded to the Braves midseason for Yunel Escobar, had 17 home runs as a Jay (only 6 so far with the Braves) and Escobar (who had 0 home runs as a Brave) has 4 home runs as a Jay. So the shortstop position for the Jays also has hit 20+ home runs, which means that the Jays possibly could have become the first team to ever have 8 players hit at least 20 home runs each (if they hadn't traded for Escobar, which I'm glad they did, and if EE hits 2 homers in the next three games).

On a related note, after Thursday night's blowout of the Twins, in which the Jays hit six home runs, they now have 253 home runs as a team so far this year. That is the 4th highest total in MLB history, trailing only the 1997 Mariners (264 HRs), the 2005 Rangers (260 HRs), and the 1996 Orioles (257 HRs). If they continue their pace of 1.591 HRs per game over the final three games, they should end up with 257.77 home runs on the season, which would be good for third all-time. It is interesting to see many of the same teams on both of these lists (the '97 Mariners also had 6 players with at least 20 home runs). One great home run hitter will not vault your team into the home run record books, the team needs at least 5-6 hitters who all can hit at least 20+ home runs, with usually one or two of those players hitting at least 30 or more (Vernon Wells with 31 and Jose Bautista with 54 so far on this year's team).

One final note of interest: also in last night's game, Jose Bautista hit his 53rd and 54th home runs of the season. The first was an upper-deck grand slam, the second an opposite field home run, the first home run to right field in his career! (Unfortunately HitTracker hasn't quite updated his home run total, so we cannot view his home run scatter plot with the one outlier to right field.) Although pitchers next year may try to beat him with outside pitches, he has again proved that he can hit pretty much anything you throw to home. Sure, he would much rather hit an inside fastball (as the first home run showed!), but if you must continue to pound the outside corner, he will either take a good swing and put the ball into right field, or simply walk. Speaking of walks, Bautista needs one more walk this weekend to become only the 14th player to ever have 50 home runs and 100 walks in one season.

So although this weekend will not bring playoff drama for the Blue Jays, there are a couple of milestones that they could reach with good performances. Hopefully they can win a couple of games, hit a few more home runs, and end 2010 with a bang.

Jose Bautista

Jose Bautista has had himself an incredible year this year, as on Friday night he hit his 51st and 52nd home runs of the season, which is already 5 better than George Bell's previous franchise record of 47 in a season. He has become the first player since 2007 to hit at least 50 home runs and is also currently second in the AL in walks. He has seemingly come out of nowhere to start hitting home runs left and right, and as a result many people have questioned the legitimacy of his season. The question that everyone wants to know is, how can a 29-year old player, with a career high of 16 home runs in a season, suddenly hit 50+? Only one player in history, Cecil Fielder, had ever hit 50 home runs in a season without having previously hit at least 20 in a season. I want to try and explain in this post how it is possible for Bautista to have such a breakout season, without involving the dreaded s-word.

Bautista himself claims that the increase in his home run total is due to regular playing time instead of being a utility player (increased confidence), better pitches to hit, and a change in his swing. I am going to mainly focus on the latter: how he could hit so many more home runs by simply making changes to his swing to maximize his strengths and minimize his weaknesses. There are basically two main factors into hitting home runs: hitting a lot of fly balls, and getting lucky (or by getting stronger and getting lucky) by hitting a higher percentage of those fly balls out of the ballpark. The first factor, Fly Ball %, is a hitter by hitter case, as some hitters are ground ball hitters, while other are fly ball hitters. The second factor, HR/FB, is mainly due to luck. Similar to BABIP, a hitter has some control over his HR/FB rate, but it fluctuates around the league average of 10.6% (if you want to read more about HR/FB, you can visit the Sabermetrics Library here).

If you have watched Bautista hit any home runs at all this year, you can see how he has changed his swing. He simply waits for a pitch in one location (almost always inside), if the pitch is not there he will not swing at it, but if it is there he will take an extremely hard swing. As a result, he is sacrificing contact for power (surprisingly enough, he is hitting for a career high in batting average this year). This means that he is very patient at the plate, drawing 98 walks so far this year, and will simply mash any mistake pitch. He has also added a pronounced uppercut to his swing, adding loft to the balls that he hits, and thus increasing his FB%. He has also increased his HR/FB rate this year, either by luck, but also because he is swinging for the fences every time he steps in the batter's box.

This picture shows all (well the first 49) of Bautista's 2010 home run landing spots, courtesy of Hit Tracker. He has not hit a single home run to the right of dead center field this year, nor any other year. This is again a critical part of Bautista's success: he looks to hit inside pitches for power to left field, while he either doesn't swing at or looks to hit for contact the pitches on the outer half of the plate.

What I want to do now is figure out how many of Bautista's home runs this year are from his conscious adjustment at the plate, and how many are mainly due to luck. To do this, I am going to compare his FB% and HR/FB rates from last year and this year, holding one constant while determine how many home runs the other rate contributed (it will make more sense when I introduce the numbers).

The first thing to figure out is Bautista's predicted 2010 numbers based on his past career numbers. Given his amount of plate appearances this year (currently 649, we are going to assume he will get to 675 by the end of the year), we can predict his home runs, FB%, and HR/FB. His career FB% (before 2010) is 42.8% - this means that 42.8% of the balls he puts in play are fly balls (as opposed to ground balls or line drives). It is important to not misinterpret this number as the % of PA that are fly balls - that would grossly inflate his predicted home runs. His career HR/FB rate (again, before 2010) is 10.4%, which is very similar to the league average of 10.6%. So if he were to have 675 PA this year with his average rates, he would hit 19.54 home runs this year.

The next thing to calculate is his predicted 2010 numbers using this year's splits (we could almost use his numbers right now, but there are 8 games left). With 675 PA, and a FB% of 54.8% and a FB/HR of 21.8%, he is predicted to hit 54.08 home runs.  In 2010 he has increased his FB% by 12 percentage points (from 42.8% to 54.8%) and his HR/FB rate by 11.2% (from 10.6% to 21.8%). These two increases result in a 34.54 increase in home runs this year as opposed to his career average. The question I want to answer is what percentage of that increase is due to skill, and what percentage is due to "luck"?

To determine how much is due to Bautista's swing adjustment, we want to hold his HR/FB rate constant (keep it at his career average) and set his FB% to 54.8%, his rate for this year. This will show the effect of Bautista increasing his fly ball rate, which is a "skill", without increasing his HR/FB rate, which is due mostly to luck. So with 675 PA, FB% of 54.8%, and HR/FB of 10.4%, Bautista would have hit 25.83 home runs this season. This means that 6.29 extra home runs (25.83-19.54) came purely from Bautista's changed swing at the plate.

To figure out how many home runs came from Bautista being "lucky", we are going to hold his FB% constant at a career average of 42.8%, and increase his HR/FB rate to 21.8%, his rate this year. In 675 PA he would then hit 41.08 home runs, which means that 21.54 more home runs came from his HR/FB rate increasing, which has a lot to do with luck. However, since he has changed his swing to become much more powerful, he would have an increase in his HR/FB rate anyway, but for the purpose of this study we will credit these home runs to luck.

Finally, as you may have figured out from the math, there are a couple of home runs which are unaccounted for. If we take the base of 19.54 home runs, and add the increases of 6.29 and 21.54, Bautista would have hit 47.37. But we previously stated that his projection is 54.08 home runs, so we are missing 6.71 home runs. These home runs are found through the "interaction" term, which is when both Bautista's FB% and HR/FB increase. We can safely credit these to "skill", as I believe that his HR/FB rate has increased because of the change in his swing.

What this all means is that overall, Bautista has hit at least 32.54 home runs this season due to his skill and new swing mechanics, while at most he has hit 21.54 home runs due to luck, although that number is probably much lower. So what he is doing this year should not be a fluke, even if his HR/FB rate drops all the way back down to his career average of 10.4% next year, he should still hit at least 30 home runs. The large majority of his home runs this year have indeed come because of his adjustments at the plate, and possibly because of other intangible measures such as improved confidence and better pitches to hit (although that could be measured in an exhaustive study).

What I would like to conclude is that: a) Bautista's season is no fluke, he should return to the 30 or 40 home run club next season, b) I cannot say that he is not taking steroids, but I can say that they are not the reason he has hit so many home runs this season, and c) this season is going to cost the Blue Jays (or some other team) a lot of money, and I do believe that Bautista has at least a couple more good years left in him. I hope this post clears up at least a little bit of the shock and disbelief at Bautista's incredible season, but it is good to know that there are ways to measure why and how he is hitting all of these home runs.

Home Run Streaks

Last night, the Jays hit a home run in their 16th consecutive game, which seems to be a fairly impressive streak. I wanted to find out if in fact it was impressive, and just how difficult is it to do?

First of all, this streak is now the second longest HR streak by the Jays in club history. It is also the second longest HR streak in the MLB so far this year. The 16 games in a row is only surpassed by the 23 games in a row in 2000. So while this isn't exactly uncharted territory, they do have a good streak going. What makes this streak really interesting is that, out of all HR streaks of at least 13 games, it is the only streak where they have a losing record (they are currently 5-11 in the streak, the next worst streak is when they went 7-7 in 14 games in 1996). Also interesting is that they have only scored 73 runs in the 16 games, which gives them a runs scored/game of 4.56, which is also the lowest of the ten times they have hit home runs in at least 13 games straight.

Another interesting fact is that while they have hit 31 home runs in the 16 games (1.94/game, as opposed to 1.51 HR/game the rest of the season), they are scoring fewer runs per game than for the entire season (they were averaging 4.65 runs/game in their first 129 games, they are averaging 4.56 runs/game in the last 16). So while they are hitting more home runs, those runs produced from the home runs seem to be just about the only runs they are scoring.

The last thing I wanted to do was figure out how difficult it is to hit home runs in 16 straight games. The Jays have hit 226 home runs so far this year, and have hit home runs in 107 of the 145 games they have played (here is a summary of every home run they have hit so far if you are so interested). So the probability of them hitting a home run in any given game is 0.738, or 73.8%. That means that the probability of them hitting a home run in n different games is simply 0.738ⁿ, as the probability of them hitting a home run in two games is 0.738*0.738, in three games 0.738*0.738*0.738, and so on up to n. So the probability of them hitting a home run in 16 straight games is 0.738¹⁶, which is equal to 0.774%. What this means is that out of 1000 "sets" of 16 games, the Blue Jays would hit a home run in each game 7.74 times, or one out of every 129.2 sets. Considering that there are 147 sets of 16 games in each season (games 1-16, 2-17, 3-18, ..., 146-161, 147-162), we can see that this should happen about 1.138 times this season.

So what we can see by looking at the math is that although this streak of home runs is impressive, it is certainly not out of the ordinary and mathematically probably should have happened at least once this season. Now, keep in mind that the Blue Jays are hitting home runs at a mindblowing pace this year (on pace for 252.5), in fact very close to the record for most home runs by a team in a single season (the 1997 Seattle Mariners hold the record with 264 HRs). So the chances of the 2010 Blue Jays to hit home runs in 16 straight games is a lot higher than the chance of any other Blue Jays team to hit home runs in 16 straight games. That is why this is the second longest streak in club history. It remains to be seen how long they can continue the streak, but don't be surprised if it ends tonight or during the weekend series with the Red Sox.

Fact of the Week II: Home Runs

We all know that Jose Bautista has been having a great season. He is currently leading the league in home runs with 37, 6 more than any other player. But we probably didn't realize just how great of a season it has been in comparison to all other Blue Jays seasons.

Only eight times has a player for the Blue Jays hit at least 40 home runs, and only six players have done it (Carlos Delgado did it three times, in 1999, 2000, and 2003). Bautista already has 37 home runs, and is on pace for 50, which would be the first time a Blue Jays batter has hit 50 home runs in a season (the highest single season total is 47 by George Bell in 1987). Bautista should at least hit 48 home runs to break the single season record.

Pretty good, right? It actually gets better. Bautista is not only hitting a lot of home runs, he is actually getting on base a ton too (he is currently tied with Daric Barton for the AL lead in walks with 73). Bautista is already only the third player in franchise history with at least 37 home runs and an on-base percentage of at least .370. If he can keep his OBP at .370 and hit just eight more home runs, he will have the most home runs by a Jays player with an OBP of at least .370.

So we can see that this season has been a very special one for Jose Bautista and the Blue Jays. Although he has not hit for a high average, he has excelled in a few key areas, such as hitting home runs, walking, and also throwing out baserunners (he is second in the MLB this year with 10 assists, only one behind Shin-Soo Choo). Hopefully he can play this well the rest of the season and for the next couple of years.