Are This Year's Home Runs Really That Different? December 22, 2010
Posted by tomflesher in Baseball, Economics.Tags: Carlos Pena, Carlos Quentin, home run distributions, home runs, Jose Bautista, kurtosis, Mark Teixeira, Miguel Cabrera, Paul Konerko, R, skewness, statistics
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This year’s home runs are quite confounding. On the one hand, home runs per game in the AL have dropped precipitously (as noted and examined in the two previous posts). On the other hand, Jose Bautista had an absolutely outstanding year. How much different is this year’s distribution than those of previous years? To answer that question, I took off to Baseball Reference and found the list of all players with at least one plate appearance, sorted by home runs.
There are several parameters that are of interest when discussing the distribution of events. Th
e first is the mean. This year’s mean was 5.43, meaning that of the players with at least one plate appearance, on average each one hit 5.43 homers. That’s down from 6.53 last year and 5.66 in 2008.
Next, consider the variance and standard deviation. (The variance is the standard deviation squared, so the numbers derive similarly.) A low variance means that the numbers are clumped tightly around the mean. This year’s variance was 68.4, down from last year’s 84.64 but up from 2008′s 66.44.
The skewness and kurtosis represent the length and thickness of the tails, respectively. Since a lot of people have very 
few home runs, the skewness of every year’s distribution is going to be positive. Roughly, that means that there are observations far larger than the mean, but very few that are far smaller. That makes sense, since there’s no such thing as a negative home run total. The kurtosis number represents how pointy the distribution is, or alternatively how much of the distribution is found in the tail.
For example, in 2009, Mark Teixeira and Carlos Pena jointly led the American League in home runs with 39. There was a high mean, but the tail was relatively thin with a 
high variance. Compared with this year, when Bautista led his nearest competitor (Paul Konerko) by 15 runs and only 8 players were over 30 home runs, 2009 saw 15 players above 30 home runs with a pretty tight race for the lead. Kurtosis in 2010 was 7.72 compared with 2009′s 4.56 and 2008′s 5.55. (In 2008, 11 players were above the 30-mark, and Miguel Cabrera‘s 37 home runs edged Carlos Quentin by just one.)
The numbers say that 2008 and 2009 were much more similar than either of them is to 2010. A quick look at the distributions bears that out – this was a weird year.
What Happened to Home Runs This Year? December 22, 2010
Posted by tomflesher in Baseball, Economics.Tags: baseball-reference.com, forecasting, home runs, R, regression, standard error, statistics, time series, Year of the Pitcher
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I was talking to Jim, the writer behind Apparently, I’m An Angels Fan, who’s gamely trying to learn baseball because he wants to be just like me. Jim wondered aloud how much the vaunted “Year of the Pitcher” has affected home run production. Sure enough, on checking the AL Batting Encyclopedia at Baseball-Reference.com, production dropped by about .15 home runs per game (from 1.13 to .97). Is that normal statistical variation or does it show that this year was really different?
In two previous posts, I looked at the trend of home runs per game to examine Stuff Keith Hernandez Says and then examined Japanese baseball’s data for evidence of structural break. I used the Batting Encyclopedia to run a time-series regression for a quadratic trend and added a dummy variable for the Designated Hitter. I found that the time trend and DH control account for approximately 56% of the variation in home runs per year, and that the functional form is
with t=1 in 1955, t=2 in 1956, and so on. That means t=56 in 2010. Consequently, we’d expect home run production per game in 2010 in the American League to be approximately
That means we expected production to increase this year and it dropped precipitously, for a residual of -.28. The residual standard error on the original regression was .1092, so on 106 degrees of freedom, so the t-value using Texas A&M’s table is 1.984 (approximating using 100 df). That means we can be 95% confident that the actual number of home runs should fall within .1092*1.984, or about .2041, of the expected value. The lower bound would be about 1.05, meaning we’re still significantly below what we’d expect. In fact, the observed number is about 3.4 standard errors below the expected number. In other words, we’d expect that to happen by chance less than .1% (that is, less than one tenth of one percent) of the time.
Clearly, something else is in play.
Home Run Derby: Does it ruin swings? December 15, 2010
Posted by tomflesher in Baseball, Economics.Tags: Baseball, baseball-reference.com, Chris Young, Corey Hart, David Ortiz, Hanley Ramirez, home run derby, home runs, Matt Holliday, Miguel Cabrera, Nick Swisher, Vernon Wells
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Earlier this year, there was a lot of discussion about the alleged home run derby curse. This post by Andy on Baseball-Reference.com asked if the Home Run Derby is bad for baseball, and this Hardball Times piece agrees with him that it is not. The standard explanation involves selection bias – sure, players tend to hit fewer home runs in the second half after they hit in the Derby, but that’s because the people who hit in the Derby get invited to do so because they had an abnormally high number of home runs in the first half.
Though this deserves a much more thorough macro-level treatment, let’s just take a look at the density of home runs in either half of the season for each player who participated in the Home Run Derby. Those players include David Ortiz, Hanley Ramirez, Chris Young, Nick Swisher, Corey Hart, Miguel Cabrera, Matt Holliday, and Vernon Wells.
For each player, plus Robinson Cano (who was of interest to Andy in the Baseball-Reference.com post), I took the percentage of games before the Derby and compared it with the percentage of home runs before the Derby. If the Ruined Swing theory holds, then we’d expect
The table below shows that in almost every case, including Cano (who did not participate), the density of home runs in the pre-Derby games was much higher than the post-Derby games.
| Player | HR Before | HR Total | g(Games) | g(HR) | Diff |
| Ortiz | 18 | 32 | 0.54321 | 0.5625 | 0.01929 |
| Hanley | 13 | 21 | 0.54321 | 0.619048 | 0.075838 |
| Swisher | 15 | 29 | 0.537037 | 0.517241 | -0.0198 |
| Wells | 19 | 31 | 0.549383 | 0.612903 | 0.063521 |
| Holliday | 16 | 28 | 0.54321 | 0.571429 | 0.028219 |
| Hart | 21 | 31 | 0.549383 | 0.677419 | 0.128037 |
| Cabrera | 22 | 38 | 0.530864 | 0.578947 | 0.048083 |
| Young | 15 | 27 | 0.549383 | 0.555556 | 0.006173 |
| Cano | 16 | 29 | 0.537037 | 0.551724 | 0.014687 |
Is this evidence that the Derby causes home run percentages to drop off? Certainly not. There are some caveats:
- This should be normalized based on games the player played, instead of team games.
- It would probably even be better to look at a home run per plate appearance rate instead.
- It could stand to be corrected for deviation from the mean to explain selection bias.
- Cano’s numbers are almost identical to Swisher’s. They play for the same team. If there was an effect to be seen, it would probably show up here, and it doesn’t.
Once finals are up, I’ll dig into this a little more deeply.
600 Home Runs: Who's Second? July 25, 2010
Posted by tomflesher in Baseball, Economics.Tags: 600 home runs, Alex Rodriguez, binomial distribution, Dodgers, home runs, Jim Thome, Manny Ramirez, quick and dirty stats, Twins
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Alex Rodriguez is, as I’m writing this, sitting at 599 home runs. Almost certainly, he’ll be the next player to hit the 600 home-run milestone, since the next two active players are Jim Thome at 575 and Manny Ramirez at 554. Today’s Toyota Text Poll (which runs during Yankee games on YES) asked which of those two players would reach #600 sooner.
There are a few levels of abstraction to answering this question. First of all, without looking at the players’ stats, Thome gets the nod at the first order because he’s significantly closer than Driving in 25 home runs is easier than driving in 46, so Thome will probably get there first.
At the second order, we should take a look at the players’ respective rates. Over the past two seasons, Thome has averaged a rate of .053 home runs per plate appearance, while Ramirez has averaged .041 home runs per plate appearance. With fewer home runs to hit and a higher likelihood of hitting one each time he makes it to the plate, Thome stays more likely to hit #600 before Ramirez does… but how much more likely?
Using the binomial distribution, I tested the likelihood that each player would hit his required number of home runs in different numbers of plate appearances to see where that likelihood reached a maximum. For Thome, the probability increases until 471 plate appearances, then starts decreasing, so roughly, I expect Thome to hit his 25th home run within 471 plate appearances. For Manny, that maximum doesn’t occur until 1121 plate appearances. Again, the nod has to go to Thome. He’ll probably reach the milestone in less than half as many plate appearances.
But wait. How many plate appearances is that, anyway? Until recently, Manny played 80-90% of the games in a season. Last year, he played 64%. So far the Dodgers have played 99 games and Manny appeared in 61 of them, but of course he’s disabled this year. Let’s make the generous assumption that Manny will play in 75% of the games in each season starting with this one. Then, let’s look at his average plate appearances per game. For most of his career, he averaged between 4.1 and 4.3 plate appearances per game, but this year he’s down to 3.6. Let’s make the (again, generous) assumption that he’ll get 4 plate appearances in each game from now on. At that rate, to get 1121 plate appearances, he needs to play in 280.25 games, which averages to 1.723 seasons of 162 games or about 2.62 seasons of 75% playing time.
Thome, on the other hand, has consistently played in 80% or more of his team’s games but suffered last year and this year because he hasn’t been serving as an everyday player. He pinch-hit in the National League last year and has, in Minnesota, played in about 69% of the games averaging only 3 plate appearances in each. Let’s give Jim the benefit of the doubt and assume that from here on out he’ll hit in 70% of the games and get 3.5 appearances (fewer games and fewer appearances than Ramirez). He’d need about 120.3 games, which equates to about 3/4 of a 162-game season or about 1.06 seasons with 70% playing time. Even if we downgrade Thome to 2.5 PA per game and 66% playing time, that still gives us an expectation that he’ll hit #600 within the next 1.6 real-time seasons.
Since Thome and Ramirez are the same age, there’s probably no good reason to expect one to retire before the other, and they’ll probably both be hitting as designated hitters in the AL next year. As a result, it’s very fair to expect Thome to A) reach 600 home runs and B) do it before Manny Ramirez.
More on Home Runs Per Game July 9, 2010
Posted by tomflesher in Baseball, Economics.Tags: Baseball, baseball-reference.com, Chow test, home runs, Japan, Japanese baseball, R, Rays, regression, replication
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In the previous post, I looked at the trend in home runs per game in the Major Leagues and suggested that the recent deviation from the increasing trend might have been due to the development of strong farm systems like the Tampa Bay Rays’. That means that if the same data analysis process is used on data in an otherwise identical league, we should see similar trends but no dropoff around 1995. As usual, for replication purposes I’m going to use Japan’s Pro Baseball leagues, the Pacific and Central Leagues. They’re ideal because, just like the American Major Leagues, one league uses the designated hitter and one does not. There are some differences – the talent pool is a bit smaller because of the lower population base that the leagues draw from, and there are only 6 teams in each league as opposed to MLB’s 14 and 16.
As a reminder, the MLB regression gave us a regression equation of
where 
Just examining the data on home runs per game from the Japanese leagues, the trend looks significantly differe
nt. Instead of the rough U-shape that the MLB data showed, the Japanese data looks almost M-shaped with a maximum around 1984. (Why, I’m not sure – I’m not knowledgeable enough about Japanese baseball to know what might have caused that spike.) It reaches a minimum again and then keeps rising.
After running the same regression with t=1 in 1950, I got these results:
| Estimate | Std. Error | t-value | p-value | Signif | |
| B0 | 0.2462 | 0.0992 | 2.481 | 0.0148 | 0.9852 |
| t | 0.0478 | 0.0062 | 7.64 | 1.63E-11 | 1 |
| tsq | -0.0006 | 0.00009 | -7.463 | 3.82E-11 | 1 |
| DH | 0.0052 | 0.0359 | 0.144 | 0.8855 | 0.1145 |
This equation shows two things, one that surprises me and one that doesn’t. The unsurprising factor is the switching of signs for the t variables – we expected that based on the shape of the data. The surprising factor is that the designated hitter rule is insignificant. We can only be about 11% sure it’s significant. In addition, this model explains less of the variation than the MLB version – while that explained about 56% of the variation, the Japanese model has an 
There’s a slightly interesting pattern to the residual home runs per game (
it isn’t as pronounced, this data also shows a spike – but the spike is at t=55, so instead of showing up in 1995, the Japan leagues spiked around the early 2000s. Clearly the same effect is not in play, but why might the Japanese leagues see the same effect later than the MLB teams? It can’t be an expansion effect, since the Japanese leagues have stayed constant at 6 teams since their inception.
Incidentally, the Japanese league data is heteroskedastic (Breusch-Pagan test p-value .0796), so it might be better modeled using a generalized least squares formula, but doing so would have skewed the results of the replication.
In order to show that the parameters really are different, the appropriate test is Chow’s test for structural change. To clean it up, I’m using only the data from 1960 on. (It’s quick and dirty, but it’ll do the job.) Chow’s test takes
where 
The critical value for 90% significance at 4 and 192 degrees of freedom would be 1.974 according to Texas A&M’s F calculator. That means we don’t have enough evidence that the parameters are different to treat them differently. This is probably an artifact of the small amount of data we have.
In the previous post, I looked at the trend in home runs per game in the Major Leagues and suggested that the recent deviation from the increasing trend might have been due to the development of strong farm systems like the Tampa Bay Rays’. That means that if the same data analysis process is used on data in an otherwise identical league, we should see similar trends but no dropoff around 1995. As usual, for replication purposes I’m going to use Japan’s Pro Baseball leagues, the Pacific and Central Leagues. They’re ideal because, just like the American Major Leagues, one league uses the designated hitter and one does not. There are some differences – the talent pool is a bit smaller because of the lower population base that the leagues draw from, and there are only 6 teams in each league as opposed to MLB’s 14 and 16.
As a reminder, the MLB regression gave us a regression equation of
where 
Just examining the data on home runs per game from the Japanese leagues, the trend looks significantly different. Instead of the rough U-shape that the MLB data showed, the Japanese data looks almost M-shaped with a maximum around 1984. (Why, I’m not sure – I’m not knowledgeable enough about Japanese baseball to know what might have caused that spike.) It reaches a minimum again and then keeps rising.
After running the same regression with t=1 in 1950, I got these results:
| Estimate | Std. Error | t-value | p-value | Signif | |
| B0 | 0.2462 | 0.0992 | 2.481 | 0.0148 | 0.9852 |
| t | 0.0478 | 0.0062 | 7.64 | 1.63E-11 | 1 |
| tsq | -0.0006 | 0.00009 | -7.463 | 3.82E-11 | 1 |
| DH | 0.0052 | 0.0359 | 0.144 | 0.8855 | 0.1145 |
This equation shows two things, one that surprises me and one that doesn’t. The unsurprising factor is the switching of signs for the t variables – we expected that based on the shape of the data. The surprising factor is that the designated hitter rule is insignificant. We can only be about 11% sure it’s significant. In addition, this model explains less of the variation than the MLB version – while that explained about 56% of the variation, the Japanese model has an 
There’s a slightly interesting pattern to the residual home runs per game (
it isn’t as pronounced, this data also shows a spike – but the spike is at t=55, so instead of showing up in 1995, the Japan leagues spiked around the early 2000s. Clearly the same effect is not in play, but why might the Japanese leagues see the same effect later than the MLB teams? It can’t be an expansion effect, since the Japanese leagues have stayed constant at 6 teams since their inception.
Incidentally, the Japanese league data is heteroskedastic (Breusch-Pagan test p-value .0796), so it might be better modeled using a generalized least squares formula, but doing so would have skewed the results of the replication.
In order to show that the parameters really are different, the appropriate test is Chow’s test for structural change. To clean it up, I’m using only the data from 1960 on. (It’s quick and dirty, but it’ll do the job.) Chow’s test takes
Back when it was hard to hit 55… July 8, 2010
Posted by tomflesher in Baseball, Economics.Tags: Baseball, baseball-reference.com, home runs, R, regression, sabermetrics, Stuff Keith Hernandez Says, talent pool dilution, Willie Mays, Year of the Pitcher
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Last night was one of those classic Keith Hernandez moments where he started talking and then stopped abruptly, which I always like to assume is because the guys in the truck are telling him to shut the hell up. He was talking about Willie Mays for some reason, and said that Mays hit 55 home runs “back when it was hard to hit 55.” Keith coyly said that, while it was easy for a while, it was “getting hard again,” at which point he abruptly stopped talking.
Keith’s unusual candor about drug use and Mays’ career best of 52 home runs aside, this pinged my “Stuff Keith Hernandez Says” meter. After accounting for any time trend and other factors that might explain home run hitting, is there an upward trend? If so, is there a pattern to the remaining home runs?
The first step is to examine the data to see if there appears to be any trend. Just looking at it, there appears to be a messy U shape with a minimum around t=20, which indicates a quadratic trend. That means I want to include a term for time and a term for time squared.
Using the per-game averages for home runs from 1955 to 2009, I detrended the data using t=1 in 1955. I also had to correct for the effect of the designated hitter. That gives us an equation of the form
The results:
| Estimate | Std. Error | t-value | p-value | Signif | |
| B0 | 0.957 | 0.0328 | 29.189 | 0.0001 | 0.9999 |
| t | -0.0188 | 0.0028 | -6.738 | 0.0001 | 0.9999 |
| tsq | 0.0004 | 0.00005 | 8.599 | 0.0001 | 0.9999 |
| DH | 0.0911 | 0.0246 | 3.706 | 0.0003 | 0.9997 |
We can see that there’s an upward quadratic trend in predicted home runs that together with the DH rule account for about 56% of the variation in the number of home runs per game in a season (
Then, I needed to look at the difference between the predicted number of home runs per game and the actual number of home runs per game, which is accessible by subtracting
This represents the “abnormal” number of home runs per year. The question then becomes, “Is there a patt
ern to the number of abnormal home runs?” There are two ways to answer this. The first way is to look at the abnormal home runs. Up until about t=40 (the mid-1990s), the abnormal home runs are pretty much scattershot above and below 0. However, at t=40, the residual jumps up for both leagues and then begins a downward trend. It’s not clear what the cause of this is, but the knee-jerk reaction is that there might be a drug use effect. On the other hand, there are a couple of other explanations.
The most obvious is a boring old expansion effect. In 1993, the National League added two teams (the Marlins and the Rockies), and in 1998 each league added a team (the AL’s Rays and the NL’s Diamondbacks). Talent pool dilution has shown up in our discussion of hit batsmen, and I believe that it can be a real effect. It would be mitigated over time, however, by the establishment and development of farm systems, in particular strong systems like the one that’s producing good, cheap talent for the Rays.
Santana the Late-Blooming Hitter July 7, 2010
Posted by tomflesher in Baseball.Tags: Brewers, Dave Eiland, home runs, Jason Jennings, Johan Santana, Mets, Pitchers batting, Yovani Gallardo
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Last night, Johan Santana hit his first home run in his 87th career game as a batter. (Granted, he’s played far more than that many games because he played a few years in the American League.) Out of curiosity, I checked Baseball-Reference.com’s Play Index to see how many home runs have been hit by pitchers in their first 87 games as batters.
Since 1961, there have been 431 home runs (although the Play Index only lists games starting at 1970, so that may or may not be accurate). Four pitchers have hit home runs in their first games, including Yankee pitching coach Dave Eiland in 1992 and Rockies pitcher Jason Jennings. Like Johan, Jennings pitched a complete game shutout for the win that night.
The all-time leader in home runs by a pitcher in the first 87 games (how’s that for esoteric?) is Yovani Gallardo, who’s in his fourth season pitching for the Brewers. He’s hit seven of them, and as of July 4 he’s only hit in 71 games. He’s got a lot of time to pick up the pace and possibly hit the triple-digit mark when he gets back from the disabled list some time after July 20.
At the other end… June 22, 2010
Posted by tomflesher in Baseball.Tags: Andre Ethier, As, Cedrick Bowers, Diamondbacks, Esmerling Vasquez, extra innings, free baseball, home runs, Joey Votto, Michael Wuertz, Ramon Hernandez, Rangers, Reds, Scott Rolen, weird lines
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Although AJ Burnett had a bad first inning last night, the Oakland As had a bad tenth inning. After taking a 2-2 game into extra innings, the Cincinnati Reds knocked three out of the park against pitchers Michael Wuertz and Cedrick Bowers. The first was hit by Ramon Hernandez; Joey Votto and Scott Rolen went deep back to back. Although extra-inning home runs aren’t very rare (there have been 35 so far this year), only three pitchers have surrendered more than one, and neither of the other two (Chad Durbin and Matt Belisle) gave them both up on the same night.
Last year, everyone’s favorite balk-off artist, Arizona’s Esmerling Vasquez, gave up two home runs in extra innings against the Texas Rangers on June 25th. Those were two of 83 free-baseball homers in 2009. Extra-innings home runs are more common in the tops of innings, because in a tied game a home run for the home team is a walk-off whereas the road team will get the chance to capitalize on their momentum, but I would have expected the proportions to be much more different than they are. In 2009, for example, of those 83, only 44 were hit by the away team with 39 hit by the home team (and 33 of those were game-enders).
So far, no batter has more than one extra-innings home run this year, but last year there were several. Andre Ethier led the pack with 3, with a bunch of batters who had 2.
