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Burnett, Hughes, and Playoff Rotations October 12, 2010

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There was much discussion of the Yankees’ specialized playoff rotation for the American League Division Series. As is conventional in the ALDS, Joe Girardi went with a three-man rotation. CC Sabathia and Andy Pettitte were locks; the third starter could have been A.J. Burnett, Javier Vazquez, or Dustin Moseley. Girardi went with young All-Star Phil Hughes in the third slot. That, of course, led to a sweep of the Minnestoa Twins to advance to the American League Championship Series.

First of all, I think it was probably the right decision. Hughes pitched 176 1/3 innings and gave up 82 earned runs, for an ER/IP of about .47. In Burnett’s 186 2/3 innings, he allowed 109 runs for an ER/IP of about .58. Surprisingly, Burnett allowed 9 unearned runs for a rate of about .048 unearned runs per inning pitched, whereas Hughes had only one unearned run for a rate of about .006, but of course those numbers probably don’t say anything significant. With 730 batters faced, he allowed about .11 earned runs per batter, or about 1 earned run every 9 batters faced, while Burnett’s 829 batters faced mean he had similar numbers of .13 earned runs per batter and 7.69 batters.

Most importantly to me, Hughes was much more predictable. Burnett faced, on average, 4.68 batters per inning pitched, with a variance of .92. Hughes faced over half a batter less per inning – 4.13 – and had a variance of .33. That means that not only did Burnett allow more baserunners, but when he was off, he was very off. Although the decision gets tougher when you have a higher BF/IP and a lower variance, Hughes was both better and more consistent in a similar number of innings, so he has to get the nod.

(That said, it’s shocking that such similar numbers produced one 18-8 pitcher and one 10-15 pitcher.)

The only question now is what order to pitch the announced four-man rotation for the ALCS. Of the choices,

OPTION 3
Sabathia
Hughes
Pettitte
Burnett
Sabathia
Hughes
Pettitte

seems clearly superior to me. It allows Burnett to start but avoids starting him twice, gets Hughes in play quite often, and puts the very reliable Andy Pettitte in play for a potential Game Seven. The linked article lists as a con that Pettitte is considered the number 2 starter, but at the Major League level a manager can’t be concerned with such frivolities. Besides, Pettitte is an established company man. I’d be surprised if he balked at a rotation that both maximized the team’s chances to win and put him in position to be the clutch hero.

Incidentally, this option lends itself to using the same rotation in the World Series. Option 2:

Sabathia
Pettitte
Hughes
Sabathia
Burnett
Pettitte
Sabathia

leaves Sabathia unavailable to start Game 1 of the World Series and might put Pettitte on short rest depending on the schedule to start Game 1. I can’t see starting the Series with Hughes or Burnett.

Jim Thome’s Marginal Value October 5, 2010

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I’ve alluded to the similarity between Manny Ramirez and Jim Thome quite a bit. They both played in Cleveland for a few years before moving on to other teams. They’re each in the DH phase of their careers. Thome is about two years older than Ramirez, but otherwise they’ve had relatively similar production. That’s why it was so odd for the White Sox to let Thome go a few years back only to pick an injured, probably going-downhill Manny for about a quarter of the season when Ramirez is making about $18 million and Thome’s maximum salary was about $15.7 million. There’s an argument that Manny still has more productive years left than Thome, of course. (I happen to think that argument is wrong, but that’s just me.)

Just for fun, let’s take a look at their production since Manny’s trade.

In the last 24 games he played, Ramirez had 88 plate appearances, a respectable .420 OBP, and a Jeteresque .261 batting average. His win probability added was -.273, for those of you who are into that sort of thing. Meanwhile, over the same number of games, the flagging, decrepit Thome had only 79 plate appearances, with a paltry .333 batting average, and only a .494 OBP.

Thome’s salary this year for the Twins was $1.5 million.

I think the winner here is clear.

Mariano’s Walk-Off Beanball September 12, 2010

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Mariano Rivera did something strange tonight: He plunked in the winning run. He hit Jeff Francoeur of the Texas Rangers to force in Nelson Cruz for the winning run in extra innings. It was his fourth hit batsman of the year and only his third loss.

A walk-off beaning requires an extraordinary set of circumstances. First of all, like all walk-off plays, it requires the home team to be at bat in the bottom of the inning. In this case, it was in extra innings rather than the bottom of the 9th. It additionally requires a tied game in the bottom of said inning. Finally, it requires the bases to be loaded when the plunking occurs.

This is all magnified by the face that Rivera does not ordinarily load the bases. Assuming his 2010 OBP against (.214) held, the probability the bases being loaded with two outs or fewer is:

p(bases loaded, 0 outs) + p(bases loaded, 1 out) + p(bases loaded, 2 outs) = (.214^3) + (.214^3 \times .786) + (.214^3 \times .706^2) = .0098 + .0077 + .0061 = .0236

Then, if that situation occurs, we still have to deal with the unlikely event of Mariano hitting a player with a pitch. Before this evening, Mo had hit three batters in 196 plate appearances, for a rate of about .0153. Thus, the probability of Mariano Rivera hitting a batter with a pitch after having loaded the bases is

.0236 \times .0153 \approx .0004

That means that in 10,000 innings, we would expect that to occur about 4 times, assuming that Mariano wasn’t removed after having walked the bases (which would obviously introduce some bias).

Oddly, the last walk-off hit by pitch also involved the Yankees, albeit on the other side, way back on July 19 of 2008. That night, the A’s’ Lenny DiNardo hit Jose Molina with a pitch to force in Derek Jeter, again in extra innings. David Robertson grabbed the win that night.

Teixeira and Cano: Picking up slack? August 5, 2010

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Michael Kaye, the YES broadcaster for the Yankees, often pointed out between July 22 and August 4 that the Yankees were turning up their offense to make up for Alex Rodriguez‘s lack of home run production. That seems like it might be subject to significant confirmation bias – seeing a few guys hit home runs when you wouldn’t expect them to might lead you to believe that the team in general has increased its production. So, did the Yankees produce more home runs during A-Rod’s drought?

During the first 93 games of the season, the Yankees hit 109 home runs in 3660 plate appearances for rates of 1.17 home runs per game and .0298 home runs per plate appearance. From July 23 to August 3, they hit 17 home runs in 451 plate appearances over 12 games for rates of 1.42 home runs per game and .0377 home runs per plate appearances. Obviously those numbers are quite a bit higher than expected, but can it be due simply to chance?

Assume for the moment that the first 93 games represent the team’s true production capabilities. Then, using the binomial distribution, the likelihood of hitting at least 17 home runs in 451 plate appearances is

p(K = k) = {n\choose k}p^k(1-p)^{n-k} = {451\choose 17}.0298^{17}(.9702)^{434} \approx .0626

The cumulative probability is about .868, meaning the probability of hitting 17 or fewer home runs is .868 and the probability of hitting more than that is about .132. The probability of hitting 16 or fewer is .805, which means out of 100 strings of 451 plate appearances about 81 of them should end with 16 or fewer plate appearances. This is a perfectly reasonable number and not inherently indicative of a special performance by A-Rod’s teammates.

Kaye frequently cited Mark Teixeira and Robinson Cano as upping their games. Teixeira hit 18 home runs over the first 93 games and made 423 plate appearances for rates of .194 home runs per game and .0426 home runs per plate appearance. From July 23 to August 3, he had 5 home runs in 12 games and 54 plate appearances for rates of .417 per game and .0926. That rate of home runs per plate appearance is about 8% likely, meaning that either Teixeira did up his game considerably or he was exceptionally lucky.

Cano played 92 games up to July 21, hitting 18 home runs in 400 plate appearances for rates of .196 home runs per game and .045 per plate appearance. During A-Rod’s drought, he hit 3 home runs in 50 plate appearances over 12 games for rates of .25 and .06. That per-plate-appearance rate is about 39% likely, which means we don’t have enough evidence to reject the idea that Cano’s performance (though better than usual) is just a random fluctuation.

It will be interesting to see if Teixeira slows down as a home-run hitter now that Rodriguez’s drought is over.

Quickie: 600th Home Run for A-Rod August 4, 2010

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Alex Rodriguez finally hit #600 deep to center field in Yankee Stadium on the third anniversary of his 500th home run. A-Rod hit the home run in his first plate appearance. There were 51 plate appearances since #599. He had a final Choke Index of .944, but luckily he won’t run into another milestone home run for at least a few years.

The ball landed in Monument Park, so the Yankees didn’t need to negotiate with a fan to get it back. (A security guard picked it up.) According to Michael Kaye, if the ball had landed in the stands, the Yankees would have been willing to pay for the person who caught the ball to have lunch with Alex Rodriguez and Cameron Diaz in exchange for getting the ball back, on top of an autographed baseball, hat, and bat. That opens interesting questions of valuation, much like those that came up after Doug Mientkiewicz attempted to keep the ball that he caught to make the final out in the 2004 World Series.

Is A-Rod’s Performance Different? August 3, 2010

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In games between milestone home runs, is Alex Rodriguez’ hitting similar to other times? (This is all a very polite way of asking, “Does A-Rod choke?”) It’s difficult to answer, because there’s so little data about those milestone home runs. A-Rod, though, has some statistically improbable results and it would be interesting to look at it a bit more closely.

Over 2008-2009, Alex played in 262 games and had 1129 plate appearances with 281 hits, 65 home runs, a triple:double ratio of 1:50, an OBP of .397, and a SLG of .553. His OBP has a margin of error of .0146, so we can be 95% confident that over those years his baseline production would be somewhere between .368 and .426 and absent any time or age effect that is the range in which A-Rod should produce for any given period.

Two recent milestone home runs come to mind as examples of Rodriguez’s reputed choking. First, the stretch between home run #499 and #500 was 8 games and 36 plate appearances. (I’m intentionally ignoring extra plate appearances on the days he hit #499 and #500.) During that time, Alex had an OBP of only .306. That’s a difference of .091 over 36 plate appearances and that performance has a standard error of about .078 when compared with his regular performance, implying a t-value of about 1.16. With 35 degrees of freedom, Texas A&M’s t Calculator gives a p-value of about .127, so this difference is marginally within the realm of chance. (The usual cutoff for significance would be .05.)

A-Rod hit his last home run on July 22. Discounting the plate appearances after his last home run, he’s played in 11 games with a paltry .255 OBP and .238 SLG over 47 plate appearances. His .255 OBP has a difference of about .142 and a standard error of about .064. That implies a t-value of about 2.21, with a p-value of about .016. That is, the probability of this difference occurring by chance is less than 2%. That gives us one result as close to significant and one as probably significant.

As a side note, A-Rod’s Choke Index continues to rise. He’s gone 48 plate appearances without a home run, and at a rate of .055 home runs per plate appearance the probability of that occurring by chance is about .066. That leaves his Choke Index at .934.

The Choke Index August 1, 2010

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It’s been quite a while since Alex Rodriguez hit Home Run #599 – nine days since July 22, but more quantifiably, 42 plate appearances. Just how much of a slump is he in? I’d like to propose a quantifiable answer: the Choke Index.

From 2000 to 2009, A-Rod was hitting approximately .064 home runs per plate appearance. In 2008 he hit .059 and in 2009 he hit .056, so it’s probably much fairer to use a slightly lower rate. I’m going to make the assumption that Rodriguez’s true production is about .055 home runs per plate appearance, since he exhibited a downward trend and his 2010 production has been very low. (It also cuts him some additional slack in the Choke Index.)

Simply, we should assume that A-Rod’s failure to produce is merely the result of chance, and not due to choking or media distraction or even Rodriguez’s discomfort with the special chipped baseballs. (A better man than I would call this the Numbered Ball Effect.) Then, we should see how likely that is.

At .055 home runs per plate appearance, the likelihood of going 42 plate appearances without a home run is (1-.055)^{42} or approximately .093. The Choke Index is simply 1-(likelihood) or, in this case, .907. As it becomes progressively less likely that Rodriguez will go another plate appearance without hitting a home run, the Choke Index number rises. A theoretical Choke Index of 1 would indicate that the player’s lack of home run hitting is nearly impossible to describe by chance alone.

A-Rod’s Choke Index between #499 and #500 was about .877. This is a man who doesn’t handle milestones well.

Another example was Gary Sheffield in 2009, when he was attempting to hit his 500th home run. In the previous two years, he hit approximately .041 home runs per plate appearance. Much was made of Sheffield’s trouble hitting #500, but since he was hitting almost exclusively as a pinch hitter, he simply didn’t have many opportunities. Between his final plate appearance on September 26 of 2008 and his only plate appearance on April 17 of 2009, Sheffield went 21 plate appearances without hitting a homer. That gives him a choke index of .556.

Barry Bonds, meanwhile, was hitting .065 home runs per plate appearance in the seasons prior to his record-breaking home run #756. #755 was hit in Bonds’ first plate appearance on August 4, 2007. Bonds made 3 more plate appearances, all walks, in that game. He hit #756 in his third plate appearance only three days later on August 7.  He had August 5 off and made 4 plate appearances on August 6, meaning that Bonds went 9 plate appearances between home runs, giving him a choke index of .454.

Rodriguez will hit his 600th home run eventually, but it’s getting painful to watch.

The Best Game Ever July 30, 2010

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Two of my favorite things about baseball happened during tonight’s game between the Yankees and the Indians.

First of all, in the top of the ninth inning, corner infielder Andy Marte pitched for the Indians. Marte pitched a perfect ninth and coincidentally struck out Nick Swisher, who was brought in to pitch for the Yankees in a similar situation last year and struck out Gabe Kapler of the Tampa Bay Rays. I can’t promise it’s true, but I think that puts Swisher at the top of the list for involvement in position player pitcher strikeouts.

Marte’s presence was necessary because the Indians used seven other pitchers. Starter Mitch Talbot went only two innings, and the Indians got another two out of Rafael Perez. Frank Hermann took the loss for the Indians during his 1 1/3 innings. Tony Sipp pitched another 1 1/3, and Joe Smith managed to give up four earned runs in 1/3 of an inning before being removed for Jess Todd for an inning. In the bottom of the 9th, Marte was all the Indians had left.

Not to be outdone, Joe Girardi gave up his designated hitter by moving his DH – funnily enough, it was Swisher – into right field as part of a triple switch. Swisher moved to right field; Colin Curtis moved from right field to left field; Marcus Thames moved from left field to third base;  finally, pitcher Chan Ho Park was put into the batting order in place of Alex Rodriguez, who came out of the game.

Finally, A-Rod is up to 33 plate appearances without a home run. Assuming his standard rate of .064 home runs per plate appearance, the likelihood of this happening by chance is .936^{33} = .113 \approx 11.3 \% . I stand by my belief that there’s something other than chance (i.e. distraction or other mental factors) causing Rodriguez’s hitting to suffer.

The 600 Home Run Almanac July 28, 2010

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People are interested in players who hit 600 home runs, at least judging by the Google searches that point people here. With that in mind, let’s take a look at some quick facts about the 600th home run and the people who have hit it.

Age: There are six players to have hit #600. Sammy Sosa was the oldest at 39 years old in 2007. Ken Griffey, Jr. was 38 in 2007, as were Willie Mays in 1969 and Barry Bonds in 2002. Hank Aaron was 37. Babe Ruth was the youngest at 36 in 1931. Alex Rodriguez, who is 35 as of July 27, will almost certainly be the youngest player to reach 600 home runs. If both Manny Ramirez and Jim Thome hang on to hit #600 over the next two to three seasons, Thome (who was born in August of 1970) will probably be 42 in 2012; Ramirez (born in May of 1972) will be 41 in 2013. (In an earlier post that’s when I estimated each player would hit #600.) If Thome holds on, then, he’ll be the oldest player to hit his 600th home run.

Productivity: Since 2000 (which encompasses Rodriguez, Ramirez, and Thome in their primes), the average league rate of home runs per plate appearances has been about .028. That is, a home run was hit in about 2.8% of plate appearances. Over the same time period, Rodriguez’ rate was .064 – more than double the league average. Ramirez hit .059 – again, over double the league rate. Thome, for his part, hit at a rate of .065 home runs per plate appearance. From 2000 to 2009, Thome was more productive than Rodriguez.

Standing Out: Obviously it’s unusual for them to be that far above the curve. There were 1,877,363 plate appearances (trials) from 2000 to 2009. The margin of error for a proportion like the rate of home runs per plate appearance is

\sqrt{\frac{p(1-p)}{n-1}} = \sqrt{\frac{.028(.972)}{1,877,362}} = \sqrt{\frac{.027}{1,877,362}} \approx \sqrt{\frac{14}{1,000,000,000}} = .00012

Ordinarily, we expect a random individual chosen from the population to land within the space of p \pm 1.96 \times MoE 95% of the time. That means our interval is

.027 \pm .00024

That means that all three of the players are well without that confidence interval. (However, it’s likely that home run hitting is highly correlated with other factors that make this test less useful than it is in other situations.)

Alex’s Drought: Finally, just how likely is it that Alex Rodriguez will go this long without a home run? He hit his last home run in his fourth plate appearance on July 22. He had a fifth plate appearance in which he doubled. Since then, he’s played in five games totalling 22 plate appearances, so he’s gone 23 plate appearances without a home run. Assuming his rate of .064 home runs per plate appearance, how likely is that? We’d expect (.064*23) = about 1.5 home runs in that time, but how unlikely is this drought?

The binomial distribution is used to model strings of successes and failures in tests where we can say clearly whether each trial ended in a “yes” or “no.” We don’t need to break out that tool here, though – if the probability of a home run is .064, the probability of anything else is .936. The likelihood of a string of 23 non-home runs is

.936^{23} = .218

It’s only about 22% likely that this drought happened only by chance. The better guess is that, as Rodriguez has said, he’s distracted by the switching to marked baseballs and media pressure to finally hit #600.

Matt Garza, Fifth No-Hitter of 2010 July 26, 2010

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Tonight, Matt Garza pitched the fifth no-hitter of 2010. He joins Edwin Jackson, Roy Halladay, Dallas Braden, and Ubaldo Jimenez in the Year of the Pitcher club.

As I pointed out when Jackson hit his no-hitter, no-hit games are probably Poisson distributed. Let’s update the chart.

The Poisson distribution has probability density function

f(n; \lambda)=\frac{\lambda^n e^{-\lambda}}{n!}

Maintaining our prior rate of 2.45 no-hitters per season, that means \lambda = 2.45. Our function is then

f(n; \lambda = 2.5)=\frac{2.45^n (.0864)}{n!}

The probabilities remain the same:

n p cumulative
0 0.0863 0.0863
1 0.2114 0.2977
2 0.2590 0.5567
3 0.2115 0.7683
4 0.1296 0.8978
5 0.0635 0.9613
6 0.0259 0.9872
7 0.0091 0.9963
8 0.0028 0.9991
9 0.0008 0.9998
10 0.0002 1.0000

And though the expectation (E(49)) and cumulative expectation (C(49)) remain the same, the observed values shift slightly:

E(49) Observed C(49) Total
4.23 5 4.23 5
10.36 11 14.59 16
12.69 8 27.28 24
10.36 17 37.65 41
6.35 1 43.99 42
3.11 5 47.10 47
1.27 1 48.37 48
0.44 0 48.82 48
0.14 1 48.95 49
0.04 0 48.99 49
0.01 0 49.00 49

The tailing observations (say, for 4+ no-hitters) don’t quite match the expected frequencies, but the cumulative values match quite nicely. There might be some unobserved variables that explain the weirdness in the upper tail. Still, cumulatively, we have 47 seasons with 5 or fewer no-hitters, which is almost exactly what’s expected. This is unusual, but not outside the realm of statistical expectation.