## Quickie: Ryan Howard’s Choke IndexOctober 25, 2010

Posted by tomflesher in Baseball.
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The Choke Index is alive and well.

Previous to 2010, Ryan Howard of the Philadelphia Phillies hit home runs in three consecutive postseasons. He managed 7 in his 140 plate appearances, averaging out to .05 home runs per plate appearance. Not too shabby. It’s a bit below his regular season rate of about .067, but there are a bunch of things that could account for that.

This year, Ryan made 38 plate appearances and hit a grand total of 0 home runs in the postseason. What’s the likelihood of that happening? I use the Choke Index (one minus the probability of hitting 0 home runs in a given number of plate appearances) to measure that. As always, the closer a player gets to 1, the more unlikely his homer-free streak is.

The binomial probability can be calculated using the formula

$f(k;n,p) = \Pr(K = k) = {n\choose k}p^k(1-p)^{n-k}$

Or, since we’re looking for the probability of an event NOT occurring,

$(1-p)^k$

or $.95^{38}= .142$

using his career postseason numbers. That means that Ryan Howard’s 2010 postseason Choke Index is .858. Pretty impressive!

## Quickie: 600th Home Run for A-RodAugust 4, 2010

Posted by tomflesher in Baseball.
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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

Posted by tomflesher in Baseball, Economics.
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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 IndexAugust 1, 2010

Posted by tomflesher in Baseball.
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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.