Pinch Hitters from the Bullpen

Pinch Hitters from the Bullpen July 6, 2010

Posted by tomflesher in Baseball, Economics.
Tags: binomial distribution, bullpen, Carlos Zambrano, Livan Hernandez, margin of error, Micah Owings, pinch hitter, sabermetrics
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Occasionally, a solid two-way player shows up in the majors. Carlos Zambrano is known as a solid hitter with a great arm (despite the occasional meltdown), and Micah Owings is the rare pitcher used as a pinch hitter. Even Livan Hernandez has 15 pinch-hit plate appearances (with 2 sacrifice bunts, 6 strikeouts, and a .077 average and .077 OBP, compared with a lifetime .227 average and .237 OBP).

Like Hernandez, Zambrano has a very different batting line as a pinch hitter than as a pitcher. In 24 plate appearances as a pinch hitter, Big Z is hitting only .087 with a .087 OBP, compared to his .243/.249 line when hitting as a pitcher. Since we see the same effect for both of these pitchers, it seems like there’s some sort of difference in hitting as a pinch hitter that causes the pitchers to be less mentally prepared. Of course, these numbers come from a very small sample.

On the other hand, Micah Owings hits .307/.331 as a pitcher, and a quite similar .250/.298 as a pinch hitter. What’s the difference? Owings has almost double Zambrano’s plate appearances as a pinch hitter with 47. That seems to show that maybe Owings’ larger sample size is what causes the similarity. How can this be tested rigorously?

As we did with Kevin Youkilis and his title of Greek God of Take Your Base, we can use the binomial distribution to see if it’s reasonable for Owings, Hernandez and Zambrano to hit so differently as pinch hitters. To figure out whether it’s reasonable or not, let’s limit our inquiry to OBP just because it’s a more inclusive measure and then assume that the batting average as a pitcher (i.e. the one with a larger sample size) is the pitcher’s “true” batting average and use that to represent the probability of getting on base. Each plate appearance is a Bernoulli trial with a binary outcome – we’ll call it a success if the player gets on base and a failure otherwise.

Under the binomial distribution, the probability of a player with OBP p getting on base k times in n plate appearances is:

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

with

${n\choose k}=\frac{n!}{k!(n-k)!}$

We’ll also need the margin of error for proportions. If p = OBP as pitcher, and we assume a t-distribution with over 100 plate appearances (i.e. degrees of freedom), then the margin of error is:

$\sqrt{\frac{p(1-p)}{n-1}}$

so that 95% of the time we’d expect the pinch hitting OBP to lie within

$OBP \pm 2\times\sqrt{\frac{p(1-p)}{n-1}}$

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

with

${n\choose k}=\frac{n!}{k!(n-k)!}$

We’ll also need the margin of error for proportions. If p = OBP as pitcher, and we assume a t-distribution with over 100 plate appearances (i.e. degrees of freedom), then the margin of error is:

$\sqrt{\frac{p(1-p)}{n-1}}$

so that 95% of the time we’d expect the pinch hitting OBP to lie within

$OBP \pm 2\times\sqrt{\frac{p(1-p)}{n-1}}$

Let’s start with Owings. He has an OBP of .331 as a pitcher in 151 plate appearances, so the probability of having at most 14 times on base in 47 plate appearances is .3778. In other words, about 38% of the time, we’d expect a random string of 47 plate appearances to have 14 or fewer times on base. His 95% confidence interval is .254 to .408, so his .298 OBP as a pinch hitter is certainly statistically credible.

Owings is special, though. Hernandez, for example, has 994 plate appearances as a pitcher and a .237 OBP, with only one time on base in 15 plate appearances. It’s a very small sample, but the binomial distribution predicts he would have at most one time on base only about 9.8% of the time. His confidence interval is .210 to .264, which means that it’s very unlikely that he’d end up with an OBP of .077 unless there is some relevant difference between hitting as a pitcher and hitting as a pinch hitter.

Zambrano’s interval breaks down, too. He has 601 plate appearances as a pitcher with a .249 OBP, but an anemic .087 OBP (two hits) in 24 plate appearances as a pinch hitter. We’d expect 2 or fewer hits only 4% of the time, and 95% of the time we’d expect Big Z to hit between .214 and .284.

As a result, we can make two determinations.

Zambrano and Hernandez are hitting considerably below expectations as pinch hitters. It’s likely, though not proven, that this is a pattern among most pitchers.
Micah Owings is a statistical outlier from the pattern. It’s not clear why.

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