## Zambrano Back on the HorseMay 27, 2011

Posted by tomflesher in Baseball.
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Last night, Carlos Zambrano pitched on one day’s rest after pinch-hitting against the hapless Mets for two RBIs on Tuesday. We’ve talked about Zambrano’s pinch-hitting prowess before, but last night he was an awesome 3 for 3 from the plate, including a double. In fact, in 26 plate appearances, Zambrano’s got 9 hits for a .375 batting average and, since he has no walks, a .375 on-base percentage. Not only is that impressive, but I hear he can pitch, too.

I figured that was pretty impressive. It can’t be often that a pitcher gets three at-bats and hits for all of them, can it? It’s happened 450 times since 1919, including, surprisingly, once already this year. The Mets’ Chris Young managed a 3-for-3 night while notching the win against the Phillies back on April 5.

In recent memory, the most at-bats by a pitcher who hit each time was Dan Haren, who grabbed a 4-for-4 as a Diamondback against the Cardinals last year (also as the winning pitcher). Haren also gave up a whopping 7 runs, so he’s lucky he was hitting.

Micah Owings has had two games where he pitched and hit in all of at least 3 plate appearances, including a 4-for-4 from 2007 in which three of his 4 hits were doubles.

Finally, Mel Stottlemyre (in 1964) and two pitchers from the 1920s had 5-for-5 games. Stottlemyre’s two-hit gem included him hitting a double and pitching to a game score of 83.

## Micah Owings and Cobb-Douglas ProductionJuly 22, 2010

Posted by tomflesher in Baseball, Economics.
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Micah Owings, who is one of the best two-way players in baseball since Brooks Kieschnick, was sent down to the minors by the Cincinnati Reds yesterday. As big a fan as I am of Micah (really, look at the blog), I think this was probably the right decision.

Owings was being used as a long reliever. For a big-hitting pitcher like Micah, that’s death to begin with. Relievers need to be available to pitch, so the Reds couldn’t get their money’s worth from Owings as a pinch hitter, since he wouldn’t be available to re-enter the game as a pitcher unless they used him immediately. They also weren’t getting their money’s worth as a pitcher, since, as Cincinnati.com notes, the Reds’ starting pitching was doing very well and so long relief wasn’t being used very often.

Letting Owings start in AAA will give him the best possible outcome – he’ll have regular opportunities to pitch, so he won’t rust, and he’ll get to bat at least some of the time. Owings needs to be cultivated as a batter because that’s where his comparative advantage is. I doubt he’ll ever be at the top of the rotation, but he could be a competent fifth starter. If he pitches often enough to get there, he’ll add significant value to the team in terms of his OBP above the expected pitcher. He’ll get on base more, so he’ll both advance runners and avoid making an out.

A baseball player is a factory for producing run differential. He does so using two inputs: defensive ability (pitching and fielding) and offensive ability (batting). In the National League, if a player can’t hit at all, he’s likely to produce very little in the way of run differential, but at the same time, if he’s a liability on defense, he’s not likely to be very useful either. Defense produces marginal runs by preventing opposing runs from scoring, and offense produces marginal runs by scoring runs. Having either one set to zero (in the case of a pitcher who can’t hit at all) or a negative value (an actively bad pitcher) would negatively affect the player’s run production. This is similar to a factory situation where labor and equipment are used to produce goods, and that situation is usually modeled using a Cobb-Douglas production function:

$Y = K^{\alpha} \times L^{1 - \alpha}$

with Y = production, z = a productivity constant, K = equipment and technology, L = labor input, and $\alpha$ is a constant between 0 and 1 that represents relatively how important the input is. K might be, for example, operating expenses for a machine to produce widgets, and L might be the wages paid to the operators of the machine. This function has the nice property that if we think both inputs are equally important (that is, $\alpha$ = .5) then production is maximized when the inputs are equal.

In general, production of run differential could be modeled using the same method. For example:

$RD = P^{\alpha} \times F^{\beta} \times B^{1 - \alpha - \beta}$

where P = pitching contribution, F = fielding contribution, B = batting contribution, and $\alpha$ and $\beta$ are both between 0 and 1 and would vary based on position. For example, David Ortiz is a designated hitter. His pitching ability is totally irrelevant, and so is his fielding ability outside of interleague games. The DH’s $\alpha$ would be 0 and his $\beta$ would be very close to 0. On the other hand, an American League pitcher would have an $\alpha$ very close to 1 since pitcher fielding is not as important as pitching and his hitting is entirely inconsequential in the AL. Catchers would have $\alpha$ at 0 but $\beta$ much higher than other positions.

The upshot of this method of modeling production is that it shows Owings can make up for being a less than stellar pitcher by helping his team score runs and be a considerably better investment than a pitcher with a slightly lower ERA but no run production.

## Pinch Hitters from the BullpenJuly 6, 2010

Posted by tomflesher in Baseball, Economics.
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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.

1. 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.
2. Micah Owings is a statistical outlier from the pattern. It’s not clear why.

## Carlos Zambrano, Ace Pinch Hitter?June 21, 2010

Posted by tomflesher in Baseball.
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Earlier this year, Chicago Cubs manager Lou Piniella experimented with moving starting pitcher and relatively big hitter Carlos Zambrano to the bullpen, briefly making him the Major Leagues’ best-paid setup man. Zambrano is back in the rotation as of the beginning of June. I’m curious what the effect of moving him to the bullpen was.

The thing is that not only is Zambrano an excellent pitcher (though he was slumping at the time), he’s also a regarded as a very good hitter for a pitcher. He’s a career .237 hitter, with a slump last year at “only” .217 in 72 plate appearances (17th most in the National League), which was 6th in the National League among pitchers with at least 50 plate appearances. He didn’t walk enough (his OBP was 13th on the same list), but he was 9th of the 51 pitchers on the list in terms of Base-Out Runs Added (RE24) with about 5.117 runs below a replacement-level batter. Ubaldo Jimenez was also up there with a respectable .220 BA, .292 OBP, but -8.950 RE24.

It should be pointed out that pitcher RE24 is almost always negative for starters – the best RE24 on that list is Micah Owings with -2.069. Zambrano’s run contribution was negative, sure, but it was a lot less negative than most starters. Zambrano also lost a bit of flexibility as an emergency pinch hitter (something that Owings is going through right now due to his recent move to the bullpen) – he’s more valuable as a reliever, so they won’t use him to pinch hit. As a result, he loses at-bats, and that not only keeps him from amassing hits. It also allows him to get rusty.

It’s hard to precisely value the loss of Zambrano’s contribution, although he’s already on pace for -6.1 batting RE24. It’s likely, in my opinion, that his RE24 will rise as he continues hitting over the course of the year. His pitching value is also negative, however, which is unusual. He’s always been very respectable among Cubs starters. It’s possible that although he was pitching very well in relief, the fact that he has the ability to go long means that it’s inefficient to use him as a reliever. This is the opposite of, say, Joba Chamberlain, who is overpowering in relief but struggles as a starter.

As a starter, Zambrano has never been a net loss of runs. He needs to stay out of the bullpen, and Joba needs to stay there.

## Pitchers with 4+ RBIs (Sorry, Mets fans)September 23, 2008

Posted by tomflesher in Academia, Baseball.
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