## Complete Game in a Non-Quality StartMay 26, 2011

Posted by tomflesher in Baseball.
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Dillon Gee of the Mets was credited with a complete game in last night’s win over the Cubs. His line: 6 IP, 4 H, 4 R, 4 ER, 2 BB, 4 K, 0 HR, and 1 HBP, for a game score of 50. He qualified for a quality start under the Game Score definition, but not under the six-inning, three-run criterion. That makes it a form of Cheap Win, where a pitcher is credited with a win even though he didn’t pitch as effectively as expected.

Since the game was shortened by rain, Gee got a complete game, even though that usually involves 8 innings for the visiting pitcher on a losing team or 9 inning for a winning pitcher regardless. That made me wonder how many pitchers from the modern era, when complete games are less common than in previous years, have pitched complete games in non-quality starts.

A quality start, under the Game Score definition, is a start with less than 50 points. That represents that a pitcher had negative value for his team. It can’t be especially common, can it?

According to this list I queried from Baseball Reference, a non-quality start complete game hasnt been pitched since 2006 when Freddy Garcia pitched a rain-shortened 5-inning complete game for the White Sox to defeat the Blue Jays 6-4, with a game score of 42. The last nine-inning complete game non-quality start was Pete Harnisch with the Reds, who won a 10-6 slugfest in August of 2000 on 124 pitches with only one walk and three strikeouts. Aside from the six earned runs (all scored in the first three innings) it wasn’t a bad performance, somewhat reminiscent of Edwin Jackson‘s ugly but effective no-hitter last year.

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

Posted by tomflesher in Baseball.
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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.

## Edwin Jackson, Fourth No-Hitter of 2010June 25, 2010

Posted by tomflesher in Baseball, Economics.
Tags: , , , , , , , , ,

Tonight, Edwin Jackson of the Arizona Diamondbacks pitched a no-hitter against the Tampa Bay Rays. That’s the fourth no-hitter of this year, following Ubaldo Jimenez and the perfect games by Dallas Braden and Roy Halladay.

Two questions come to mind immediately:

1. How likely is a season with 4 no-hitters?
2. Does this mean we’re on pace for a lot more?

The second question is pretty easy to dispense with. Taking a look at the list of all no-hitters (which interestingly enough includes several losses), it’s hard to predict a pattern. No-hitters aren’t uniformly distributed over time, so saying that we’ve had 4 no-hitters in x games doesn’t tell us anything meaningful about a pace.

The first is a bit more interesting. I’m interested in the frequency of no-hitters, so I’m going to take a look at the list of frequencies here and take a page from Martin over at BayesBall in using the Poisson distribution to figure out whether this is something we can expect.

The Poisson distribution takes the form

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

where $\lambda$ is the expected number of occurrences and we want to know how likely it would be to have $n$ occurrences based on that.

Using Martin’s numbers – 201506 opportunities for no-hitters and an average of 4112 games per season from 1961 to 2009 – I looked at the number of no-hitters since 1961 (120) and determined that an average season should return about 2.44876 no-hitters. That means

$\lambda = 2.44876$

and

$f(n; \lambda = 2.44876)=\frac{2.44876^n (.0864)}{n!}$

Above is the distribution. p is the probability of exactly n no-hitters being thrown in a single season of 4112 games; cdf is the cumulative probability, or the probability of n or fewer no-hitters; p49 is the predicted number of seasons out of 49 (1961-2009) that we would expect to have n no-hitters; obs is the observed number of seasons with n no-hitters; cp49 is the predicted number of seasons with n or fewer no-hitters; and cobs is the observed number of seasons with n or fewer no-hitters.

It’s clear that 4 or even 5 no-hitters is a perfectly reasonable number to expect.

 2.44876