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# No Hitter Added

The Mariners threw a multi-pitcher no-hitter against the Dodgers yesterday. Sadly, starter Kevin Millwood injured his groin and had to leave after the sixth inning. Millwood had been terrific, facing the minimum number of batters and only allowing one batter to reach base (Juan Rivera walked but was subsequently out on a double play). Five relievers pitched the last three innings; they got into one significant jam via two walks and a sacrifice, but they also gave up no hits while on the mound. Happiness and hugs ensued.

Six pitchers in total. So who deserves the most credit for this no-hitter?

Well, obviously Millwood does. He pitched six of the nine innings and faced 18 batters without giving up a hit. No other pitcher faced more than three batters. So if you were to divide credit between all the pitchers based on batters faced, it would go like this:

```K Millwood          60%
T Wilhelmsen        10%
S Pryor             10%
C Furbush           10%
B League             7%
L Luetge             3%```

But there’s another way to look at this. I’m going to call it No Hitter Added.

As you might guess, the logic is the same as Win Probability Added, only I’m going to look at the probability of a hit instead of the probability of a win. Here’s my logic…

If you apply the Odds Ratio to the current major league average of hits per plate appearance, as well as the average of the Mariners’ pitchers and Dodgers’ hitters, you can estimate that the probability of a Dodger getting a hit in any single plate appearance against the Mariners is 0.222. Next, you figure that the average number of non-hit plate appearances in a nine-inning game is going to be about 30 (outs, walks, reached on error and other silly things), which is perfect because that’s exactly the number of batters the Dodgers sent to the plate last night.

At that rate, the probability of 30 consecutive batters not getting a hit is 0.05%, or one game out of every 1,864 (932, if you count both teams). If the first batter doesn’t get a hit, the probability of the next 29 batters not getting a hit is 0.07%, so non-hitting that first batter has improved your no-hitter probability 0.02 points.

At these rates, the probability of a full no-hitter after 18 batters—when Millwood left the game—is still only 5%. Said differently, Millwood’s awesome performance increased the probability of a no-hitter from 0.05 to 4.92, or roughly five points. We’ll call that his No Hitter Added.

That was the lowest total of any Mariner pitcher last night. Here are the No Hitter Added figures for all pitchers:

```T Wilhelmsen        53%
B League            19%
S Pryor             12%
L Luetge             6%
C Furbush            6%
K Millwood           5%```

When Tom Wilhelmsen entered the game in the bottom of the ninth, the probability of a complete-game no-hitter was still less than 50%. His one-inning performance took the game from less-than-likely to sure-thing.

Let me be clear before someone misquotes me: I’m not saying that Wilhelmsen deserves the bulk of the credit for the Mariners’ no-hitter. Millwood obviously does. But I’ve created this stat to capture something else: to quantify which pitcher was on the mound for the most no-hit drama last night. I’m guessing that most fans were not on the edge of their seats when Charlie Furbush retired the side in order in the seventh. I’m guessing most of them were when Wilhelmsen was on the mound in the ninth. According to this stat, there should have been nine times more no-hit excitement during Wilhelmsen’s time than Furbush’s time.

The parallels with Win Probability Added are obvious. I use Win Probability a lot, because it tells me things that no other stat does. Specifically, it captures the story, the drama, of the field. When describing games and rallies, strategy and tactics, I think it’s the best stat we have. When applied to batters in particular, I think it’s fascinating.

But this application of the WPA concept also shows its weaknesses. It so values late innings that it undervalues the contribution of earlier innings. You really shouldn’t use WPA to compare the value of starting pitchers and relievers. Time is still more important than timing.

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Dave Studeman was called a "national treasure" by Rob Neyer. Seriously. Follow his sporadic tweets @dastudes.
Guest
kds

That was well done, and shows how WPA-like numbers can strongly prejudice how we look at things.

A different way of looking at it would observe that the odds on a no-hitter were 1864-1 when Millwood started and almost 20-1 when he left.  So they were about 93 times as likely.  Wilhelmsen came in with the odds almost even, and completed the no-hitter to double improve their likelyhood.  I think this way of looking at it is much more like using WPA/LI to rate a player’s effect upon the game.

Guest
studes

Excellent point, kds!  LI is easy to calculate here, because the outcome is a simple binomial yes/no.  And, when you use NHA/LI in this case, the credit is divided up according to the original table—exactly as it should be.  I need to write this up separately, because I think it’s a good insight into why WPA/LI works.