I posted an idea yesterday, a made-up stat just for fun. I called it No-Hitter Added because it was a “real time” stat that apportioned credit among Mariners’ pitchers for their Friday no-hitter. Tom Wilhelmsen was given a lot more credit for the no-hitter than Kevin Millwood, because Wilhelmsen pitched the critical ninth inning when the no-hitter was most in reach … despite the fact that Millwood faced six times as many batters.

As a reminder, if you were to break out credit for the no-hitter based on batters faced (which is obviously the best way to do it), it would look like this:

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

The No Hitter Added stat was just for fun, an exercise to show how WPA works, as well as its strengths and its flaws. kds posted a great comment on the article, noting that an approach that divides the No Hitter Added stats by the Leverage Index would be fairer. And that got my head spinning.

See, it’s easy to calculate the Leverage Index of each batter faced in this situation, because the only outcomes we care about are a hit or not. For instance, with two out in the ninth, the Leverage Index is 1.0, because an out will result in a no-hitter (a 1) but a hit will result in no chance of a no-hitter (a 0). 1-0 equals one. In another example, the Leverage Index of the first batter is 0.0007, because a hit results in a 0, but an out leads to the next at-bat, at which point the team has a 0.07% probability of throwing a no-hitter (not giving up a hit to the next 29 batters). 0.0007 minus zero is … well, you get the idea.

In all cases, the “No-hitter Leverage Index” of an at-bat is the probability of throwing a complete-game no-hitter starting with the next at-bat.

So now we have No Hitter Added, No-Hitter Leverage Index, and one divided by the other (NHA/LI). Let’s apportion credit among Mariner pitchers for that no-hitter using NHA/LI:

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

Yeah, the results are exactly the same as those using batters faced. In other words, NHA/LI works.

So what, you say? Did I just waste your time? Was that just a bag of gas for nothing? Well, yes, it mostly was except for one thing. Now you know why WPA/LI works.

Most people get WPA and LI, but WPA/LI is maddeningly difficult to explain. Tangotiger has tried several times, as have I, but it’s hard to do. People have just had to take it on faith that it works.

But No Hitter Added is a much simpler concept (and the math is much simpler) than Win Probability Added. And here you can see that dividing the “Added” stat by the Leverage Index results in the exact numbers it should. I’m not going to try to explain why it works; I’ve tried that before. I’m hoping you can see that the math, which works perfectly in the no-hitting example, will work just as well within the framework of winning games.

WPA/LI is available at both Fangraphs and Baseball Reference.

studes said...

Colin tweeted me yesterday, saying that he isn’t sure this is a good model for WPA/LI, “in that you can have assuming state before ninth inning.” Not sure what he means, other than perhaps you can reach a non-no-hitter before the ninth, but you can’t win a game before the ninth. That said, I don’t know why this would undermine the approach. I’m hoping Colin stops by to discuss it.

Leverage Index is basically the range of probabilities in the subsequent at-bat. When you think about it, NHA/LI is going to equalize because the probability of a no-hitter increases at a steady (albeit exponential) rate. So the change in state from one at-bat to another will increase at the same rate the probability does.

Win probability also changes at an underlying steady pace, though it is much harder to see because of WPA’s complexity. Probabilities go up and down, unlike the no-hitter probability (which went steadily up). Does this make the no-hitter example a bad one for WPA/LI. I don’t think so—I think the underlying mathematical relationship applies throughout the game on an at-bat basis. But I’m no mathematician.

Colin Wyers said...

That’s what I mean, yeah. In that sense it’s not an apples-to-apples comparison.

Now does that matter? I don’t know. I haven’t had a chance to actually do any math (reading articles on a smartphone, while an astonishing technological marvel, is not very conducive to in-depth review and comment).

Detroit Michael said...

Studes wrote: “…if you were to break out credit for the no-hitter based on batters faced (which is obviously the best way to do it)…” It seems to me that a better way is to credit the no-hitter based on outs recorded, not batters faced. While walking a batter means a hit was not yielded, it doesn’t get the team any closer to the end of the game.

studes said...

Given the way I approached this, I do think batters faced is best. I assumed there would be 30 non-hit plate appearances in all, an average number of other non-hit events. Given that was how exactly the way this game played out, it fits well and it makes the math much easier to implement.

Overall, your point is a good one, though I’d argue that *some* credit should be given to anyone who faces a batter and doesn’t yield a hit. More credit should be given to someone who gets an out and doesn’t yield a hit. I have no idea what those proportions should be.