Tuesday, October 13, 2009

There's a Stat for That: Adjusted Runs Prevented

It's hard to argue for or against relievers based on statistics. Relievers can give up other pitcher's runs without their own ERA taking a hit. Hits, walks and strikeouts don't necessarily depict how successful a player was. And of course saves are borderline silly.

What would really be valuable would be a stat that somehow gave credit for stranding inherited runners, but cost a reliever when he got the team into a jam.

Turns out, there's a stat for that. It's called Adjusted Runs Prevented, or ARP.

I've become hesitant to introduce any new stats, because this is when the vast majority of eyes glaze over. I'm going to try it anyway with ARP, because it's just so darn simple. It's also ultimately very, very powerful when talking about relievers. But first, the simple:

ARP = expected runs when a reliever enters a game - expected runs when he left the game.

So say a reliever comes in with runners on the corners and no outs. From studying tens of thousands of previous MLB innings, we know, on average, how many runs will score that inning. It's about 1.9. So if the reliever strands those runners, he gets +1.9 ARP. If he gets out of the inning with just one run scoring, he gets +.9. If two runs score, he gave up more runs then the average, so his ARP is -.1.

Or say a reliever comes in to pitch the ninth inning. The average runs scored in an inning are .56. So if he pitches a perfect ninth, he gets +.56 ARP. And if he gives up a 2-run home run to Alex Rodriguez (for example) his ARP would be -1.44.

If you have some more questions about ARP, leave them in the comments section. We can do some more complicated examples there if you like.

Anyway, ARP is exactly what it is named; it is how many runs a reliever prevented. And those runs aren't random - they're the average runs from decades of major league games. Finally, there is no goofy extra credit for pitching late innings - really valuable middle relievers or setup men often lead the league in ARP. It is an EXCELLENT statistic for showing which relievers have had the best year. It's a MVP stat for relievers.

So who were the top relievers this year? Of the 729 relievers tracked by Baseball Prospectus, here's the top five:
Andrew Bailey (OAK) +29
Michael Wurtz (OAK) +28.3

Joe Nathan (MIN) +28

Matt Guerrier (MIN) +26.8

Mariano Rivera (NYY) +25.8

That third name might surprise some Twins fans. Turns out Joe Nathan had a hell of a year. Remember, he lost ARP runs in those games where he collapsed, but overall he was extremely effective. You can't find three guys in the majors that were more effective than him.

What's more, this was not a one-year thing. Nathan ranked #7 in overall ARP last year. By the way, there was only one other guy who was in the top 10 both years. It's Mariano Rivera.

But it's also not just a two-year thing. Because in 2007 Nathan was #8. Rivera finished 21st that year. And it keeps going...
In 2006, Nathan was #6 and Rivera was #8
In 2005, Nathan was #15 and Rivera was #2.
In 2004, Nathan was #8 and Rivera was #6.

As you go through this list, year by year, there are two names that are on the list over and over - Nathan and Rivera. Those two, and ONLY those two, have been the model of consistency, effectiveness and health since Nathan joined the Twins in 2004.

Now, this doesn't mean that Nathan can't be traded. For the right package of players, just about nobody on the Twins should be untouchable.

But to say that Nathan is slipping, or nearly finished, is almost indefensible. His handful of failures have been more memorable because of when they happened and who they happened against, but objectively they barely dent the successes he had.

However, there is something that is even more indefensible, and that is to say that he can be easily replaced. Or even replaced at all:
- Matt Guerrier? Great this year. But his ARP last year was negative.
- Jose Mijares? His ARP was 12.8, less than half of Nathan's.
- Pat Neshek? Um, you might want to look at Francisco Liriano this year before you commit to someone coming off of Tommy John surgery?
- And the most popular - someone else? After all, Nathan wasn't anything special when we got him...

Wrong again. Nathan was already on that list of ARP leaders back when he was throwing middle relief for the Giants, ranking 24th out of 676 relievers in 2003. He was just two slots below Francisco Rodriguez that year, and that was when KRod was exploding onto the scene.

The doubters can bitch about the stat, but they're going to need bring something tangible to the table besides a couple of bad nights. The dreamers can can talk about a trade, but they're going to need to bring some realistic player packages. And both of them are going to need to bring a replacement for someone who is arguably the second best relief pitcher of this decade.

And that includes 2009.


Late addition: For those of you saying we should trade Nathan for "the right package", I really would like to hear what you consider the right package to be in the comments section. Give me teams and names that you think are realistic and would improve the team. Thanks!


Mike Whitaker said...

So, just to check....

If say, Joe Reliever starts the ninth, and quits after a bad day with no outs and runners on the corners, his ARP is 0.58 - 1.9... i.e -1.32

Mike (from 4000 miles away)

GM said...


"Or say a reliever comes in to pitch the ninth inning. The average runs scored in an inning are .56. So if he pitches a perfect ninth, he gets +.56 ARP. And if he gives up a 2-run home run to Alex Rodriguez (for example) his ARP would be -1.44.

isn't quite right, I don't think. Or at least is incomplete. It would be right if the pitcher finished the inning, but that automatically gives full-inning credit to a closer who loses in a final at bat, and that pitcher never pitches a full inning. ARP for an appearance should depend on the number of outs recorded. If a guy comes in and gives up two runs while recording no outs, the ARP should be .56 - 2 -.56 (because the run expectation is unchanged when he leaves ---- still .56). This needs to be done for closers who lose in a walk off, or they are forgiven some negative ARP not forgiven to other pitchers.

Maybe this IS covered by ARP, and the problem is just with how you wrote your example, but if your example is really how ARP works, then that's a problem.

For Nathan, an off-the-cuff guess is that correctly accounting for walk-off losses lowers his ARP by maybe 2. Also a note: this is a counter-effective bias (it lowers the ARP of anyone who loses in a walkoff, but the more often a pitcher loses in a walkoff, the more this bias helps that pitcher).

TwtinsTarget said...

Great stat, John. A must for those evaluating relievers.

GM, from my knowledge the reliever who starts an inning (.56), gives up two runs (-2), and is pulled before making an out but leaves the bases empty (.56), his ARP would be .56-2-.56... or an even -2.

For those curious, here is the complete formula:

(ER(sS,P) - (ER(sF,P) + IF*ER(sO,P) - R) / pe(P)

ER is the runs expected to score in sS (base/outs state at beginning of inning) including a park factor of P.

ER is the runs expected to score in sF,P (base/outs state at end of inning) including a park factor of P.

IF*ER(sO,P) is the number of innings pitched (IF) times the expected number of runs to score in a certain park (P) at the beginning of an inning.

R is the total number of runs scored while the reliever was on the mound.

pe(P) is, again, the park effect.

Anonymous said...

Interesting, John--thanks!

Just to clarify the formula, I'm guessing that sS and sF refer to the state when the reliever starts and finishes, not at the beginning and end of the inning.

The definition of the "state" as the number of outs and location of baserunners raises a question for me: has research ever shown whether or not the inning has a significant effect on expected runs, holding runners and outs constant?

On Nathan: I read a post on the Strib web site suggesting that perhaps Nathan's arm was worn about by this time of the year (and noting that Jeff Reardon used ot have a big year saving something like 35 games). Makes me wonder how Nathan's ARP looks month by month, or as a function of cumulative innings (or perhaps pitches, or appearances).

Jeff B said...

Geek, THIS is exactly why I've been a fan of yours since the email list days way back when. Nice work objectifying the issues.

Now, how bout a stat that can identify WHEN a reliever (or other pitcher) is about to go in the tank?? And I'm not suggesting Nathan is heading in that direction unless an injury surfaces.

And for those that say his velocity has decreased...he did hit 94 in Game 3 according to the Dome's gun...and has anyone heard of Trevor Hoffman?

Here's the link to the 2009 ARP leaders.

John said...

Thanks gang.

Yes, I simplified this for the story, but you're exactly right in that you are subtracting the actual runs given up AND the expected runs they leave in.

So let's go back to the second example. First of all, yes, I did assume that Nathan finished the whole inning,, which is why his ARP for that game was -1.45. But let's say the following happens.

1. Nathan starts an inning
2. Gives up a walk and a two-run home run.
3. Then he gives up two singles taht results in runners on first and third and no out.

Ok, so when he started the inning the expected runs was .55.

When he left he had given up two runs AND it's expected the other team will score another 1.9 runs in the inning. So Nathan's ARP is:

.55 - (2 + 1.9) which is -3.35

Does that make it clearer?

(Oh, and I should mention, I'm not including anything about park effect in that example. This is straight from Run Expectancy Matrix. I'll let someone a little more anal add the park effect.)

TwinsFanc1981 said...

According to Brooksbaseball.net and their pitchf/x database, he actually top out a 95-mph in Game 3. This is down about 2-mph from his max velocity of 97.5-mph in September of 2007.

I'm not advocating a trade, I'm on the side that believes if a team presents us with a stack of prospects, you need to seriously consider it. The Twins have been bad in recent years at maximizing player's value in trades and now we are left without much MLB-ready positional players in the system.

At 35 in 2010, you wonder how much longer Nathan will be an elite closer. Will he start to show wear next year? In the final year prior to his option in 2012? Could it be that he has an extended career into his late-30s/early-40s like Rivera or Hoffman? Unlike that duo, Nathan lacks the dominate out-pitch like Rivera's cut fastball or Hoffman's change that has helped them prolong their careers even as their velocity declined.

Trading Nathan would hinder the team in 2010 but may provide long-term solutions in other problem areas. Then again, if nobody in the FO is wowed by any offerings, you can go into 2010 knowing you have one of the league's best closers at your disposal.

John said...

To me, you start the coversation with "We need to get an elite bullpen arm.". Then, after that's taken care of, you can trade Nathan. But you don't make that move without a net.

TT said...

The Twins have one of the best core groups of young players in the major leagues. They are buyers, not sellers. Nathan is worth more to them than he is to any other team in baseball. They can't possibly get back even equal value.

I understand the "sell high" idea, but the Twins are in the baseball business, not the investment business. Its like the guy who wanted to sell his house at the peak of the housing market. Until his wife asked "I know we will make a lot of money honey. But where will we live?"

BeefMaster said...

Really interesting stuff - I like it. I would mention that since this is a counting stat, it might also be useful to have a version that is scaled to IP in the event of a guy who only played a half season, or a LOOGy (who only gets one out at a time, so he isn't knocking off many expected runs even when he's effective). Do you know if anyone tracks this somewhere online, so we can find it during the season?

Anonymous - I saw an article recently (I think by Tom Verducci) mentioning that, in general, the ninth inning is more difficult to score in, because of closers. He didn't have any specific statistics, though, so I don't know by how much, or if he was even using actual numbers rather than just stating it.

Anonymous said...

I was going to say something similar to beefmaster. It seems like a lot of this is out of the reliever's control, but more based on how the manager chooses to use him. A LOOGY is more likely to come in with runners on to face a lefty, right? And a mop up guy is more likely to come in with the bases empty?

Dave_MN said...

Did the calculator melt when you tried to figure out Jesse Crain's numbers?

Anonymous said...

To Beefmaster's point, you could just divide ARP by inning's pitched to get run's prevented per inning.

But seeing the total runs is actually in itself a correction for overvaluing people who are used sparingly in low-impact situations. A LOOGY who pitches to one guy at a time, and never faces the many people who could batter him, is valuable, but he wouldn't prevent many runs in total because of the limitations on his use. Nathan has to face all comers in the 9th inning of every game (or every close game with a lead).

The other way to correct for high impact situations of course is to talk about Win Probability Added, which Nathan traditionally rocks in too.

As to the other interesting point Beefmaster raises, that fewer runs tend to be scored in the 9th inning, I'm not sure it would be fair to correct for that. If the reason is that most teams use a better reliever in the 9th than in the 8th, it doesn't mean their job was easier, just because fewer runs are scored in total.

David.Wiederich said...

A friend suggested I read this article, mainly bc he loves Nathan. After a quick read through the article and some of the comments posted I think one huge factor is that "forecasting" drives this stat. Not saying it isnt a good stat that could be used for further trends, also a forecasting termenology, but not an actual stat- being it is subjective.