What Leverage Index Do Relievers Deserve?
by Sky Kalkman on Apr 20, 2009 2:00 PM EDT
57 comments
Mariano Rivera pitches 65 innings a year and makes $15M. Sure, they're really really good innings, but the other reason he's worth so much in only 65 inning is because his manager can pick the spots when he pitches. In geek-speak, he pitches in a lot of high-leverage situations, making his innings worth as much as 115 "regular" innings.
Nobody's going to argue that Mo deserves to be a closer and rack up a 1.8 Leverage Index (LI) year after year. But not all managers are rational and not all relievers are used in the proper situations. Take, for example, Joe Borowski and Heath Bell. Over the past few years, Borowski's been a closer with a FIP over 4.00, while Bell's been a setup man with a FIP close to 3.00. Giving each pitcher credit for the average LI in which they were used makes them look similarly valuable, while almost anyone in their right mind would rather have Bell.
One way to solve that dilemna is to still give relievers credit for the leverage of important late innings (so they can be compared fairly to other positions), but to use an LI based on the role they deserve instead of based on when their managers chose to use them.
I first discussed the idea of "deserved leverage index" (which Harry wants to call deLI, mmm) with Justin Inaz and Tom Tango over a year ago, but it got dropped. I'm bringing it back. Tom's best idea was to use a model that looks like deLI=(lgFIP/FIP)^1.5, where the 1.5 could be messed around with. Using a 4.10 average FIP for relievers, you'd get:
FIP deLI
2.50 2.0
2.75 1.7
3.00 1.5
3.25 1.4
3.50 1.2
3.75 1.1
4.00 1.0
4.25 1.0
4.50 .9
4.75 .8
5.00 .8
(deLI should probably be capped at the high end at 1.8ish, by the way.)
Now, Joe Borowski's 4.25 FIP spits out a 1.0 deserved LI, and Heath Bell's 3.25 FIP deserves a 1.4 LI, a step above setup man, but not an elite closer. To me, that seems a lot more fair when we attempt to quantify their performance, at least in a forward-looking analysis.
That's where I'm at. How does it look? Any other approaches you'd use?
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Sky -- How does the number of pitchers in each bracket compare
Basically, do you have too many high deLI pitchers compared to the actually pitchers in the league?
by Jeff Zimmerman (TucsonRoyal) on Apr 20, 2009 2:05 PM EDT reply actions 0 recs
Good question. Definitely something to check. Shouldn't be too hard...
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 2:13 PM EDT up reply actions 0 recs
Are you sure that 1.8 is a reasonable cap?
I was taking a look at this last year and if you limit it to all closers with 20+ saves, the pLI of the whole group was 1.93.
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by Jack Moore on Apr 20, 2009 2:32 PM EDT reply actions 0 recs
No, I'm not sure.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 2:48 PM EDT up reply actions 0 recs
I collected relevant reliever data for 2009 from Fangraphs.
I then divided them up into 7 bins (assuming 72 IP per each of the seven spots in the bullpen) in two ways: sorting by pLI and sorting by FIP. Here’s how the bins work out, with deLI in the third column:
FIP pLI deLI
2.65 1.90 1.93
3.40 1.42 1.33
3.76 1.20 1.14
4.07 1.02 1.01
4.42 0.84 0.89
4.95 0.66 0.75
6.32 0.37 0.52
Really not too shabby, actually.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 3:27 PM EDT up reply actions 0 recs
That should read "for 2008"
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 3:27 PM EDT up reply actions 0 recs
Here's the spreadsheet, for those interested in playing around with it
You’ll need to re sort by FIP and/or pLI and re run the pivot table in order to make it useful for that stat. To use smaller bins, change what the “2160 group” function is divided by in column F on the Useful tab.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 3:34 PM EDT up reply actions 0 recs
Here's using 14 bins
FIP pLI deLI
2.20 2.07 2.54
3.07 1.72 1.54
3.31 1.49 1.38
3.49 1.34 1.27
3.69 1.26 1.17
3.84 1.15 1.10
3.98 1.07 1.05
4.15 0.98 0.98
4.33 0.88 0.92
4.52 0.81 0.86
4.78 0.71 0.79
5.13 0.60 0.72
5.50 0.48 0.64
7.18 0.24 0.43
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 3:37 PM EDT up reply actions 0 recs
It's mostly the high end that has issues.
In the 3.00 to 3.40 range, deLI is a bit low. Below 3.00, deLI is high.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 3:40 PM EDT up reply actions 0 recs
Jason Isringhausen had an average LI of 2.32 in 2006
the same year he had a 5.70 FIP.
by erik on Apr 20, 2009 3:29 PM EDT up reply actions 0 recs
Oops.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 3:34 PM EDT up reply actions 0 recs
Wow
Tony LaRussa sure is a genius, isn’t he?
by Brendan Scolari on Apr 23, 2009 2:41 AM EDT up reply actions 0 recs
I like this
Either don’t use leverage at all or use deserved leverage. Reliever A shouldn’t be more valuable than Reliever B just because Pitcher A’s manager pitches him in the 9th and Pitch B’s manager pitches him in the 7th.
by VictorW on Apr 20, 2009 2:37 PM EDT reply actions 0 recs
Well, if you're doing a retro-value thing, you still might want to use actual LI. But yes, I like this better, too.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 2:46 PM EDT up reply actions 0 recs
I'm too dumb for this thread
SuperStar Fred McGriff gives this post his "full endorsahment"!
by Molly Flogger on Apr 20, 2009 3:59 PM EDT reply actions 0 recs
No you're not. You just might not be familiar with the lingo.
Leverage index is a way to quantify the importance of a game situation. The more you’re sitting on the edge of your seat, the higher the LI.
Some closers are worth what they’re paid because they pitch in really important situations (close and late) and are really good. But some closers aren’t actually very good, yet still rack up saves and score well in analyses that take into account their LIs, like Joe Borowski. He was a shmuck with the Indians. Deserved LI is an attempt to figure out how a reliever SHOULD have been used.
Guys with a 3.00 FIP (think deserved ERA) should be closers, with a 1.8ish LI. Guys with a 3.50 IP are setup men, with a 1.3ish FIP. Guys with a 4.10 FIP are league average relievers and should be middle of the bullpen guys. Anyone worse will be a long reliever with an LI under 1.0
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 4:18 PM EDT up reply actions 0 recs
my thoughts
I remember disagreeing with you about doing this when we were talking about it in the past, but that was mostly because I was focused at the time on trying to determine historical value and you were, as you said above, focused on forward thinking analysis.
Where I like pLI the best is in the context of projections. Dusty Baker rarely ever used Jeremy Affeldt in high leverage situations last year (his pLI was something around 0.5-0.6). But you wouldn’t want to just use a regressed weighted average of his past pLI in a projection of future performance, because a manager that is less prone to trying to fit players into preconceived stereotypes would use Affeldt more effectively than he was used last season. You could ignore pLI altogether (as most do), but then you massively undervalue ace relievers. So, deserved LI makes a lot of sense in that context.
I think my preference, though, would be to base it as much as possible on true (average) usage tendencies. So, I’d prefer to go with a function fit to match your 14-bin distribution (or a more careful study) rather than a variant of the model that you credited to Tango above. It is true that sometimes pitchers will get higher pLI’s than this would project, but that’s also part of the point—higher pLI’s aren’t something we should necessarily expect (or project), even for relievers with kick-ass FIP’s like you posted above, because they don’t always get those kinds of opportunities.
-j
by JinAZ on Apr 20, 2009 4:54 PM EDT reply actions 0 recs
Well put.
I think deLI’s ideal use is for grading players in some sort of ranking system (HoF, for example) or figuring out what they should be paid. Of course Tango would argue that chaining needs to be used, which I sort of see, but also somewhat disagree with. See here, comments 6, 9,17,18, and 41.
I’m up for a more in-depth study, I’m just not sure of needed details. More than just more bins?
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 5:04 PM EDT up reply actions 0 recs
Here's a chart of using 30 bins (plus a little left over)
And the table:
FIP pLI deLI
1.94 2.21 3.08
2.41 1.94 2.22
2.87 1.80 1.71
3.17 1.70 1.47
3.26 1.59 1.41
3.32 1.48 1.37
3.39 1.39 1.33
3.49 1.35 1.27
3.61 1.31 1.21
3.67 1.27 1.18
3.75 1.22 1.14
3.81 1.17 1.11
3.88 1.12 1.08
3.95 1.08 1.06
4.00 1.05 1.04
4.10 1.02 1.00
4.18 0.97 0.97
4.23 0.91 0.95
4.35 0.87 0.92
4.46 0.83 0.88
4.51 0.81 0.87
4.60 0.76 0.84
4.74 0.72 0.80
4.93 0.68 0.76
5.08 0.62 0.72
5.21 0.56 0.70
5.40 0.51 0.66
5.60 0.45 0.63
6.10 0.36 0.55
7.56 0.18 0.40
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 5:32 PM EDT up reply actions 0 recs
Not sure I buy the results on the high end.
Mashing the best 500 IP LI bin with the best 500 FIP bin might be implying you could actually give the best FIP pitchers the best LI roles, when in fact they are just constructs of flukes/luck. For example, the Angels gave their closer a huge LI, which would have existed had KRod been there or someone else not as good. You probably want to look at all thirty closer roles as a whole, which would be 30*72 IP bins, or seven total bins, which is what my first chart in the comments showed. 2.65 FIP = 1.90 deLI, and not go any higher on the deLI side of things.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 5:38 PM EDT up reply actions 0 recs
How does deLI look as you increase or decrease the 1.5 constant
You could base the constant value on where the deLI and pLI values have the highest r-squared value.
by Jeff Zimmerman (TucsonRoyal) on Apr 20, 2009 5:42 PM EDT up reply actions 0 recs
I'm working on that now. Well, using Excel's solver to get the sum of the absolute errors to be minimized.
I can get the shapes to match quite well, with almost all bins (using 30 bins) within .05 points of LI.
But the top two bins are WAY off, and the third best bin is .2 points off.
I think I’m ok with that, if we cap deLI at a certain number (in the 1.9 to 2.0 range, it appears).
The exponent is 2.07 right now.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 5:55 PM EDT up reply actions 0 recs
Here. Exponent is 2.07
FIP pLI deLI
1.94 2.21 4.71
2.41 1.94 3.01
2.87 1.80 2.09
3.17 1.70 1.70
3.26 1.59 1.61
3.32 1.48 1.55
3.39 1.39 1.48
3.49 1.35 1.40
3.61 1.31 1.30
3.67 1.27 1.26
3.75 1.22 1.20
3.81 1.17 1.16
3.88 1.12 1.12
3.95 1.08 1.08
4.00 1.05 1.05
4.10 1.02 1.00
4.18 0.97 0.96
4.23 0.91 0.94
4.35 0.87 0.89
4.46 0.83 0.84
4.51 0.81 0.82
4.60 0.76 0.79
4.74 0.72 0.74
4.93 0.68 0.68
5.08 0.62 0.64
5.21 0.56 0.61
5.40 0.51 0.57
5.60 0.45 0.52
6.10 0.36 0.44
7.56 0.18 0.28
Going to need more than just a cap, as the best three bins need to increase in deLI from 1.70 much more slowly.
The actual data also takes a bit of a dip in bins 5 through 7. Justin might have a point about not using a continuous model, and instead program a bunch of different points and interpolate linearly in between.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 5:59 PM EDT up reply actions 0 recs
Running some quick numbers
I get as the best equation:
deLI = -0.4116(FIP)+2.7873
which has an r-squared of .9117
by Jeff Zimmerman (TucsonRoyal) on Apr 20, 2009 6:09 PM EDT up reply actions 0 recs
One that is better on the top end, but breaks down on the highest FIP
deLI = -1.7540log(FIP)+3.5148
by Jeff Zimmerman (TucsonRoyal) on Apr 20, 2009 6:20 PM EDT up reply actions 0 recs
I doubt we want something linear here.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 6:29 PM EDT up reply actions 0 recs
This might need differential equations
This sort of reminds me of the logistic equation sort of thing that we’re trying to do here. Instead of trying to explain what that does I’ll just post the wiki
http://en.wikipedia.org/wiki/Logistic_function
---
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http://www.rightfieldbleachers.com
by Jack Moore on Apr 20, 2009 6:10 PM EDT up reply actions 0 recs
Actually, the shape does appear to match the logistic function.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 6:30 PM EDT up reply actions 0 recs
Any tips on finding a logistic model given some data?
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 6:33 PM EDT up reply actions 0 recs
I think Excel may be able to do it.
I see editgrid has a function called LOGREG, and I bet excel has it or something similar.
---
Juuuust a bit outside!!
http://www.rightfieldbleachers.com
by Jack Moore on Apr 20, 2009 6:39 PM EDT up reply actions 0 recs
LOGEST
In regression analysis, calculates an exponential curve that fits your data and returns an array of values that describes the curve. Because this function returns an array of values, it must be entered as an array formula.
The equation for the curve is:
y = b*m^x or
y = (b*(m1^x1)(m2^x2)_) (if there are multiple x-values)
where the dependent y-value is a function of the independent x-values. The m-values are bases corresponding to each exponent x-value, and b is a constant value. Note that y, x, and m can be vectors. The array that LOGEST returns is {mn,mn-1,…,m1,b}.
Syntax
LOGEST
Known_y’s is the set of y-values you already know in the relationship y = b*m^x.
If the array known_y’s is in a single column, then each column of known_x’s is interpreted as a separate variable.
If the array known_y’s is in a single row, then each row of known_x’s is interpreted as a separate variable.
Known_x’s is an optional set of x-values that you may already know in the relationship y = b*m^x.
The array known_x’s can include one or more sets of variables. If only one variable is used, known_y’s and known_x’s can be ranges of any shape, as long as they have equal dimensions. If more than one variable is used, known_y’s must be a range of cells with a height of one row or a width of one column (which is also known as a vector).
If known_x’s is omitted, it is assumed to be the array {1,2,3,…} that is the same size as known_y’s.
Const is a logical value specifying whether to force the constant b to equal 1.
If const is TRUE or omitted, b is calculated normally.
If const is FALSE, b is set equal to 1, and the m-values are fitted to y = m^x.
Stats is a logical value specifying whether to return additional regression statistics.
If stats is TRUE, LOGEST returns the additional regression statistics, so the returned array is {mn,mn-1,…,m1,b;sen,sen-1,…,se1,seb;r 2,sey; F,df;ssreg,ssresid}.
If stats is FALSE or omitted, LOGEST returns only the m-coefficients and the constant b.
For more information about additional regression statistics, see LINEST.
Remarks
The more a plot of your data resembles an exponential curve, the better the calculated line will fit your data. Like LINEST, LOGEST returns an array of values that describes a relationship among the values, but LINEST fits a straight line to your data; LOGEST fits an exponential curve. For more information, see LINEST.
When you have only one independent x-variable, you can obtain y-intercept (b) values directly by using the following formula:
Y-intercept (b):
INDEX,2)
You can use the y = b*m^x equation to predict future values of y, but Microsoft Excel provides the GROWTH function to do this for you. For more information, see GROWTH.
Formulas that return arrays must be entered as array formulas.
When entering an array constant such as known_x’s as an argument, use commas to separate values in the same row and semicolons to separate rows. Separator characters may be different depending on your locale setting in Regional and Language Options in Control Panel.
You should note that the y-values predicted by the regression equation may not be valid if they are outside the range of y-values you used to determine the equation.
by laxtonto on Apr 20, 2009 10:23 PM EDT up reply actions 0 recs
Here is a even better equation
deLI = 7.0029-0.6195(FIP)
r-squared of .9870
by Jeff Zimmerman (TucsonRoyal) on Apr 20, 2009 6:18 PM EDT up reply actions 0 recs
wrong equation
deLI = 7.0029-0.6195^(FIP)
by Jeff Zimmerman (TucsonRoyal) on Apr 20, 2009 6:33 PM EDT up reply actions 0 recs
A negative to this...
…is that we’re essentially giving very bad performances a free pass. I’m not sure if that’s desirable or not. I worry if the consequence of this is that we’ll be minimizing the difference between absolutely horrible relievers and merely below average relievers.
Then again, maybe that doesn’t matter—replacement level for a reliever is, what, a 4.90-5.00 ERA? The ones who will have their value severely discounted by this system are sub-replacement anyway.
-j
by JinAZ on Apr 20, 2009 9:31 PM EDT up reply actions 0 recs
Riders On The Storm
Couple of thoughts/questions, some perhaps off the wall.
1) Are you using park and league adjusted FIP?
2) Do some teams have significantly more higher leverage late inning situations than other teams?
3) A LOOGY with a low FIP due to platoon splits, may be better off not promoted to a higher leverage role if the platoon split will be less advantageous.
4) Overall, I like this project. Keep up the good work. Fun reading.
by Xeifrank on Apr 20, 2009 4:55 PM EDT reply actions 0 recs
Thanks.
1) No, I don’t believe FG’s data are park adjusted.
2) Yes, although I’m not sure this is relevant to most uses of deLI, though. Teams who play closer games would be more inclined to have a stud reliever or two, though.
3) I could actually see LOOGY’s having higher FIPs than their usage LIs would suggest. They face lefties in high-leverage situations, but are allowed to get beat up in low leverage situations against righties. But I’m not sure. And I agree tha LOOGY’s and ROOGY’s should be treated differently.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 20, 2009 5:06 PM EDT up reply actions 0 recs
I like it.
Though I won’t lie, the math is beyond me. But once you have it settled in and explain to me why things are the way they are, I’ll be all over it. :-)
by Marc Normandin on Apr 21, 2009 11:32 AM EDT reply actions 0 recs
Shouldn't you use FIP/LI or something like that?
there should be an adjustment for how they’re already used in order to talk about where they should be used.
it’s entirely possible i already missed part of that discussion, though…
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by colintj on Apr 21, 2009 4:09 PM EDT reply actions 0 recs
Are you saying that pitchers in high LI situations currently are facing more difficult hitters?
If not, can you clarify?
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 21, 2009 5:26 PM EDT up reply actions 0 recs
more difficult circumstances, in any case, which should probably be accounted for, no?
conversely if we’re assuming leverage is merely a multiplier of effect rather than cause, then you can go about your business.
but shouldn’t leverage have SOMETHING to do with quality of batters faced? a manager knows which the best and worst hitters are and he can choose relievers based on that information. he will use a LOOGY specifically on David Ortiz, for example. so it seems to me that they would in fact be facing a somewhat different batter set.
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by colintj on Apr 21, 2009 5:41 PM EDT up reply actions 0 recs
I wonder what the connection between LI and batter difficulty is.
I’m sure there’s something there, as there will be more PHs and fewer scrubs in high LI innings, but well, I’ve never really seen anything on that. Using something like BPro’s quality of batters faced too would be useful for both starters AND relievers.
LOOGYs and ROOGYs are a strange beast, though. They deserve a separate analysis.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 21, 2009 5:48 PM EDT up reply actions 0 recs
the whole thing is somewhat convoluted
but my guess is that high end relievers are even more valuable. anecdotally, the reliever market this past winter was pretty frickin’ decent despite the general decline in salaries. i know omar minaya isn’t particularly well regarded, but mark shapiro certainly is and i think in general we regard ourselves as intellectually superior to a lot of clubs when i doubt that’s the case. so if we do in fact trust a limited market to work out proper evaluations of talent, perhaps we ought to start making deductions about market trends.
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by colintj on Apr 21, 2009 5:54 PM EDT up reply actions 0 recs
I agree with this.
my guess is that high end relievers are even more valuable
One large problem with BPro’s database is that they don’t divide things up by batter handedness. Facing a big hitter like ARod as a lefty is a lot tougher than facing him as a righty.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 21, 2009 6:00 PM EDT up reply actions 0 recs
PECOTA does
vs. LHP: .293/.389/.527 – 916 – .394 wOBAestimated
vs. RHP: .278/.364/.490 – 854 – .372 wOBAestimated
that’s an 11.5 run difference over 600 PA
estimated by: I ran a quick regression from fangraphs leaderboard in ’08 w/ OPS and wOBA. They correlate extremely well (r = .95). The trend line gave me wOBA = .351*OPS + .072.
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by colintj on Apr 21, 2009 6:10 PM EDT up reply actions 0 recs
I meant BPro's report for quality of batters faced doesn't treat "ARod against LH" different from "ARod against RH"
http://baseballprospectus.com/statistics/sortable/index.php?cid=204018
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 21, 2009 6:19 PM EDT up reply actions 0 recs
and then there's Trey Hillman
but let’s assume a rational actor model.
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by colintj on Apr 21, 2009 5:42 PM EDT up reply actions 1 recs
I blame the GM for giving Hillman his Farnsworth toy.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 21, 2009 5:47 PM EDT up reply actions 0 recs
it's got to be more like a mancrush or something like that
or cro magnon.
FASTBALL GOOOOD!
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by colintj on Apr 21, 2009 5:55 PM EDT up reply actions 0 recs
bah.
third time’s the charm:
basically, the leverage index has no knowledge of who’s at the plate, right? it assumes an average hitter, which may or may not be the case specifically, but may also not be the case in aggregate.
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by colintj on Apr 21, 2009 5:45 PM EDT up reply actions 0 recs
Yes, true.
You’d have to tackle that directly to “correct” something like WPA or anything based on actual LI.
But for something like deserved LI, I’d rather attack the problem separately from leverage, looking at quality of opponent overall and not PA by PA.
Beyond the Boxscore // Calling BJ Upton lazy is lazy.
by Sky Kalkman on Apr 21, 2009 6:02 PM EDT up reply actions 0 recs
i think what you did mostly does the job
and yields good results. i guess i’ve discovered i have a few issues with leverage, if anything.
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by colintj on Apr 21, 2009 6:11 PM EDT up reply actions 0 recs
just to add to that
we already know more or less what to expect when a starter is converted to any reliever. but how about adding leverage adjustments for that too? we could start talking in a more general replacement level for pitchers if we can predict performance based on leverage index.
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by colintj on Apr 21, 2009 4:12 PM EDT reply actions 0 recs
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