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Finding Replacement Offensive Winning Percentage

Or: Fun With Math

Or: The Unified Theory Of Baseball

Or: Winter Break Is Boring As Hell

Star-divide

 

With my recent delve into Offensive Winning Percentage, I thought I'd take a look at whether or not the replacement value of .350% was correct. In order to do this, we must reconcile the OWP baseline with the winning percentage baselines set by Tango for SPs and RPs. Allow me to take you through my mathematical process. First, we have the major assumption: A replacement team will win 30% of its games - that is, win% = .300. Now, some more assumptions. Any numerical data is taken from the 2008 season, and is averaged between NL and AL.

Win% = from Pythagenpat - repR = replacement runs scored, repRA = replacement runs allowed.

lgRA = lgR = 4.61

x = this is our pythagopat exponent

So we have everything we need to make the Pythagenpat calculation. Now, recall the formula for Offensive Winning Percentage

so

OK, now we need to figure out how we'll find runs per out. Well, to make it simpler, we only need to find league runs per out because we will use repOWP as our independent variable. So let's find lg(R/outs). In order to do this, we need runs/162 and outs/162. These are both pretty easy, as long as we make the assumption that each game is 27 outs. This isn't necessarily true, due to only having anywhere from 24 to 27 outs as the home team, but I feel that this is offset enough by extra inning games for any error to be significant.

lgR/G = 4.61

lgR/162 G = 4.61*162 = 746.82

Outs/G = 27

lgOuts/162 G = 27*162 = 4374

So then we have lg(R/Outs) = 746.82/4374 = .1707. Now let's plug this into our repOWP equation.

Great. Now we're down to just R/outs as our variable on the right side. Let's eliminate outs by multiplying the top and bottom of the fraction by outs^1.89. Remember, outs = 4374 from above.

 

 

Yes, that's 269317.23 in the denominator. A little more rearranging...

 

There. repR has been isolated. So let's see what we get when we use the .350 that Bill James gives as replacement level as our repOWP.

 

Alright! We're done with step 1! Now, we can finally use our first assumption - win% = .300. Using Pythagenpat we have the following (note that 538.158 R/162 = 3.32 R/G):

 

 

So we now have our runs allowed for a replacement team where OWP = .350. Let's convert this to Defensive Winning Percentage (DWP). (Note that repRA/g = 842.57/162 = 5.20)

 

lgRA = 4.61, repRA = 5.20

The current accepted values for DWP are .380 for starters and .470 for relievers. In order for this to add up to .442, relievers would have to pitch 68% of innings. Clearly, that means that our repOWP is too low.

I created this spreadsheet where you can just change the OWP baseline in the lower right and it'll spit out RS, RA, and DWP numbers, as well as allowing you to change innings pitched by starters and relievers. I assume that a replacement team would average 5 IP per start, resulting in 810 IP from starting pitchers and 648 from relievers (again, this is assuming 9 IP per game, 9*162 = 1458).

Using the above values for starter and reliever DWP and our inning distribution, we can find what our expected DWP is.

Finally, we can figure out what our OWP should be based on this. This is where the spreadsheet comes in. It's a real pain to figure out repRA using the DWP formula, and since DWP and OWP are dependent on each other, I just input OWP values until I got the correct DWP. It turns out that the correct OWP is EDIT: .370%, which gives 562.9 RS and 881.3 RA, or 3.47 RS/g and 5.44 RA/g. Now, applying this to my data from my last post, I get the following (I also fixed the data using the current simple WAR estimator - I had the average wOBA set too low).

Player staticWAR dynamicWAR
Rivera ,Mike 0.39 0.44
Kendall, Jason -0.2

-0.38

Nelson ,Brad 0.17

0.20

Fielder, Prince 4.68

4.62

Weeks, Rickie 2.29 2.66
Hall, Bill 1.12 1.19
Lamb, Mike 0.35 0.39
Hardy, J.J. 2.44 2.68
Braun, Ryan 4.61 4.44
Cameron, Mike 1.69 1.87
Hart, Corey 2.39 2.57
Gwynn, Tony 0.09 0.08
Duffy, Chris 0.09 0.07
Total
20.11 20.82

Basically, what we see here is that the WAR values we get here are much closer, overall, to what we get using the simple WAR estimation.  It should be noted that the replacement wOBA that I get using OWP is .304, which strikes me as being a bit high (but maybe not that high), but that's why I'm looking for some discussion on this.

Edit: So I realized that when I was figuring replacement wOBA I forgot to divide by 1.15.  It turns out that replacement wOBA works out to .306, which I think makes a little bit more sense.  This correction is also reflected in the table (total WAR is lower because I had forgotten the 1.15 in the players projections as well).

Edit 2: Yet another stupid mistake:  Due to a computing error, the correct OWP% should be .370%.  This leads to a replacement wOBA of.304The necessary changes were made to the article.

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wBOA for Replacement

I agree wOBA seems a bit off here… assume lgwOBA=.332 (as it was in 08) we get wRAA=((.309-.332)/1.15)=-.023 value per pa… assuming 600 PA we get wRAA=-13.8 for the year. I have read many places that a crude estimate of Replacement is 20 under average.

You can compare this to the PECOTA MLVR projections and see that an average player with a VORP of 0 has an MLVR=-.137 per game… At a rate of 4.3 or so PA per game, we get 140 games for a total of 600PA (as above), yet we -19.18.

I am going to look at the math more carefully to see if I notice anything.

by RiverMise on Jan 12, 2009 9:22 PM EST reply reply actions actions   0 recs

A few other things to think about.

1 – It’s possible that the values we have for SP/RP replacement defensive winning percentage are wrong. There would of course be adjustments to make in that case, but it would be relatively simple to find.

2 – If OWP is lower than .381 and the SP/RP replacement benchmarks are correct, then perhaps replacement FIELDING is lower than average, to account for the difference in total DWP and pitcher DWP. For example, at .350 OWP, the DWP for a replacement team is .442. The difference between a .442 DWP and a .420 DWP is 38.88 runs over the course of a season, or over 600 PAs, it’s approximately 3 runs.

3 – OWP is necessarily based on BOTH wOBA and OBP, because in order to find wRC/out, you need to find plate appearances per out, and then finally to find WAR from OWP, you need to find PA/27 outs. I can go more in depth on this if anybody would like.

---
Juuuust a bit outside!!
http://www.rightfieldbleachers.com

by jhmoore on Jan 13, 2009 2:04 PM EST reply reply actions actions   0 recs

I really wish I could contribute more

Advanced math isn’t my strong suit, I’m on the English/Social Science side of the spectrum. Thanks for helping me understand the method, though— it makes a lot of sense, and it would seem to be extremely accurate.

Scored three times and detonated an indisputable in four visits to the batting box.

by Jordan M on Jan 13, 2009 11:21 PM EST reply reply actions actions   0 recs

This is more being creative with math as opposed to actually knowing advanced mathematical techniques

I didn’t use anything that I learned after Algebra II/Advanced Algebra/whatever your high school wants to call it. The advanced part is being able to make the connections.

---
Juuuust a bit outside!!
http://www.rightfieldbleachers.com

by jhmoore on Jan 14, 2009 1:39 AM EST to parent up reply reply actions actions   0 recs

wow

am I glad my major made me take 9 hours of stats, great work here this such a complex issue thanks for going over the math with us.

'talkin 'bout practice?

by CubFanRaysaddict on Jan 16, 2009 1:37 AM EST reply reply actions actions   0 recs

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