Scoring Self-Indulgence, pt. 7: Field Locations and Ball Trajectories

When the ball is put into play, I track its location using a set of field location codes. What I use is not nearly as intricate as the system developed by Project Scoresheet and used or altered by STATS and subsequent data compilers. The entire exercise is inherently subject to error, and it is next to impossible to make accurate determinations about whether balls are hit within the borders of zones when it close. And one’s perspective can certainly skew judgment even in cases where the location or trajectory appear to be clear. Still, I find it a useful thing to record, recognizing the biases inherent and the particular fallibility of my judgment.

For the infield, I do not break the zones up at all. Everything on the infield is 1, 2, 3, 4, 5, or 6, corresponding to the fielder closest to the play (although often in practice this winds up being the fielder that makes or attempts to make the play, regardless of where the ball was actually hit). I do use combinations for balls that are truly in between--64 would be a ball right up the middle, perhaps a bit to the left of second base, while 46 would be just to the right of second base; 13, 15, 23 and 25 (or the reverse) are often helpful for bunts that are in between the area of responsibility for the two positions. Other than that, the location codes stay simple.

If the ball is hit where you’d expect it to be hit given the outcome of the play as seen on the scoresheet, I make no redundant location note. A “63” does not need a note explaining that the ball was hit in the vicinity of the shortstop unless something out of the ordinary happened. If there is something that needs to be noted, I tack the location on as a subscript.

Suppose for instance that there is a popup on the pitcher’s mound; generally, the pitcher will move out of the way and let a real infielder take it. If it is the third baseman, the scoring would look like this:



In the outfield, I use divide things up more extensively. In terms of depth, there is “S” for shallow, “D” for deep and “W” for warning track/wall. Medium depth is the default if none of the other options is listed, but could be indicated by “M” if one was so inclined.

From left to right, I have 7l (left field line; written with a cursive “l”, which makes it much easier to distinguish on the scoresheet), 7 (left), 78 (left-center), 87 (center-left), 8 (center), 89 (center-right), 98 (right-center), 9 (right), and 9l (right field line, again with the cursive). This diagram is poorly drawn, but should give you the gist of it:



The difference between center-right(left) and right-center is razor thin to begin with, and I’m positive that I’m biased by the identity of the fielder who makes the play. That is, I’m much more likely to record the location of a ball as “89” if the center fielder makes the play; if the right fielder makes the play, it’s more likely to go down as “98”.

Let me offer a few examples of flyouts scored using this system, which hopefully will be sufficient for anyone who actually cares (and I know full well that no one does) to understand. A flyball caught by the centerfielder in medium deep center would simply be “8”. But suppose he made the play in shallow center:



A catch made by the right fielder on the warning track along the right field line would be:



A catch made in deep center-left by the left fielder:



A foul fly caught by the left fielder in deep left:



Hopefully, those examples are sufficient to allow one to figure out the many possible combinations themselves. The other element that needs to be added before completing the scoring of hits is the ball trajectory. The biases here are even worse than those for hit location, but as long as you remember it’s just one observation from a particular vantage point, I don’t see what the harm is in taking a stab at this on your scoresheet.

For hits, I combine the location code with a trajectory code. The basic trajectory categories are: bunt (seen earlier with outs), chop (seen earlier, and only used on infield hits), flyball, groundball, and line drive. An infield hit is always assumed to be a groundball unless otherwise noted; an outfield hit is always assumed to be a flyball unless otherwise noted. Using default trajectory types helps to remove some of the clutter that otherwise would be present on the sheet.

The trajectory for a hit is generally noted above or below the field location of the hit. The symbol for a bunt is a squiggly line below; for a chop, a “v” shape below; for a flyball, an arch above; for a groundball, a straight line below; and for a line drive, a straight line above. This is best shown through some examples. First, examples of infield hits, where a groundball is assumed unless otherwise noted. Thus, the following play is a groundball single to the first baseman:



The next example is single on a ball chopped to third base. I use the chop symbol sparingly; if a ball is a tweener between a chop and a grounder, I go with the latter description:



This is a bunt single right in front of the plate to the catcher:



This is a line drive hit to the pitcher; picture the pitcher knocking the ball down but having no play (it’s better than picturing Herb Score or Willie Blair):



A flyball infield hit (remember, I do not distinguish between a flyball and a popup) can happen when the ball drops between fielders and no error is assigned; this one drops on the pitchers’ mound:



Scoring outfield hits is a little more involved because there is a wider variety of locations and a greater diversity of trajectories. Again, if there is no trajectory included for an outfield hit, it is assumed to be a flyball, so this is a flyball single to medium left, along/near the line:



Here is a flyball single to shallow center-right:



There is no need to use any depth modifiers for groundball hits to the outfield. This one is a classic grounder right up the middle:



A line drive single to left-center:



A flyball double to the warning track or wall down the left field line:



A line drive double to deep center:



A flyball triple to deep right-center:



I usually don’t use any trajectory codes on home runs--they are generally considered to be flyballs, but some real laser beams get a line drive tag. This one is hit down the right field line:



Of course, inside-the-park home runs provide more fodder for location codes. A flyball off the wall in deep center-left that ends up a homer:

Scoring Self-Indulgence, pt. 6: Outs on Base

I previously demonstrated how I score batting outs, but did not touch on outs that occur once there is a runner on base. Again, the examples will assume that the runner reached first by being hit by a pitch. I will begin by looking at plays in which the runner is thrown out in his attempt to advance before the ball is put into play, then look at those that occur after the ball has been put in play (which often are forces for which the runner is not culpable).

Caught stealing is given the obvious code “CS”. The most common is the catcher-to-shortstop putout, although there are obviously many possible combinations. This runner was nailed on the last pitch of a PA (indicating that the PA must have ended with a strikeout; otherwise the runner would have been free to advance on a free pass event or the ball would have been in play) taken by the #3 hitter:



The CS symbol can pop up when an out is not actually recorded; for example, "2CSRE6" is a play in which the catcher gets an assist and the runner is charged with a caught stealing, but is safe due to a receiving error by the shortstop.

I divide pickoffs into three different classes: those that are pure pickoffs (PO), those that are caught stealing/pickoffs on which the play is made at the subsequent base (CP), and those that are caught stealing/pickoffs on which the play is made at the original base (PC). The first is the most straightforward--the runner is picked off when not attempting to advance, in this case on the fourth pitch to the #7 hitter, with the play going 13:



Caught stealing/pickoffs are common when the runner starts for the next base before the pitcher starts his motion, then he steps off and throws. This example is a 136 putout made on the first pitch to the #9 hitter:



I score it as a pickoff/caught stealing when the runner is charged with a caught stealing because he was leaning, but the play is made at the base from which he started. This should be pretty easy to visualize--the runner gets just a touch too far away from the bag, the pitcher whirls to throw, he dives back...and gets tagged out. On the 7th pitch to the #5 hitter:



One extra note I sometimes include, which can be applicable to a number of different outs on base, is the use of the symbols OVS and OVR. OVS is for overslide, and OVR for overrun. This just gives the reader of the scoresheet a better idea of what actually happened on the play. If a runner is caught stealing, but it’s because he popped off the base after initially being sage, you can include the OVS symbol:



The most common ways of getting retired once the ball is put into play are on fielder’s choices and double plays. I do not score any play that technically is considered a fielder’s choice as such; I only use “FC” for a forceout. I’ll show an example of those cases in a minute; first, a standard fielder’s choice. This one goes pitcher to shortstop and forces the runner at second. There is no need to record which batter was responsible, because it will be evident from an examination of the scoresheet:



If the runner is retired as part of a double play when forced (or when doubled off his base on a flyout), I use “DP”. In the latter case, it is necessary to note which batter was the catalyst for the play in question, as I’ll demonstrate in a moment. In the first example, the runner is forced at second third baseman to second baseman as part of a double play:



Of course, there is always the possibility of a triple play, but those are so rare I won’t bother with an example, and the scoring is conceptually similar to that of a double play. I use the obvious symbol “TP” in those cases.

This example is a double play in which the runner is double off first, right fielder to first baseman, after a flyout hit by the #8 hitter:



When a runner chooses to advance of his own volition, I record it as a fielder’s choice provided that the batter does not receive credit for a hit on the play, or it is not an error or flyout. Suppose that a runner at second is thrown out at third by the shortstop when he attempts to advance, unforced, on a groundball. Since the batter will not be credited with a hit, I record the runner’s out as “FC65”. In the corresponding batter’s scorebox, the play will be scored as “FC6”:



If the runner is thrown out attempting to take an extra base on a hit (or on an error), then I scored it as an out advancing. It is not necessary to note which batter’s PA the out occurred on as long as there is some other evidence of it in the runner’s scorebox. In this example, the runner advanced on some play initiated by the #1 hitter, but then tried to an extra base and was throw out at third, center fielder to third baseman. Since the advancement to second is already noted as having occurred during the PA of the leadoff hitter, it is assumed that the out on third also occurred on that play.



If the advancement occurred on a play in which the runner did not take another base noted on the scoresheet, then there would have to be some note of it. In this case, the runner at third was thrown out at the plate by the left fielder while trying to advance on a flyball hit by the ninth hitter:



When a batter is thrown out attempting to take an extra base after reaching safely on a hit or an error, I score it as “OS” (Out Stretching). In this example, the batter singled to right, then was gunned out right fielder to shortstop attempting to take second:



One unusual way in which a runner can be wiped out is when he is hit by a batted ball. For the batter, this automatically becomes a single. Let’s suppose that a runner on first is hit by a ground ball hit by the batter in the vicinity of the second baseman. First, the scoring for the batter: he gets credit for a groundball infield single to second base, with the fact that his ball hit a runner noted with the use of the code “HBB” (Hit by Ball):



For the runner, the out is noted as HBBx, where x is the position number of the fielder that gets credit for the putout (so, in this case, HBB4). There is no need to note which batter hit the ball because it will be clear from examining the other scoreboxes:



Another modifier along the lines of OVR/OVS is the use of OBL for “Out of Baseline” (not the world’s most wanted man). This code is used when the runner is ruled out for going out of the baseline. There will still be standard scoring for this play, so the notation is made parenthetically (I use brackets). Suppose the play is scored catcher to first base:



The above example was for a batter-runner, but it could also be used for runners already on base.

Other modifier codes that are rarely used but that can pop up are “LE” (left early--I use this if a runner is called out for tagging before the ball is caught, followed by the appropriate credit for the putout and a circled indication of which batter initiated the play) and “MB” (missed base, used when a runner is called out for failing to touch one of the bases, and again followed by the putout credit and indication of which batter was at the plate if necessary).

2012 Predictions

Every year I try to disavow these predictions as a serious exercise--this is just me as a baseball fan, having fun. Every year I stress that picking a team first in a rank order doesn’t mean that I necessarily think they necessarily have even a 30% chance to actually win their division. Every year I try to disassociate these predictions from sabermetrics--sure, my thinking on them is influenced by sabermetrics, as is everything else I think about baseball-- but these are decidedly not “sabermetric” predictions, not even in the sense that the PECOTA or CAIRO or Davenport projected standings might be. But despite my efforts, every year some random yahoo on the internet links this post and demonstrates no understanding of any of these points.

So this year I will dispense with all of it, recognizing the lost cause for what it is, and get down to business--no, strike that, fun:

AL EAST

1. Boston
2. New York (wildcard)
3. Tampa Bay
4. Toronto
5. Baltimore

I’m sure the Yankees will be the consensus pick, but I’ll be a contrarian and stick with the Red Sox. Yes, their starting pitching is shaky, but at least the guys at the back of the rotation have some upside. Their offense is as good as anyone’s on paper. The Yankees are certainly a force to be reckoned with; however, I think one could very easily overstate the pitching difference between the two teams. Sabathia and Pineda is a good duo (oops!), but the rest of New York’s rotation is hardly rock solid with Kuroda’s age and Nova’s ordinariness. It’s an edge, but it isn’t overwhelming. Tampa Bay remains a threat, and the second wildcard certainly brightens their 2012 outlook, but their offense does not inspire confidence. If Toronto were in the Central or the NL West, I’d pick them second at worst, but such is the nature of the AL East. If things go right for them (Bautista maintains his level, Lawrie plays at a high level, they get some production out of first and left, rotation potential in Alvarez, Morrow, and/or Drabek steps forward), they could surprise. Baltimore...yeah.

AL CENTRAL

1. Detroit
2. Chicago
3. Cleveland
4. Minnesota
5. Kansas City

I would not pick Detroit to win any other division save the NL West, and even there I’d consider them vulnerable. But this is not any other division, and their foes appear incapable of mounting the mid-to-high 80s win total that could topple the Tigers. A lot of things went right for Detroit last year, and outside of Verlander the rotation isn’t special. Outside of Fielder and Cabrera (which is admittedly a big qualifier given the fact that they are both among the best hitters in the game), the offense doesn’t feature any proven high-end performers, so overall it projects as good not great. Alas, no one in the division appears up for the challenge. Chicago has been written off as rebuilding, but there’s still a pretty good pitching staff there, and you can always dream on Dunn and Beckham coming to life and boosting the offense to contention-level. Cleveland is a team that went from solid rebuilder to adrift without a plan in the span of about ten days, although admittedly some of that sentiment may be over emotionalism as a fan. I’ve written about them more in-depth, and while I’m not downgrading them over their spring training woes, there’s no new information that’s come to light since I wrote that piece that has given me reason for optimism (in fact, the Sizemore injury, Chisenhall’s flailing, various minor pitching injuries, and a bullpen that looks shakier than I’d thought have had the opposite effect). I think it’s more likely the Tribe finishes last than first. I’m picking Minnesota ahead of Kansas City on the hope that Mauer and Morneau return to even 75% of their 2010 production, but the Royals certainly have the brighter looking future. Then again, if there’s going to be a team that comes out of nowhere in MLB this year, this is the division that offers the best opportunity.

AL WEST

1. Texas
2. Los Angeles (wildcard)
3. Seattle
4. Oakland

Texas is a terrific team, of course. They lack any huge stars (sorry, Josh Hamilton isn’t consistent enough for this label, and Ian Kinsler’s 2010 power outage makes me pause) but are solid everywhere except perhaps center field. They have enough minor league depth that they should be able to plug leaks as they emerge about as well as anyone in the league. Los Angeles got rid of Tony Reagins (a definite plus), then finally made the huge splash. And it was huge. They arguably have the league’s best pitching staff, shaky fifth starter notwithstanding, and should make this an interesting race. I’m not really sure why I picked Seattle over Oakland, but it shouldn’t matter--neither team has much of a chance. I’d guess that Oakland has a higher variance of expected wins.

NL EAST

1. Miami
2. Philadelphia (wildcard)
3. Atlanta
4. Washington
5. New York

My predictions make no claims to accuracy, but there are two divisions I have been consistently wrong about--the AL Central and the NL East. Assuming that there’s a cause for those mistakes other than chance, I’ve chalked up the former to the fact that I’m a Cleveland fan and tend to pick them when I think it’s defensible (which does not include 2012). In the NL East, the reason would be a tendency to predict the demise of the dynasty too early. I picked against the Braves consistently near the end of their run, then picked them over the Phillies in recent years.

I am not learning from past mistakes and am picking Miami to win their first division title. The Marlins are really easy for me to hate, with the uniforms and the home run fountain and Jeff Loria and Ozzie Guillen. The top three are all very close and so I am picking what would annoy me the most. I also think they are the most balanced between offense and defense, which doesn’t translate to wins but also means it’s harder to point out the Achilles heel. Do you like that segue? Ryan Howard is the least of the Phillies concerns, as Chase Utley is and always has been a more valuable player, and now a bigger loss to injury. The offense is old and was only average in 2011. The starting pitching is tremendous, but the bullpen nothing special. Atlanta would be easier to like if they had a shortstop or another big bat, but I wouldn’t count out Jason Heyward in the latter role. They should be in the hunt. Washington still looks more like a .500 team than a contender to me, but they’re close enough that good fortune could put them in the playoffs. New York remains a mediocre team more than a bad one, but that won’t stop us from having to read the lamentations of Mets fans. I realize it’s tough to see the crosstown Yankees win consistently and the Dodgers escape ownership purgatory, but toughen up guys.

NL CENTRAL

1. Cincinnati
2. St. Louis (wildcard)
3. Milwaukee
4. Chicago
5. Pittsburgh
6. Houston

The Reds stood pat after making the playoffs in 2010, which not surprisingly resulted in a step backwards. This year, they decided to go for it, trading for Mat Latos and signing Ryan Madson. The latter move has flopped thanks to no fault of the team, but in this case, it really is the thought that counts. They have a capable offense and if they are willing to make hard choices (like sticking with Aroldis Chapman and relegating Bronson Arroyo to long reliever if need be), I think they can do it. In other words, I’m putting my division pick in the hands of Dusty Baker. Gulp. St. Louis lost Pujols, but signing Beltran is about as good of a response as one could expect, and I wasn’t penciling in Carpenter and Wainwright for more than 350 combined innings anyway. Milwaukee is obviously a weaker offense without Prince Fielder, but in this division they remain firmly in the contenders tier. I was (relatively) bullish on Chicago in 2011; that was a mistake but mediocrity is good for the top of the second division in the NL Central. It must be really frustrating to be a Pirates fan; not for the obvious reasons, but for the little things. The team had a hot three and a half months last year which gave their fans a semblance of hope and fun, and they finally have a divisional rival that is much worse off than they are. So of course MLB strongarms that rival to move to the other league. The Astros new front office is easy to like, but would be more so if they hadn’t moved Brett Myers to the bullpen, a move that I don’t understand on any level.

NL WEST

1. San Francisco
2. Arizona
3. Colorado
4. Los Angeles
5. San Diego

Picking the Giants feels wrong, as I object to picking an organization that seems to view scoring runs with contempt. But this division isn’t very strong and the terrific pitching has overcome this punchless offense before. Arizona’s starting pitching has the potential for serious regression from Kennedy, Collmenter, or Cahill and their offense while solid doesn’t seem to offer a lot of upside. Jamie Moyer is a great story and I wish him all the best (who wouldn’t love to see a legitimate 50 year old non-knuckleball pitcher in 2013?), but his presence in the rotation really encapsulates what you need to know about Colorado. The Dodgers exceeded expectations last year and the ownership change should foster optimism for the future, but Ned Colleti’s bizarre offseason does the opposite for the immediate future. San Diego is not a horrible team, and the trade for Carlos Quentin indicated that Josh Byrnes may not be as committed to a rebuild as Jed Hoyer was.

WORLD SERIES

Boston over Miami

AL Rookie of the Year: SP Matt Moore, TB
AL Cy Young: David Price, TB
AL MVP: 1B Albert Pujols, LAA
NL Rookie of the Year: C Devin Mesoraco, CIN
NL Cy Young: Zack Greinke, MIL
NL MVP: 3B Hanley Ramirez, MIA

First manager fired: Jim Tracy, COL...just kidding, he’s manager for life (Dan O’Dowd’s life, at least). So, in a mercy firing, Brad Mills, HOU.
Best pennant race: NL East
Worst pennant race: AL Central
Worst team in each league: BAL, HOU
Most likely to go .500 in each league: CHA, WAS
Team in each league most likely to disappoint mainstream consensus: CLE, ARI
Team in each league most likely to surprise mainstream consensus: BOS, MIL

Great Moments in Yahoo! Standings

Ubaldo and Tulo

In the first inning of today’s Cleveland/Colorado game, Ubaldo Jimenez hit Troy Tulowitzki with his first pitch. The two have had some sort of silly squabble in the press this spring, which you can read about elsewhere.

Jimenez claims that hitting Tulo was an accident. Should we believe him? We can’t know for sure, but this is a fun application of some simple Bayesian estimates. We are interested in estimating the probability that Jimenez was intentionally throwing at Tulo given that he hit him; I’ll call this P(I|HB).

Based on Bayes theorem, we can write:

P(I|HB) = P(HB|I)*P(I)/(P(HB|I)*P(I) + P(HB|NI)*P(NI))

So there are four unknowns we need to estimate:
* P(HB|I) -- the probability of a hit batter given that Jimenez was intentionally throwing at Tulo. I’ll estimate this as 50%; my intuition is that it’s higher, but Ubaldo’s control this spring has been terrible and the lower this is set, the better the end probability will look for him.
* P(I) -- the probability that Jimenez was intentionally throwing at Tulowitzki. Obviously, we can’t know this. Let’s be very generous and assume that it was only 1%.
* P(HB|NI) --the probability that Jimenez would hit Tulo given that he was not intentionally throwing at him. In his ML career, Jimenez has hit 44 batters and thrown 15,218 pitches, which is about .3%. Some of those may have been intentional, and his control is not a constant, but I’ll use .003 as the estimate here.
* P(NI) -- the probability that Jimenez was not throwing at Tulo. This is unknowable, but it is just the complement of P(I), so we’ll start it out at 99%.

Given these assumptions:

P(I|HB) = .5*.01/(.5*.01 + .003*.99) = .627

So given that we observed Jimenez hitting Tulo and the other assumptions, there is a 62.7% chance that he intended to hit him.

The following chart varies P(I) and presents the associated probabilities for three P(HB|I) values--50%, 25%, and 75%. As you can see, P(I) is the dominant factor here; once you establish a reasonable probability of intent, the probability of succeeding in plunking Tulo doesn’t matter much. Of course, once you have a very high estimate of intent, you are pretty confident and the observation that Tulo was actually hit isn’t that important:



One can get carried away with this type of analysis, though. As you can see, any assumption that there may have been intent involved will result in a very high probability that intent was in fact present. I have no doubt that sometimes pitchers with grudges hit batters by accident, and wouldn’t want to presume that such innocent coincidences are beyond the realm of possibility. When it’s a direct hit on the first offering in a spring training game...I’m with Bayes.

EDIT: See this thread on Inside the Book for some additional points. MGL's language in #1 does a much better job of expressing what P(I) represents than I did.

BA/OBA/SLG Calculus, pt. 2

The ease of figuring OPS given OBA and SLG, along with the fact that the metrics are reasonably well-accepted by the general baseball community and the fact that, when taken alone, they do at least approximate fundamental baseball values (especially in the case of OBA), has made equations of the form a*OBA + b*SLG common in sabermetrics. OPS is the most obvious example, with a = b = 1.

The way OPS+ does its weighting is a little different; rather than placing a fixed coefficient on OBA and SLG, it adds relative OBA and SLG, which essentially makes the league SLG/OBA ratio the weighting for OBA. For any given league OBA and SLG:

x = 1/LgOBA
y = 1/LgSLG
OPS+ = x*OBA + y*SLG - 1
dOPS+ = x*dOBA + y*dSLG = x*1 + y*TBw/AB% = x + y*TBw/AB%

For the 2010 majors taken as a whole, OBA was .3421 and SLG was .4028, so x = 2.923 and y = 2.483. The subtraction of one is a constant, and thus has no derivative and has no effect on the intrinsic weight of each event. It still needs to be subtracted at the end of the process, though.

Given the x and y values for the 2010 majors and the 88.59 AB%, OPS+ can be written as:

OPS+ = (2.92W + 5.73S + 8.53D + 11.3T + 14.1HR)/(AB + W) - 1

Just as with the derivatives for standard OPS, the coefficients vary for each possible AB%. The extreme AB%s in the 2010 AL were turned in by AJ Pierzysnki and Jim Thome:



It is helpful to restate these values relative to each other; I will dispense with using walks as the reference point as I did last time and go straight to singles. Also included are the relative coefficients in wOBA, the most comparable alternate metric, and for OPS with a 88.59 AB% (OPS avg):



As you can see, OPS+ values walks more highly than OPS, and also reduces the extent to which extra base hits are valued relative to singles. However, the adjustment is fairly small in magnitude (usually equivalent to a weight of around 1.2 on OBA), and thus doesn't significantly improve the relative weights.

It is only natural, then, that other constructs of the form a*OBA + b*SLG have put additional weight on OBA. A typical value, like that used in GPA, is 1.8; there are various other similar OBA weights in use, generally established through regression against team run scored or attempts to match linear weight values. I'll refer to the general form as mOPS. The differentiation is cleaner than for OPS+:

mOPS = a*OBA + b*SLG
d(mOPS) = a*dOBA + b*dSLG = a(1) + b(TBw/AB%) = a + b(TBw/AB%)

If a = 1.8 and b = 1, then the weight for any event is 1.8 + TBw/AB%. For the 2010 average AB%, this is the resulting equivalent equation:

mOPS = (1.8W + 2.93S + 4.06D + 5.19T + 6.32HR)/(AB + W)

Here are the values for the extremes, and the relative weights:





The relative weights for mOPS match wOBA fairly closely, except for the walk. The difference in weighting caused by extreme AB%s is also reduced greatly, since more weight is given to OBA which uses PA as the denominator. Unfortunately, the match for the hit weights is so close that it would be difficult to keep the form a*OBA + b*SLG and get much improvement...which of course speaks to why sabermetricians have settled on an OBA multiplier around 1.8.

If you are hellbent on matching the wOBA weights, one can get pretty close by introducing a third statistic into the mix--batting average. Including BA allows the intrinsic weight of each hit to vary by a fixed amount without having any effect on the BA rate. To get a rough estimate of the optimal weights, I ran a regression of dBA, dOBA, and dSLG for each event against the coefficient from the pared-down wOBA version discussed in part one:

wOBA = (.62W + .77S + 1.08D + 1.37T + 1.70HR)/(AB + W)

I've already discussed how to find the derivatives for OBA and SLG, but not BA:

dBA = Hw/AB%, where Hw = 1 if a hit, 0 otherwise
= 1/AB% for a hit, 0 otherwise

I ran this regression only on the results at one particular AB% (the 2010 major league average). Obviously it would be preferable to examine the function across the range of observed AB%s, but I'm just doing this as a demonstration--I certainly don't actually to propagate the usage of OPS and OPS-like metrics. The regression to fit a function to x*OBA + y*SLG + z*BA gave this result:

.620(OBA) + .273(SLG) - .160(BA)

I fiddled with this a little bit to 1) force the SLG coefficient to 1 so that it is the form of mOPS and 2) make the OBA and BA coefficients multiples of .05:

bOPS = 2.25(OBA) + SLG - .5(BA)
d(bOPS) = 2.25(dOBA) + dSLG - .5(dBA)

or generally bOPS = x(OBA) + y(SLG) + z(BA)
d(bOPS) = x(dOBA) + y(dSLG) + z(dBA)

For the 2010 average AB%, this differentiation results in:

(2.25W + 2.81S + 3.94D + 5.07T + 6.20HR)/(AB + W)

Here are the intrinsic weights relative to a single, with Pierzynski and Thome-level AB% as well:



Now the weights are pretty good matches for wOBA, although introducing BA, yet another metric with a different denominator, has apparently pushed the values for extreme AB%s further away from each other than was the case with mOPS.

While I find the "linear weight test" a more compelling way to evaluate a metric than a test on aggregate seasonal team batting stats, I'll throw in the correlation with R/(AB - H) for OPS, mOPS, and bOPS in 1955-2008:



This goes to show that just about any reasonable metric will give acceptable results when applied to team-seasons. bOPS performs a little worse than mOPS, which performs a little better than OPS. Whatever added accuracy comes from using mOPS or bOPS will be at the player level, where the distribution of outcomes is more extreme, and will be most evident for players with high walk rates.

It is tempting to look at the bOPS formula, see the negative coefficient for BA, and make a statement about the merits of BA as a metric. While we could all spend the rest of our lives taking potshots at BA (and rightly so), that should not be the takeaway from this exercise. The introduction of BA here allows us to improve the intrinsic weights by changing each hit weight by an equal amount while leaving walks untouched; the alternative of changing the weight on SLG affects all hit types differently (based on how many bases they are worth, of course).

In doing so, the OBA weight has been raised above its optimum level (around 1.8), but this has been offset by removing .5*BA so that walks can get an extra boost. It is therefore much more of a statement about how metrics based only on linear combinations of OBA and SLG are incapable of valuing walks properly without distorting the relationship between intrinsic weights for the other events.

The writing in the last two posts has been dry and choppy even by my usual standards, which is largely because the material is math-y without telling us much about baseball. In writing them, it was not my intention to in anyway encourage the use of OPS-type metrics for anything other than quick-and-dirty uses. Hopefully, looking at the mathematical properties of OPS in a slightly different manner than I have before will add to the body of evidence of why OPS fails when it is used for more involved purposes.

BA/OBA/SLG Calculus, pt. 1

I wrote a series of posts about OPS four years ago entitled "Beating a Dead Horse". Among serious sabermetricians, there is a consensus that OPS is flawed, and the only disagreement is whether the consequence of its flaws should be banishment to the dustbin of history or mere consignment to back of the envelope computations. In writing about the odd construction of OPS again, it may look as if I'm trying to rebuke someone. That is not the case--while it's more math than baseball, I find what follows interesting and perhaps some of you will too.

The easiest way to examine how a non-linear metric values each event is to take the partial derivative of the metric with respect to each event. I have gone through this exercise for OPS in the past, but in doing so have focused on looking at OPS from the perspective of a complete counting stat line. This time, I am looking at it from a rate perspective while holding the ratio of at bats to plate appearances constant.

If you look at the totals and attempt to differentiate, you have to account for the fact that every change in an offensive event also changes the PA total and (unless it is a walk) the AB total. By treating AB/PA (which I'll call AB%) as a constant, we can focus solely on how the change in rates of positive events effect OPS. In doing so, we'll be able to restate OPS for a given AB% as a mathematically equivalent PA-weighted metric.

Let me define s, d, t, hr, and w as the respective rates of singles, doubles, triples, homers, and walks per PA. For the purpose of this post PA = AB + W. One could easily include hit batters and sacrifice flies if they were so inclined.

Given these new definitions, we can write:

OBA = (H + W)/(AB + W) = s + d + t + hr + w
SLG = TB/AB = (s + 2d + 3t + 4hr)/AB%

Then we can differentiate each of these equations with respect to s, d, t, hr, and w. In the case of OBA, the derivative is 1 for each event. For SLG, the derivative is equal to AB% (remember, AB% is being treated as a constant) times the total base weight of the event (1 for a single, 2 for a double, etc.). Let me call the total base weight of an event TBw:

dOBA (s, d, t, hr, w) = 1
dSLG (s, d, t, hr, w) = TBw/AB%

So it follows that dOPS = 1 + TBw/AB%, and using this formula we can write OPS in the form of (coeff*S + coeff*D + coeff*T + coeff*HR + coeff*W)/PA where coeff is each event's respective dOPS.

The practical implication of this is that the sole variable factor in determining the OPS weight of an event is the player's AB%. Or, put another way, his walk rate (since 1 = AB% + w). To put this in practice, let's consider the OPS weights for three AB%s--the player with the highest AB% in the 2010 AL (AJ Pierzynski, 96.93%), the player with the lowest (Jim Thome, 82.14%), and the major league average for 2010 (88.59%).

First, the OPS weights for the major league average:
W coeff = 1 + 0/.8859 = 1
S coeff = 1 + 1/.8859 = 2.129
D coeff = 1 + 2/.8859 = 3.258
T coeff = 1 + 3/.8859 = 4.387
HR coeff = 1 + 4/.8859 = 5.515

As mentioned above, this means that a precise alternate formula for OPS given a fixed AB% of 88.59% is:

OPS = (W + 2.129S + 3.258D + 4.387T + 5.515HR)/(AB + W)

This table gives the coefficients for the AB% level represented by Thome and Pierzynski. It also includes calculations of what each player's OPS would be if the weights from the other AB% level were applicable:



The calculation of OPS using weights for a different AB% is a misleading exercise, since AB% is not actually a constant. I include it to illustrate that each possible AB% results in a different set of intrinsic event weights.

There is nothing inherently wrong with different weights for a different set of inputs. It can be a very useful property of a metric, as is the case with Base Runs, or with a Markov model of run scoring, or other dynamic estimators. However, I believe that it is crucial that if there is dynamism at work in a metric, it must serve some purpose. What could be called a "false dynamism" adds nothing and requires metrics to be employed with additional caveats.

An example of false dynamism was the old Clay Davenport Equivalent Runs methodology. EqR is fully linear if a player attempts no steals, but stolen base attempts alter the intrinsic weights of all events.

The false dynamism of EqR was of a much lesser magnitude than the OPS case, and thus was more of an annoyance than a damning flaw. In the case of OPS, the damning flaw of the metric is a failure to properly weight OBA, but the false dynamism caused by the different denominators is another serious black mark against it.

Given the intrinsic weights of OPS, we can compare OPS' relative weighting of each event to the weighting used by a good linear weights-based metric. This is a tricky subject because one must first determine what purpose OPS will be used for. If you plan to convert OPS into an estimate of runs scored, then you should perform this type of analysis after the conversion has been made, and the comparison should be to standard linear weight values (this is the type of OPS analysis I have published previously). However, if you plan to use straight OPS as a measure of overall batter productivity, it makes more sense to contrast it to a metric that similarly attempts to measure overall batter productivity with a denominator of plate appearances.

Tangotiger's wOBA is such a metric. The relative weights of events look a little different than those procured from standard linear weights, but that is a result of re-casting the value of each event as the standard linear weight value plus the absolute value of an out. As such, it uses a R+/PA perspective on the ultimate batter rate stat rather than the more common R/O approach. The differences between the two schools of thought is a series of posts unto itself, but suffice it to say that this is the necessary approach if you'd like to express overall productivity with a denominator of PA and without making any debits for outs in the numerator. Since that is the construction of the alternate OPS derivation here, wOBA is the proper comparison.

A basic wOBA formula (without the 15% boost to match the OBA scale) is:

(.62W + .77S + 1.08D + 1.37T + 1.70HR)/PA

The chart below recasts those weights as relative to that of a walk (.62) and compares it to the three different 2010 OPS results discussed above (average, Pierzynski high AB% extreme, Thome low AB% extreme):



This chart makes what we've observed already stick out like a sore thumb--OPS severely undervalues walks. Unfortunately, that makes the relationships between the hit types harder to see at a glance, so allow me to recast the chart as each event's weight relative to that of a single:



This chart suggests that while the most egregious weighting centers around walks, OPS still overvalues extra base hits, and thus slugging average. This is of course a common criticism of the metric and one that is addressed by a number of fixes. Also note that while a lower AB% results in even higher weighting for power, it results in lower relative weighting for walks. In this manner, the pro-power and anti-walk biases of OPS offset each other to some extent for players with high secondary averages. The players that really have the potential to lose out are those that draw walks without hitting for power. The players that stand to benefit the most are those that hit for power but do not draw walks (I'm going to refrain from naming names, but there is a certain, much-debated, recent inductee to the Hall of Fame that fits this description nicely).

In re-reading this a year after I originally wrote it, I think I need to repeat myself a little bit to emphasize something. The formulas in this post are not estimates of OPS. They are precise, equivalent formulas to calculate OPS given that AB% is held constant. In the past, the work I’ve done with OPS derivatives has always attempted to capture the dynamic nature of AB%. Every plate appearance (excluding sacrifices) moves AB% one way or the other--down with a walk/hit batter/interference, up with anything else, and thus the equivalent formula for OPS is never constant for any entity. But at any moment in time, the equations in this post are mathematically equivalent formulas for OPS.

Frankly, I’m amazed I never thought of differentiating OPS this way before. It results in a much more easily explained and useful in practice result, and the math is actually easier (although none of the calculus I’ve ever posted on this blog is hard if you know what you’re doing).

So keep in mind that the intrinsic weight OPS places on each event changes every plate appearance. Unlike a dynamic run estimator that changes intrinsic weights continuously in an attempt to model the dynamics of the run scoring process, OPS changes weights just because someone draws a walk or doesn’t.

In the next installment, I will apply this differentiation approach to a couple of the OPS variants (such as OPS+ and GPA) which attempt to address these issues by weighting OBA more heavily.

BA/OBA/SLG Algebra, pt. 2

OPS

OPS = OBA + SLG = (H + W)/(AB + W) + TB/AB = ((H+W)*AB + TB*(AB + W))/(AB*(AB + W))

I believe the technical term for that is gobbledy-gook.

The only argument in OPS favor is that it is easy to calculate, given that you already have OBA and SLG. The argument that it is simple in theory is only true to math simpletons, who don’t bother to ask why two statistics with different denominators should be added and what the algebraic result of doing so looks like.

Suppose that OBA and SLG did not exist in their present forms as commonly used statistics. Would anyone ever propose

((H + W)*AB + TB*(AB + W))/(AB*(AB + W))
as a measure of a hitter’s productivity?

Or, alternatively, assume that when you were first introduced to OPS, you were not told it was OBA + SLG. Rather, it was introduced to you as the mess above. How long would it have taken you to do the algebra that would pull it apart until into OBA + SLG? Wouldn’t you have asked, “Why are times on base weighted by at bats while total bases are weighted by plate appearances? What is the reasoning behind it?”


OTS/Basic Runs Created

On Base Times Slugging (OBA*SLG) has gone by a few different names (BRA and SLOB) the most frequent, but it’s most easily recognizable when it’s multiplied by at bats:

(H + W)/(AB + W)*(TB/AB)*AB = (H + W)*TB/(AB + W)

Which of course is Bill James’ basic Runs Created formula, which means that OTS is equal to Runs Created/At Bat. We can convert it to Runs Created/PA by multiplying by AB% (since RC/AB*AB/PA = RC/PA), which we have already seen is (1 - OBA)/(1 - BA):

RC/AB*AB% = OBA*SLG*(1 - OBA)/(1 - BA) = RC/PA

We can also endeavor to figure RC/Out from BA/OBA/SLG. Since OBA*SLG is RC/AB:

(RC/AB)*(AB/Out) = RC/Out

Since we’re only considering batting outs, AB/Out is AB/(AB - H) = 1/(1 - BA), leading to:

RC/Out = (RC/AB)*(AB/Out) = OBA*SLG*(1/(1 - BA)) = OBA*SLG/(1 - BA)

I have a lengthier post I want to write about this equation, comparing it to Euler’s identity (in elegance only). It captures the three most commonly used rate statistics and relates them elegantly to the fundamental (at least on the team level) measure of offensive productivity (run/out). It would be perfect if not for the fact that it doesn’t hold up theoretically (minor details, you know).

Bases per Plate Appearance

Bases/PA (which I’ll call BPA) and Bases/Out are intuitive enough ideas for an overall offensive statistic that they have been “invented” many times over. Leaving aside baserunning events, BPA = (TB + W)/(AB + W), which can easily be calculated from BA/OBA/SLG:

(TB + W)/(AB + W) = TB/(AB + W) + W/(AB + W)

We already know the formula for W/(AB + W), so:

TB/(AB + W) = (TB/AB)*(AB/(AB + W))

The second term is AB%, so:

BPA = SLG*(1 - OBA)/(1 - BA) + (OBA - BA)/(1 - BA)
= (SLG*(1 - OBA) + OBA - BA)/(1 - BA)

I’m not sure which of those equations looks better, but take your pick.

Bases per Out

The batting-events version of bases/out is (TB - H)/(AB - H). That of course can be split into TB/(AB - H) + W/(AB - H), and we can start with TB/AB and W/AB and convert to per (AB - H) by multiplying by AB/(AB - H). Again, AB/(AB - H) is 1/(1 - BA), so:

TB/(AB - H) = (TB/AB)*((AB - H)/AB) = SLG*1/(1 - BA) = SLG/(1 - BA)
W/(AB - H) = (W/AB)*((AB - H)/AB) = ((OBA - BA)/(1 - OBA))*1/(1 - BA) = (OBA - BA)/((1 - OBA)*(1 - BA))

Bases/Out = SLG/(1 - BA) + (OBA - BA)/((1 - OBA)*(1 - BA)) = (1/(1 - BA))*(SLG + (OBA - BA)/(1 - OBA)) = (SLG + (OBA - BA)/(1 - OBA))/(1 - BA)

Estimated Runs Produced

If you’re still awake, you should be starting to realize that BA, OBA, and SLG can be converted into just about any components that rely only on some combination of AB, H, TB, and W. This means that it’s quite possible to get some pretty decent linear run estimators from the slash stats; you just have to avoid those that break out doubles, triples, and homers rather than treating all extra bases equally. This constraint does reduce the possible accuracy a bit, but still leaves you with some pretty decent options.

A quick skeleton-style linear weight formula I use a lot is essentially Paul Johnson’s Estimated Runs Produced:

ERP = (TB + .8H + W - .3AB)*.324

It is easiest to start by writing this in its per AB form:

ERP/AB = (TB + .8H + W - .3)*.324/AB

Which can be rewritten as:

ERP/AB = (SLG + .8*BA + (OBA - BA)/(1 - OBA) - .3)*.324

Runs/at bat is not a particularly useful form if you intend to just use the result as a rate; per plate appearance or per out are much more meaningful, and of course we can convert:

(ERP/AB)*(AB/PA) = ERP/PA = (SLG + .8*BA + (OBA - BA)/(1 - OBA) - .3)*(1 - OBA)/(1 - BA)

= (SLG*(1 - OBA)/(1 - BA) + .8*BA*(1 - OBA)/(1 - BA) + (OBA - BA)/(1 - BA) - .3*(1 - OBA)/(1 - BA))*.324

= ((OBA - BA + .8*BA*(1 - OBA) + SLG*(1 - OBA) - .3*(1 - OBA))/(1 - BA) * .324

ERP/PA = ((OBA - BA) + (1 - OBA)*(.8*BA + SLG - .3))/(1 - BA)*.324

These formulas are certainly less elegant than their RC counterparts when expressed in terms of the slash stats, but they are much more sound, and arguably just as easy to calculate when using their components rather than the slash stats.

To get ERP/Out, it’s easiest to start with ERP/AB and multiply by (AB - H)/AB. There’s not much simplification to be had:

ERP/Out = (ERP/AB)*((AB - H)/AB) = (SLG + .8*BA + (OBA - BA)/(1 - OBA) - .3)*.324*1/(1 - BA)

= (SLG + .8*BA + (OBA - BA)/(1 - OBA) - .3)/(1 - BA)*.324

If you’d like that in a convenient R/G format, you can multiply by 25.2, which would leave you with:

ERP/G = (SLG + .8*BA + (OBA - BA)/(1 - OBA) - .3)/(1 - BA)*8.16

Please remember that it’s much more straightforward to just figure ERP or other similar linear weight methods from the inputs and not to bother with BA/OBA/SLG at all. I certainly wouldn’t want these more complex equations to encourage the usage of “simple” formulas like OBA + SLG.