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.

BA/OBA/SLG Algebra, pt. 1

This post is mathematical in nature and wholly unoriginal. All it does is walk through the algebraic relationships between BA, OBA, SLG and some other related metrics, with a little bit of my commentary sprinkled in. I believe there is nothing here that I haven’t written about before, let alone others, but I thought it would be nice to have it all in one place for easy reference. There is also a certain quickie method that involves two of the big three slash stats (other than OPS!) that drives me nuts, and occasionally I feel compelled to fire off a missive against it.

I will assume here that BA = H/AB, OBA = (H + W)/(AB + W), and SLG = TB/AB--in other words, ignoring the presence of HB and SF in the OBA formula. If you prefer, just think of HB as being combined with walks in a single category, so that walks in the formulas actually represent W + HB. Since they are essentially the same and none of the metrics here draw any distinction between them, such treatment won’t cause any distortion and will clean up the clutter. Sacrifices are a mess, but don’t have a major effect in any event.

This post would look much better with numerators written over denominators, but it is so much more time effective to just type everything out.

Isolated Power (ISO)

ISO is very easily calculated as SLG - BA:

SLG - BA = TB/AB - H/AB = (TB - H)/AB

Which is equivalent to (S + 2D + 3T + 4HR - (S + D + T + HR))/AB = (D + 2T + 3HR)/AB

At Bat Percentage (AB%)

That is, at bats as a percentage of plate appearances (AB + W). This is not a particularly interesting statistic on its own, but it is the complement of W/(AB + W), and knowing how to calculate it makes it easier to do later manipulations:

AB% = (1 - OBA)/(1 - BA)

(1 - OBA)/(1 - BA) = (1 - (H + W)/(AB + W))/(1 - H/AB)

Anytime you see a one, you can get rid of it by replacing it with denominator/denominator of the fraction you’re interested in:

= ((AB + W)/(AB + W) - (H + W)/(AB + W))/(AB/AB - H/AB)
= ((AB - H)/(AB + W))/((AB - H)/AB)

Of course, dividing by something is the same as multiplying by the reciprocal:

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

OBA - BA

You don’t see OBA - BA used in many formal situations, but it is sometimes used as a quick indicator of plate discipline. It’s easy to see why people might use it that way--it resembles isolated power, and since it is walks that cause OBA to differ from BA, it seems fairly sensible.

However, the reason ISO works so nicely is because both SLG and BA have the same denominator. That is not the case of OBA and BA, which makes the subtraction messy:

OBA - BA = (H + W)/(AB + W) - H/AB

In order to get a common denominator:

(H + W)/(AB + W)*(AB/AB) - (H/AB)*(AB + W)/(AB + W)

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

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

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

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

Since (AB - H)/AB = 1 - BA, we can write the above equation as:

= W/(AB + W)*(1 - BA)

In other words, OBA - BA is walk rate times outs/at bat. This chart shows OBA - BA for three walk rates at four different BA levels. As you can see, the lower the BA, the higher OBA - BA is for a given walk rate.



The distortion increases as walk rate increases. The differences are not enormous except for extreme cases, but they’re still plenty large enough for me to avoid using OBA - BA as a stand-in for walk rate in any circumstance.

Some people may object to my discussion here and counter that OBA - BA is not intended to be a measure of walk rate so much as it is a measure of the amount of on base ability the player brings above and beyond his batting average. My objections to this position are many, but one that deserves mention here is that it is a position that starts with the inferior, less fundamental statistic (BA) and attempts to build up to the statistic that pretty much defines a fundamental baseball measure (assuming runs and wins are off the table). Being forced to use OBA - BA to measure “additional” on base ability just points out the folly of making BA the building block rate in the first place.

Walks to At Bat Ratio (W/AB)

The ratio of walks to at bats is not a particularly meaningful ratio, but it will produce the same rank order list as walks per plate appearance given the assumptions of this post (ignoring SF and treating HB as zero or walks, walks per plate appearance by definition would simply be W/AB divided by itself plus one). It is also a component of secondary average, and it’s instructive to see how it can be pulled out of OBA and BA.

We know from above that OBA - BA = W*(AB - H)/(AB*(AB + W)). We also know (or can easily demonstrate) that 1 - OBA = (AB - H)/(AB + W). Divide by (OBA - BA) by (1 - OBA) and you have:

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

= W/AB

So if you want an acceptable (in that it orders players by the frequency with which they actually draw walks) stand-in for walk rate from OBA and BA, you can just figure (OBA - BA)/(1 - OBA). But there’s an equally easy way to get to the more useful and properly denominated walks per plate appearance.

Walks per Plate Appearance (W/PA)

Knowing the formulas for OBA - BA and 1 - BA, we can quickly see:

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

So please scrap the silly, unitless, distortive OBA - BA and replace it with (OBA - BA)/(1 - BA), which actually is walk rate.

Secondary Average (SEC)

Secondary Average without considering steals is (TB - H + W)/AB. This is easily recognizable as ISO + W/AB, and from the equations here it can be easily converted to a BA/OBA/SLG equivalent:

SEC = SLG - BA + (OBA - BA)/(1 - OBA)

Pessimism on the 2012 Indians

The 2011 Indians were not expected to do much; even within the organization, 2011 was likely viewed as a transitional year, with the goal of sorting out some nagging questions about the major league roster (Will Carlos Santana and Grady Sizemore be able to return from their injuries? Can Justin Masterson be an effective stating pitcher? Will Matt LaPorta ever amount to even a small fraction of his potential? Will Fausto Carmona ever come close to matching what he did in 2007?) and breaking in some younger talent as the season went along (Alex White, Lonnie Chisenhall, Jason Kipnis). Minnesota, Chicago, and Detroit all seemed to be better clubs on paper. Instead, the team started 30-15 and built a relatively sizeable lead in the American League Central, making 2011 a surprise contention season.

Of course, the team was not as good as their gaudy start, and while Chicago and Minnesota never recovered, Detroit came on strong as the summer progressed. Still, the Indians made moves that were consistent with a team that fancied itself a real contender. They promoted Lonnie Chisenhall to try to inject some life into a slumping offense, despite the fact that he wasn’t really tearing up AAA. They appeared to dabble in the trade market, being linked to names like Carlos Beltran and Josh Willingham, but given the organization’s historical reluctance to pull the trigger on a major deadline deal, even in seasons when they were real contenders, this observer discounted the rumors. But on July 30, the Tribe jumped into the trade game, sending three of the organization’s top pitching prospects to Colorado for Ubaldo Jimenez.

It didn’t do any good--Jimenez was terrible for Cleveland, and a three-game sweep at Comerica Park August 19-21 (including an excruciating 8-7 loss in the finale, a game started and seemingly lost by Jimenez before a valiant comeback effort) put the Indians 4.5 back; they’d never get closer and wound up 80-82 and fifteen games out.

The Indians were aggressive at the deadline, but mostly passive in the offseason while Detroit went out and signed Prince Fielder, a move which greatly diminished any residual optimism about Cleveland’s divisional hopes for 2012.

The Indians will once again go with the duo of Carlos Santana and Lou Marson behind the plate. Santana will also see time at first base and DH, although perhaps less than he may have initially seemed in line for as the Indians signed Casey Kotchman. It is quite possible that Santana will play first against left-handed pitchers, as Kotchman’s career platoon OPS split is 754/668. In Marson’s 199 PA against lefties, he has posted a 763 OPS, so it seems likely the Tribe will hope that even with some platoon split regression, Marson will be an asset.

At second base, Jason Kipnis will be handed the starting job after an impressive power display in his first 150 major league PA. Kipnis’ track record would indicate that his production should be tilted less towards power and more towards getting on base than it was in Cleveland, but he has the potential to be an offensive plus at the keystone.

Third base will be an open competition between Lonnie Chisenhall and Jack Hanahan. Hanahan was the opening day starter in 2011 and actually was a league average hitter (albeit fueled by a hot April) in addition to playing excellent defense. However, with the offense faltering in late June, Chisenhall was recalled and given a chance to take over the position. Poor strike zone judgment (8/49 W/K) held back his production, and Chisenhall hadn’t exactly been tearing up AAA prior to his promotion (.267/.353/.431 in 292 PA). Chisenhall will have to perform extremely well in the spring to win the job, and my money is on Hanahan.

Asdrubal Cabrera will return at shortstop after a season that included some great highlights. A surprising power surge early in the year plus several spectacular web gem plays ensured that his season would be remembered in Cleveland. While defensive metrics may dispute his fielding prowess and he’s no offensive star, there aren’t many AL shortstops I’d prefer.

The outfield consists of three question marks. In left field, it remains unclear if Michael Brantely can hit at an acceptable level to be anything more than a fourth outfielder. Playing him in left field is suboptimal, and there was even idle talk of giving him some time at first prior to the Kotchman. In center, Grady Sizemore returns on a one-year deal. Upon Sizemore’s return, there was a brief period in which he looked like the dynamic player he had been from 2006-08. Sadly, Sizemore was again bogged down by injuries and his once outstanding walk rate was nowhere to be found (18 walks in 295 PA and a .285 OBA). It will be interesting to see whether Brantley or Sizemore leads off. My guess is that Sizemore will start in the position but be on a short leash.

In right, Shin-Soo Choo seemed poised for another big year with his exemption from the Korean military secured. Instead, he got off to a slow start, was arrested for DUI, and suffered a broken thumb when hit by a pitch in San Francisco. DH Travis Hafner was productive when in the lineup, but again without the power he once showed (just 13 home runs).

Marson will be the backup catcher, but there are no locks for the rest of the bench. If Chisenhall wins the third base job, Hanahan is a good bet as a corner backup; if not, Chisenhall will go to Columbus. Russ Canzler, Shelley Duncan, and Andy LaRoche are corner backup options, with Jason Donald, Cord Phelps, and Jose Lopez as potential middle infield backups. Donald would seem to have the edge as he at least masquerades as a shortstop, and Hannahan would be the only other player on the roster (besides Cabrera, of course) remotely qualified to play short. Aaron Cunningham, Fred Lewis, Felix Pie, and Ryan Spilborghs are possible outfield reserves. Matt LaPorta will likely be ticketed to AAA with the Kotchman signing.

The Indians will try the interesting pairing of a groundball heavy rotation with infield defense which is considered less than brilliant. Justin Masterson had a terrific season, justifying Cleveland’s faith in his ability to start. Ubaldo Jimenez must pitch better than he did in 2011 to prevent the trade from going down as a disaster, and the Indians will bet that Derek Lowe’s peripherals will triumph over his advancing age and poor 2011. After that, Josh Tomlin is a good bet for the rotation (and a good bet to implode at any given moment, given his low strikeout rate and flyball tendencies. Any spike in his walk rate, which was just 1.1 per nine in 2011, will endanger his ability to pitch in the majors).

With Roberto Hernandez Heredia nee Fausto Carmona unavailable for the start of the season, the fifth starter spot will be up for grabs between Kevin Slowey, Jeanmar Gomez, Zach McAllister, and David Huff. Slowey’s experience may give him the upper hand, but the other three are all well-known by the organization and have started for the team in the past. Slowey has a decent chance to win a bullpen job if he’s shut out of the rotation. Additional depth comes from lefty Scott Barnes, but after missing a good chunk of last season with a knee injury he is likely to start in Columbus.

The Indians seem to have the foundation for a solid but unspectacular bullpen. Chris Perez is a power arm but a shaky closer, while Vinnie Pestano had a very effective rookie season as his set up man. Rafael Perez and Tony Sipp both are used in a more expansive role than simple LOOGY. Joe Smith serves the ROOGY role with his sidearm delivery. The other two bullpen spots are up for grabs, with Frank Herrmann and lefty Nick Hagadone as the 40-man roster options. Non-roster invitees Jeremy Accardo, Chris Ray, Robinson Tejada, Dan Wheeler, and Chen Lee will also be in the mix.

I would like to be optimistic about the Indians 2012 season, but I can’t force it. Detroit is the class of the division, and it would take a lot going wrong for the Tigers and right for the Indians to wipe out the gap. While many people will pick the Indians to reprise their second place finish, they don’t tower over the White Sox on paper, and both the Twins and the Royals have enough potential areas for improvement that it’s not hard to envision them jumping Cleveland. My median expectation is 77 wins and third place, with a nagging feeling that it will be worse than that.

Personally, a lot of my confidence in the organization was shaken by the Jimenez trade. While there is a case to be made that 2 1/3 years of control of Jimenez at a reasonable salary is worth the package of pitching prospects the Tribe surrendered, it wasn’t the kind of deal that the organization has made, even with better teams in more of a win-now mode. Had the trade landed a pitcher at the top of his game I might have been inclined to shrug it off, but Jimenez was not having a particularly good season for the Rockies and has just one season in his career in which he has been a top-flight starter. Jimenez *looked* horrible with the Indians, unable to command his secondary pitches and getting crushed on straight fastballs in hitters counts.

The Indians have a decent group of young position players in Santana, Kipnis, and Chisenhall (not to mention older players still under club control for multiple seasons like Cabrera and Choo), but frustratingly, the organization has been unable to produce any even league average performers at the corner positions (Choo was a trade acquisition). With production from first base and left field, one could feel pretty good about the offense.

Starting pitching has been a similar stumbling block. With Alex White and Drew Pomeranz shipped out in the Jimenez trade, there is little hope for even a mid-rotation type of homegrown starter on the horizon. The Indians have produced scores of back-end starters but have relied on trades to fill out the front end of the rotation.

Overall, I respect the front office: Mark Shapiro and Chris Antonetti, and of course I have a great deal of respect for the analytical arm which includes Keith Woolner and Sky Andrecheck. I consider Manny Acta to be a pretty competent manager and have been impressed by his thoughts on Twitter. I like them and I want them to be successful, but I don’t think it will happen in 2012.

Year Two

2011 was the first season for Greg Beals as OSU coach, replacing the retired Bob Todd who served in that role from 1988-2010. It is obviously unfair to blame a new coach for everything that happened in his first year, but the facts must still be presented--the Buckeyes went 26-27, their first losing campaign since 1987. A couple of facts balance that--the team managed to go 13-11 in conference play and earn the #4 seed in the Big Ten Tournament, while the 2010 team failed to qualify for the tournament. And Beals played a more adventurous pre-conference schedule than had been the norm in the last decade or so of Todd’s career, including a predictably poor 1-5 trip through California. Still, Boyd Nation’s ISR ranking pegged OSU as #160 nationally, the program’s worst showing for the years (1997 on) for which the ISRs have been published (the previous low was #130 in 1998).

The 2012 team will not look radically different than 2011. Junior Greg Solomon will serve as catcher and figures to get the overwhelming majority of playing time once again. Solomon hit will early in the season, but his horrific strike zone judgment (42/4 K/W) caught up with him and by the end of the season he contributed little to Ohio’s offense. Nonetheless, Beals rode his hot hand and had Solomon hitting in the middle of the lineup as his slump began. Apparently, Solomon may also see significant time at third base, where his bat will be a major liability, with true freshman Aaron Gretz taking over some of the catching chores.

At first base, sophomore Josh Dezse will start. Dezse looks like a power hitter, but his main offensive contribution as a freshman came from a .332 BA (his SEC was only a respectable .280). The hope here is that Dezse will build on his Big Ten Freshman of the Year campaign and emerge as the true all-around batting star that OSU has lacked since Ronnie Bourquin. At second base, junior Ryan Cypret is a captain and a solid defender. His offensive game is BA-centric, but nonetheless figures to hit in the middle of the lineup.

It is not clear who will play third as incumbent Matt Streng graduated. Junior Brad Hallberg showed no power (.042 ISO) and didn’t hit for average (.254), but led the team with 28 walks and showed more promise as a freshman in 2010. However, it appears as if Beals is more comfortable using Hallberg as the DH rather than at third base; in fact, one of Beals first major personnel decisions was to flip Streng and Hallberg from one corner to the other. With Hallberg as DH, true freshman Ryan Leffel or Solomon could man third.

Four-year starter Tyler Engle is gone, leaving a gaping hole at shortstop which the Bucks plan to fill with junior college transfer (junior eligible) Kirby Pellant. Pellant played his freshman season at Marshall, then transferred to Chandler-Gilbert Community College, and is expected to bat second in the lineup.

In the outfield, only one starter returns in the same position--sophomore Tim Wetzel will leadoff and play center field. Wetzel was another Buckeye who flashed no power (three doubles and two triples in 176 at bats), but a little improvement in his OBA would allow him to contribute in his role. Sophomore Mike Carroll, who sat out 2011 after transferring from Duke, will take over left field and will be expected to provide a jolt to the offense. That will allow the Buckeyes to shift senior David Corna from left to right. Corna was only an average offensive performer last year, with his team-leading 16 doubles the most impressive part of his game.

Key reserves include Gretz (if he doesn’t break into the lineup as the regular catcher), Ryan Leffel at third, senior Brad Hutton as a 1B/DH/PH, redshirt freshman Phil Jaskot on the middle infield, and junior Joe Ciamocco and sophomore Blake Hutton in the outfield (corners for the latter). Freshman who don’t look to be in the mix include catcher Ryan Wonders and outfielders Austin Achter and Pat Porter (who incidentally hails from my hometown).

On the mound, junior Brett McKinney should assume the role of Friday starter. McKinney was only average last season, but seems to have good stuff and is easily the most experienced starter on the roster. The Saturday assignment will go to Brian King, a junior-eligible transfer from Paradise Valley Community College. Baseball America rates King as the best newcomer to the Big Ten, so the lefty figures to have a big impact. Righty Greg Greve, who struggled mightily at the beginning of his freshman season but started to pitch better as the year went on, is slotted in as the Sunday starter.

A pair of sophomore right-handers, John Kuchno and Jaron Long (a JUCO teammate of Pellant), figure to serve as the mid-week starters and could certainly push Greve for conference assignments. Junior Brad Goldberg, who transferred from Coastal Carolina a year ago, could get starting nods if he is able to get an eligibility situation resolved.

Three key relievers return: junior righty sidearmer David Fathalikani, senior lefty Andrew Armstrong, and Dezse. Dezse will serve as the closer, but hopefully Beals will only utilize him in that role if he is effective. Dezse throws hard, but had several epic meltdowns as a freshman, including one in the Big Ten Tournament that essentially was the death knell for the season. Fathalikani and Armstrong posted much better statistics with much less flash in situational roles that they figure to reprise. Another freshman sidearmer, Trace Dempsey, is expected to get significant relief innings. Other pitchers on the roster include senior Paul Guey (who got some mopup work in 2011), junior Tito Nava (a transfer from Duke), and freshmen Matt Panek, Robert Sakowsky, and Sam Shafer (all except Panek are right-handed). Tyler Giannonatti, another junior eligible transfer from Gilbert-Chandler CC who may have figured in the relief mix, was lost for the season to injury.

My primary interest in watching this season is continuing to learn more about Greg Beals and his philosophy. Beals did not impress me with his tactics in his first season, but while ideally the coach would be someone whose approach to the game I can embrace, the ultimate test is whether he can build a winning program. Even two seasons is far too soon to begin to form a judgment about that, so my focus will be on his game-level decisions rather than his program-building decisions.

The Buckeyes open the season Friday-Sunday with the Big Ten/Big East Challenge in the Tampa Bay area. The following weekend they travel to Atlanta for a three game series at Georgia Tech. The first weekend in March will see OSU in Port Charlotte, Florida for the Snowbird Classic, which includes a non-conference tussle with the forces of evil, who are one of two Big Ten opponents not on the conference schedule (the other is Iowa). Then OSU goes to Myrtle Beach for the Coastal Carolina tournament.

In a departure from past seasons, the Coastal Carolina tournament (which concludes March 11) will be an early conclusion to the typical Southern swing, as the Buckeyes play their home opening series March 16-18 against Austin Peay. The optimistic scheduling of March baseball in Ohio is made possible by the installation of FieldTurf in Bill Davis Stadium, which itself was made possible by a generous donation from OSU’s most notable recent big leaguer, Nick Swisher. The new Nick Swisher Field may not be aesthetically pleasing to purists, but given the harsh conditions of spring baseball in the North, it makes sound business and baseball sense to play on the fake stuff.

Midweek opponents include Ohio University, Akron, Xavier, Cincinnati, Bowling Green, and Youngstown State at home, with trips to Louisville, Dayton, Miami, and Oklahoma State (two games in a return of a 2011 series in Columbus).

Big Ten opponents are, in order: Purdue, @Michigan State, Minnesota, Nebraska, @Illinois, @Penn State, Northwester, and @Indiana. Thanks to the new eleven team Big Ten alignment (plus Nebraska, still without baseball at the joke of an institution in Madison), OSU will be off from Big Ten play on the weekend of May 11-13. That creates an unusual five game stretch of non-conference games (the trip to Oklahoma State and a three game weekend home series against Seattle) in the heart of the Big Ten schedule.

If OSU qualifies for the Big Ten Tournament, it will once again be played in friendly territory, at the Clippers’ Huntington Park from May 23-26.

I always find it difficult to make a prognostication for the season, as my knowledge is limited to just the Buckeyes and does not extend to the conference opponents. I can only go based on the general standard of the Big Ten and how a given year’s OSU team compares to previous ones. On that basis, I think OSU is probably about as good or better as they were in 2011. But when you go 13-11 in a bunched conference and have a below .500 record, and the season is so short, what does that really tell you? I think OSU should qualify for the Big Ten Tournament, but contending for the regular season title is probably not in the cards. The coaches’ preseason poll ranks Ohio 5th which seems reasonable.

Projected lineup:

1. 8 Tim Wetzel (SM)
2. 6 Kirby Pellant (JR)
3. 7 Mike Carroll (SM)
4. 3 Josh Dezse (SM)
5. 4 Ryan Cypret (JR)
6. D Brad Hallberg (JR)
7. 9 David Corna (SR)
8. 2 Aaron Gretz (FM)
9. 5 Greg Solomon (JR)

SP #1: Brett McKinney (JR)
SP #2: Brian King (JR)
SP #3: Greg Greve (SM)
SP #4 (midweek): John Kuchno (SM)
SP #5 (midweek): Jaron Long (SM)

RP: R Trace Dempsey (FM)
RP: L Andrew Armstrong (SR)
RP: R David Fathalikhani (JR)
Closer: R Josh Dezse (SM)

Note: I do not have any inside information--details that are not obvious have either come from my supposition as a long-time follower of the program, the information published on the official athletics website, or the B1G Baseball blog.

Crude Team Ratings, 2011

Anyone can throw together a spreadsheet and declare that they have a ranking system for teams. It’s not particularly hard to construct a reasonable method by which to take an initial estimate of team strength, adjust for strength of schedule, recalculate each team’s ranting, adjust for SOS again, rinse, repeat. I have done just that, and will present the 2011 ratings here.

If you want the full details, please refer to the linked post. The gist of the system is:

1) Start with a win ratio figure for each team. It could be actual win ratio, or an estimated win ratio.

2) Figure the average win ratio of the team’s opponents.

3) Adjust for strength of schedule, resulting in a new set of ratings.

4) Begin the process again. Repeat until the ratings stabilize.

The resulting figure is in the form of an adjusted win ratio; I force the average team to a rating of 100. The ratings can be plugged directly into an odds ratio--a team with a rating of 120 should win about 60% of the time against a team with a rating of 80 (120/(120 + 80)).

I’ll present four different sets of ratings here, each using a different win ratio as the input. It’s overkill to run this many, but if for some reason you prefer a certain estimate of win ratio, it may be represented.

Since 2011 is in the past, there’s no particular value in predictive ratings, so I’ll focus on the CTR based on actual wins and losses:



aW% is the adjusted W% based on CTR; SOS is the weighted average CTR of the team’s opponents; rk is the team’s ranking among the thirty teams; and s rk is the SOS rank.

The results aren’t particularly surprising; the teams are ranked pretty close to how they would be in W%. In some recent years, the results would favor AL teams much more than just looking at pure W%, but the National League held its own with the AL in 2011 as seen from the league/division ratings (simply the average rating for each member team):



That makes for a nice rank order of divisions, with East > West > Central, and AL > NL in each case. Still, the overall AL/NL rating difference of 103/97 is a lot smaller than previous seasons, including 108/93 in 2010. While the NL Central remained the weakest division, 89 was an improvement over the 82 rating in 2010. If Houston was in the AL rather than the NL (and assuming all the ratings stayed constant), the leagues would have each had a CTR of 100.

The next set of CTRs is based on Game Expected W% as described in this post. Basically, gEW% assumes independence between runs scored and runs allowed in a given game, and uses the 2011 empirical W% for teams scoring or allowing X runs in conjunction with each team’s actual game-by-game distribution of runs scored and runs allowed to estimate their W%. The resulting CTRs:



Using classic Pythagenpat as the input:



Finally, using Pythagenpat estimated win ratios based on runs created and runs created allowed:



Obviously there exist any number of possible combinations of win ratio estimates one could use, regression can be mixed in, etc. What I’ve presented here is just the most straightforward ratings based on obvious single inputs.

Run Distribution and W%, 2011

A couple of caveats apply to everything that follows in this post. The first is that there are no park adjustments anywhere. There's obviously a difference between scoring 5 runs at Petco and scoring 5 runs at Coors, but if you're using discrete data there's not much that can be done about it unless you want to use a different distribution for every possible context. Similarly, it's necessary to acknowledge that games do not always consist of nine innings; again, it's tough to do anything about this while maintaining your sanity.

All of the conversions of runs to wins are based only on 2011 data. Ideally, I would use an appropriate distribution for runs per game based on average R/G, but I've taken the lazy way out and used the empirical data for 2011 only.

This post also contains little in the way of "analysis" and a lot of tables. This is probably a good thing for you as the reader, but I felt obliged to warn you anyway. I’ve cut out a lot of what I listed last year simply because I don’t have that much free time right now. The data was not particularly useful in any event—knowing how many runs teams scored and allowed in their wins and losses, or what percentage of their games fell into arbitrarily defined classes might offer some trivia but is not exactly essential material.

The first breakout is record in blowouts versus non-blowouts. I define a blowout as a margin of five or more runs. This is not really a satisfactory definition of a blowout, as many five-run games are quite competitive--"blowout” is just a convenient label to use, and expresses the point succinctly. I use these two categories with wide ranges rather than more narrow groupings like one-run games because the frequency and results of one-run games are highly biased by the home field advantage. Drawing the focus back a little allows us to identify close games and not so close games with a margin built in to allow a greater chance of capturing the true nature of the game in question rather than a disguised situational effect.

In 2011, 75.8% of games were non-blowouts and 24.2% were blowouts. The teams sorted by non-blowout record:



The standard deviation of W% in non-blowouts was .064, which as expected is less than the standard deviation for blowouts (.114) and all games (.070).

Records in blowouts:



Obviously the sample size on these games is pretty small, but Kansas City and Oakland at .500 in blowouts caught my eye.

This chart shows blowout W% less non-blowout W%, along with the percentage of games that were blowouts and non-blowouts for each team:



This is the second year in a row in which San Diego has ranked high in terms of difference between blowout and non-blowout record. Usually teams with large differences are the better teams; that description may have fit the Padres in 2010 but not in 2011. Cleveland was the most extreme team in either direction in the majors. Florida played in the smallest proportion of blowouts while Texas played in the most.

A more interesting way to consider game-level results is to look at how teams perform when scoring or allowing a given number of runs. For the majors as a whole, here are the counts of games in which teams scored X runs:



The “marg” column shows the marginal W% for each additional run scored. The second and third run were both worth about .15 wins on average in 2011, while scoring four runs was the cutoff point between winning and losing (on average, of course).

I use these figures to calculate a measure I call game Offensive W% (or Defensive W% as the case may be), which was suggested by Bill James in an old Abstract. It is a crude way to use each team’s actual runs per game distribution to estimate what their W% should have been by using the overall empirical W% by runs scored for the majors in the particular season.

Using the empirical distribution rather than a theoretical distribution has the upside of being simple (modeling the runs per game distribution is fairly messy), but the benefits are outnumbered by the drawbacks. A non-comprehensive list of said drawbacks:

1. The empirical distribution is subject to sample size fluctuations. In 2011, at least, each additional run increased W%. This is often not the case given the low frequency of high scoring games. Even so, the marginal values don’t necessary make sense--for instance, the marginal value of a tenth run is implied to be .006 wins while the marginal value of an eleventh run is implied to be .040.

2. Using the empirical distribution forces one to use integer values for runs scored per game. Obviously the number of runs a team scores in a game is restricted to integer values, but not allowing theoretical fractional runs makes it very difficult to apply any sort of park adjustment to the team frequency of runs scored.

3. Related to #2 (really it’s root cause, although the park issue is important enough from the standpoint of using the results to evaluate teams that I wanted to single it out), when using the empirical data there is always a tradeoff that must be made between increasing the sample size and losing context. One could use multiple years of data to generate a smoother curve of marginal win probabilities, but in doing so one would lose centering at the season’s actual run scoring rate. On the other hand, one could split the data into AL and NL and more closely match context, but you would lose sample size and introduce quirks into the data.

I will not go into the full details of how gOW%, gDW%, and gEW% (which combines both into one measure of team quality) are calculated here, but full details were disclosed in this post. The “use” column here is the coefficient applied to each game to calculate gOW% while the “invuse” is the coefficient used for gDW%. For comparison, I have looked at OW%, DW%, and EW% (Pythagenpat record) for each team; none of these have been adjusted for park to maintain consistency with the g-family of measures which are not park-adjusted.

For most teams, gOW% and OW% are very similar. Teams whose gOW% is higher than OW% distributed their runs more efficiently (at least to the extent that the methodology captures reality); the reverse is true for teams with gOW% lower than OW%. The teams that had differences of +/- 2 wins between the two metrics were (all of these are the g-type less the regular estimate):

Positive: BAL, PIT, ATL, FLA, HOU, SEA
Negative: BOS, NYA, TEX, COL

You'll note that the positive differences tended to belong to bad offenses; this is a natural result of the nature of the game, and is reflected in the marginal value of each run as discussed above. In the four years that I’ve been looking at these figures, I can’t recall a difference as large as the Red Sox’ deviation in 2011--a standard OW% of .610 and a gOW% of .572, a 6.2 win difference. Boston led the majors in OW%; their gOW% was still excellent and good enough for third in the majors, but they did not spread their runs across games in an efficient fashion. The Sox scored ten or more runs 25 times; Toronto was second with 19 and the major league average was 9. Boston scored 36% of their runs in that 15% subset of games; the major league average was 15%, and next on the list was Texas at 28%.

Differences in for gDW%:

Positive: DET, BAL
Negative: PHI, SD, TB

I combine gOW% and gDW% through some Pythagorean math to produce gEW%, which can then be compared to a team’s standard Pythagorean record (EW%). Of course, it could also be compared to actual W%, but I think the comparison to a method that also uses runs is more interesting than a comparison to the actual win totals:

Positive: BAL, PIT, CHA, DET, MIN, HOU, OAK, FLA
Negative: BOS, PHI, COL, NYA, SD, TB, LA, KC

There are so many large differences that I’m a little worried that I may have made a spreadsheet error somewhere along the way, although I have double-checked and can’t find anything. Below is a table with all of the metrics discussed in this post for each team, sorted by gEW%: