Rate State Series, pt. 8: Rate Stats for Linear Weights III

This installment will tie up a few miscellaneous loose threads pertaining to rate stats to be used within a linear weights framework. The first relates to R+/PA, which I demonstrated is a “correct” rate stat to be used in this framework; I prefer it to RAA/PA, which produces identical linear differences between players, because it can be meaningfully compared on a ratio basis as well.

One issue with R+/PA for users is that R/PA is not a quantity that most people have intuitive feel for. We could still use RAA/650 PA, or we could use R+/650 PA, or we could convert R+/PA to a “team game basis” (along the lines of the previously discussed runs/25.2 or runs/27 outs) by multiplying by the league PA/G, or a long-term average thereof. As long as it’s a scalar multiplier consistently applied, there’s no real distortion imposed on the results – it’s just a matter of user preference as to which scale should be used.

A thornier issue is the implications of RAA/PA or R+/PA for teams. I have encountered sabermetricians for whom I have a great deal of respect whose ideal rate stat would be equally applicable to individuals and teams. I do not think - for reasons that were discussed earlier in this series - that this is possible. Of everything I’ve asserted in this series, I’m most confident that R/O is the proper rate stat for teams. Applying it to the normal range of major league players causes distortions, which while not disastrous are  too frequent for me to just ignore.

Let’s look at all of the teams in the 1994 AL through these rate stats. For this exercise, we can greatly simplify the calculation of R+/PA – we don’t need to estimate extra PA, we know exactly how many PA each team has generated. We can also calculate RAA simply by using R/O. While we could always look at the number of runs a given run estimator produces for a team, it is also cleaner to just use actual runs scored:

In this case, the rank order using R/O and R+/PA is the same – this does not have to be the case, and I’m a little disappointed it worked out this way as it would have been nice to have a real-life example from this league-season. Again, the fundamental reason why plate appearances are not the correct rate state denominator for teams is that since team plate appearances are solely a function of team OBA, it is of no consequence how many plate appearances they need to produce the same R/O rate as another team. An offense that every inning produced four walks and three outs would score 1/3 R/O, the same as a team that every inning produced one home run and three outs, but the former would score .167 R/PA and the latter .250. In a league where the average team scored 1/6 R/O and .123 R/PA, each would get .5 RAA per inning, but the former would have a R+/PA of .194 and the latter .248.

To make the point again, if we use actual team runs scored or an estimate of team runs created as the starting point for a team rate stat, there is no need to attempt to measure the impact of their PA generation. In fact, making an additional adjustment would be inappropriate double-counting. The team’s totals already incorporate this impact, as the team’s actual plate appearances were a function of their on base average.

What if someone decided they wanted to construct a rate stat that didn’t include any subtraction in the numerator, and instead would just be a positive quality per plate appearance? If you follow this path, you will no longer have a final result expressed in a unit of runs, but perhaps due to your desire for a different scale this is preferable. Or maybe you want a rate stat that can never assume a negative value, as any stat that has a negative linear weight coefficient for an event has the potential to do; as we’ve discussed, even our absolute runs variant of linear weights will produce negative values for pitcher-level hitters. Maybe you want to make one of these alterations and them impose a different scale for some reason.

One way you could do this is to eliminate outs from the LW_RAA/PA equation. If you just ignore them, you will distort all of the values, and be left with a rate stat that is not only unitless, but also is just plain wrong. However, recognizing that outs are simply the complement of the events with the positive coefficients (since we’ve restricted our list of events to mutually exclusive and exhaustive components of plate appearances given our simplified definitions of terms for this series), we could rectify this by just subtracting the absolute run out value from each positive event. To put this in practice, start with our LW_RAA/PA equation:

LW_RAA/PA = (.5069S + .8382D + 1.1695T + 1.4970HR + .3495W - .3150(outs))/PA

Now we will subtract -.3150 from each of the events in the numerator that comprise the complement of outs (again, in this case this is all of the other variables in the numerator) to get a rate stat that for the moment I will call U (for “unitless”, although it’s really just for ease of reference) :

U = (.8219S + 1.1533D + 1.4846T + 1.8210HR + .6646W)/PA

This looks weird, since these weights are no longer in units of R or even R+, but what value does it take on for the league average? By definition, LW_RAA/PA for the league is zero, so this resolves to zero for the league:

LW_RAA/PA = (.5069S + .8382D + 1.1695T + 1.4970HR + .3495W - .3150(outs))/PA 

The manipulation we did to get U was just to subtract -.3150*PA from everything, so:

(.5069S + .8382D + 1.1695T + 1.4970HR + .3495W - .3150(outs) - (-.3150*PA) )/PA = (0 + .3150*PA)/PA = .3150

So the league average of U is by definition equal to the negative of the outs coefficient (.3150 for the 1994 AL). So what happens if we compare a player’s U to the league average?

(.8219S + 1.1533D + 1.4846T + 1.8210HR + .6646W )/PA – LgU 

multiply U by PA/PA to get

(.8219S + 1.1533D + 1.4846T + 1.8210HR + .6646W)/PA – (LgU *PA)/PA

= (.8219S + 1.1533D + 1.4846T + 1.8210HR + .6646W – LgU*PA)/PA

plug back in that LgU = .3150 in our case to get:

(.8219S + 1.1533D + 1.4846T + 1.8210HR + .6646W – .3150PA)/PA

If we distribute the -.3150 to all of the events already in the numerator that are plate appearances, then the leftover PA are outs, and so we are left with:

(.5069S + .8382D + 1.1695T + 1.4970HR + .3495W - .3150(outs))/PA

So U - LgU = LW_RAA/PA. Even though the coefficients have been manipulated into unitless, non-negative values, U maintains the necessary condition to be a proper rate stat for a linear weight framework because it produces the correct RAA/PA.

Why would someone want to express an equivalent to RAA/PA in these terms? This is not a question that I am qualified to answer, since I don’t personally use this form – but there’s a decent chance that you, as a consumer of baseball analysis circa 2021 have been exposed to it, whether you realize it or not. 

Since U is now unitless, one might wish to manipulate it in order to put it on a scale that they find useful, perhaps the scale of a fundamental statistic. If you think about what U is, the numerator includes all on-base events, weighted by their run value – the RAA out coefficient. The denominator is just plate appearances. There is a fundamental statistic that we have been using throughout this series that takes a similar form, except instead of weighting the on-base events, it gives them all an equal weight of one. The fundamental form is the same though:

U = (.8219S + 1.1533D + 1.4846T + 1.8210HR + .6646W)/PA

other stat = (S + D + T + HR + W)/PA

The other stat is of course On Base Average. If one were to tweak the numerator coefficients for each event in U, it could be put on the same scale as OBA, but instead of giving each on-base event a weight of one, it would give each on-base event a weight based on its run value. It would be a weighted On Base Average. Enter Tom Tango and wOBA, which is now one of the two most ubiquitous rate stat in general sabermetric use thanks to its prominence at Fangraphs (with the other being OPS+ thanks to Baseball-Reference).

In order to convert what I have called U to wOBA, we just need to multiply it by the ratio between the league OBA and U, or equivalently the ratio between the league OBA and the negative of the LW_RAA out coefficient. For the 1994 AL, the average OBA was .3433 and U is .3150, so the ratio is 1.0896 (I will define V = LgOBA/LgU just for ease of writing formulas) which produces:

wOBA = U*V = (.8219S + 1.1533D + 1.4846T + 1.8210HR + .6646W)/PA*1.0896

wOBA = (.8956S + 1.2566D + 1.6176T + 1.9744HR + .7241W)/PA

I think there is a lack of understanding among even engaged consumers of sabermetric analysis as to what wOBA actually represents, and how it relates to linear weights. I admit that even as someone who is deeply interested in anything related to run estimators or rate stats, as I don’t actively use wOBA, I have to remind myself why it works. I have fielded multiple email questions from people who obviously think much more deeply about the construction of metrics than the average consumer, and they don’t know the connection and don’t find it intuitive.

I don’t consider this is the fault of Tango, who has explained this. Dave Studeman has written a good article explaining this. But for whatever reason it feels like a secret despite the best efforts of the people who are responsible for it.

So I will take a second here and spell out the relationship between the wOBA family of metrics, which includes wOBA, wRC (Weighted Runs Created, which will be equivalent to what we called R+), and wRAA (Weighted RAA, which is equivalent to LW_RAA), and R+/PA and LW_RAA/PA as we’ve discussed in this series. Keep in mind that I am speaking generally about these relationships using the metrics as I have defined them in this series, and not specifically about the implementation at Fangraphs or anywhere else, although the basic relationships hold. Some of the algebraic demonstrations can be simplified.

If you wish to convert wOBA into RAA (which Fangraphs would call wRAA), we’ve already demonstrated that we can compare U directly league U, so when using wOBA instead of U you just need to remember to remove V:

wRAA = (wOBA – LgwOBA)/V * PA

To calculate wRC (which will be equivalent to R+),  you need to add in the league average R/PA times the batter’s PA:

wRC = wRAA + LgR/PA*PA = (wOBA – LgwOBA)/V*PA + LgR/PA*PA = ((wOBA – LgwOBA)/V + LgR/PA)*PA

wOBA can be easily converted to wRAA/PA, which is equal to LW_RAA/PA:

wRAA/PA = LW_RAA/PA = (wOBA – LgwOBA)/V

and since R+/PA = LW_RAA/PA + LgR/PA, we can also write:

R+/PA = (wOBA – LgwOBA)/V + LgR/PA

Let’s run a leaderboard using the “w” stats, although we’ve already seen most of these values in different guises:

Personally, I prefer R+/PA to wOBA as a rate stat, as the former is directly comparable both as a difference and a ratio, while the latter has to be manipulated in order to be compared either way – both differences and ratios of wOBA have no intrinsic meaning. However, the advantages of having a scale similar to that of OBA, where no negative values are possible, where each event has a clean positive weight, and the “natural” denominator of plate appearances is used. Tango et. al. also took advantage of its structure to more easily apply statistical techniques in The Book, so there are certainly reasons why a user might prefer it, and it is well-constructed by which I mean that the scaling does not cause issues for derivative metrics as long as you know how to account for it.

Finally, the third loose thread I wanted to address in this post. Prior to introducing wOBA, Tango developed a rate stat version of linear weights he called Linear Weight Ratio. It was somewhat conceptually similar to wOBA in that it sought to eliminate the negative weight for outs. Instead of adding the out value to the positive events in the numerator and thus being able to dispense with the negative value of the out as is the case for wOBA, LWR was constructed by making outs the denominator. In order to make the relationship between the positive events more clear, the value of a single was fixed at 1.0, with the coefficients for the other events defined as the ratio of the LW value for the given event to the LW value for a single. If you let the LW value of a single be s, a double be d, etc., then the formula for LWR is:

LWR = (s/s*S + d/s*D + t/s*T + hr/s*HR + w/s*W)/Outs

Which for our 1994 AL weights of  .5069S + .8382D + 1.1695T + 1.4970HR + .3495W becomes:

LWR = (S + 1.6536D + 2.3072T + 2.9532HR + .6895W)/Outs

If we define “x” as the absolute LW value of the out (-.1076 in this case), there are some very straightforward relationships between LWR and LW_RC/O:

LW_RC/O = s*LWR + x

LWR = (LW_RC/O + x)/s

If we define “y” as the average LW value of the out (-.3150 in this case), we can define similar relationships between LWR and LW_RAA/O:

LW_RAA/O = s*LWR + y

LWR = (LW_RAA/O + y)/s

Of course, LW_RAA is the building block for our “correct” rate stat (be it LW_RAA/PA, R+/PA, or wOBA) for the linear weights framework, so you could with some additional algebra convert LWR to those, although LWR sans manipulation does not meet the differential or ratio comparability standards. 

Rate State Series, pt. 7: Rate Stats for Linear Weights II

In the last post, we ended with a dilemma: we have a metric (linear weights RAA/PA) that we believe is correct for a linear framework, but it can’t be compared using ratios. We also have an approach (linear weights RC/O) that produces the right RAA but as a rate is inconsistent with RAA/PA. I asked whether we might be able to make adjustments to one or both of these that would get us to the right place.

Let’s start with trying to manipulate R/O to get a proper rate stat. We know that using R/O for individual players (even when using a linear weights estimate for runs created) overstates their value by treating them as a team with respect to their plate appearance generation. We know that using R/PA fails miserably because it doesn’t account for PA generation at all. What if, instead of trying to manipulate the denominator to get an acceptable rate stat, we attempted to manipulate the numerator?

Linear weight runs created with the “-.1 type out value” doesn’t take into account the extra opportunities created (or not created) for a batter’s teammates by his avoidance of outs, but we could make an explicit adjustment that would take care of this. In part 2, we laid out the math to calculate plate appearances as a function of OBA:

PA/G = (LgO/G)/(1 – OBA)

To compare a player’s rate of PA generation to the league, we can calculate the PA/G he would generate as a team, less the league average. For the sake of simplicity, let’s use the variable X to represent league O/G, and the variable EPA to mean “Extra PA” relative to what would be produced by a hitter with a league average OBA. (What follows is a needless bunch of algebra, but I wanted to demonstrate that the final result is tied back to the equation that relates PA/G, O/G, and OBA):

X/(1 – OBA) – X/(1 – LgOBA) = EPA/G

where “games” are defined as X outs. Since this is the case, we can factor out X from the left side of the equation and divide both sides by X, which converts EPA/G to EPA/Out:

1/(1 – OBA) – 1/(1 – LgOBA) = EPA/O

For ease of notation, I’m going to switch to using O/PA to represent the hitter’s 1/(1 – OBA); the complement of OBA is O/PA, and it’s reciprocal is thus PA/O. I will leave LgOBA as a variable since it is a constant from the perspective of calculating the individual’s PA generation:

PA/O – 1/(1 – LgOBA) = EPA/O

As this is an individual metric, I’d rather express it with a denominator of PA than O, so I will multiply both sides by O/PA to get:

(PA/O – 1/(1 – LgOBA))*O/PA = EPA/PA = 1 – 1/(1 – LgOBA)*(1 – OBA) since O/PA is just 1 – OBA

= 1 – (1 – OBA)/(1 – LgOBA) 

replace 1 with (1 – LgOBA)/(1 – LgOBA) to get:

(1 – LgOBA)/(1 – LgOBA) – (1 – OBA)/(1 – LgOBA) = (1 – LgOBA)*(1 – OBA)/(1 – LgOBA)

= (OBA – LgOBA)/(1 – LgOBA) = EPA/PA

This simple equation yields the number of additional PA generated for a batter per PA, beyond what a batter with a league average OBA would have generated. Of course, if we multiply by PA, we will get the raw number of extra PA generated. As an example, Frank Thomas had a .4921 OBA in a league where the OBA was .3433, and he had 508 PA, so he created an additional 115.1 PA for his team beyond what an average hitter would have contributed:

(.4921 - .3433)/(1 - .3433)*508 = 115.1

In order to use this with our modified R/PA, we need to convert it to runs, which can be done by simply multiplying by the league average of .1363 R/PA to get 15.7. As we’ve computed previously, Thomas had 131.4 LW_RC, which is his direct run contribution as a result of his own PA, but without considering the impact he had on his team by creating additional PA (or, more precisely, on a league average team since the entire linear weights framework we’ve been working with this in this series is built from the league average). That was worth an additional 15.7 runs, so his total run contribution for the numerator was about 147.1 runs.

Before moving on, credit is due to the developer of this approach. This methodology, at least expressed in terms of absolute runs plus the PA impact, was first developed by a BaseballBoards.com poster with the moniker “Sibelius”; if he was not the first, he is certainly the person who introduced me to this concept. Sibelius’ original construct combined this with Jim Furtado’s XR (Extrapolated Runs), so he called it XR+ and the resulting rate XR+/PA. I will be using the more generic R+/PA to describe this metric going forward.

Additionally, Sibelius calculated the “+” portion due to PA generation in a mathematically equivalent but quicker way than I did. This later tripped me up, as I had been influenced by his ideas but approached the problem in the manner I did above, starting from extra PA and converting to runs. Thus I posted it on the board as if it was a new, alternative approach rather than simply a different way of calculating the same thing. My faux paus was quickly pointed out and I gladly concede it, although I personally still find the “proof” above to be the most straightforward way to understand the math. But Sibelius’ calculation is more straightforward and I will use it going forward. Instead of messing around with PA, he jumped straight to outs, calculating the number of outs that the player avoided beyond those an average OBA hitter would have made in the same number of PA. Then he multiplied by the league average R/O to get the runs from PA generation/out avoidance. Six of one, half a dozen of the other as of course R/PA = R/O * (1 – OBA), and so Sibelius' equation for the “+” portion would look like this:

(OBA – LgOBA)*PA*LgR/O

For Thomas, this is (.4921 - .3433)*508*.2075 = 15.7

So our formula for R+/PA will be:

R+/PA = (LW_RC + (OBA – LgOBA)*PA*LgR/O)/PA

Note that this is equivalent to:

R+/PA = LW_RC/PA + (OBA - LgOBA)*LgR/O

In order to compute the associated RAA:

RAA = (R+/PA – LgR/PA)*PA

This works as of course the league average R+/PA is equal to the league average R/PA by definition.

Here are the top and bottom five hitters from the 1994 AL:

The first column for RAA is RAA based on R+/PA, and it is an exact match for LW_RAA. Thus R+/PA meets the necessary condition for being an acceptable rate stat. However, so did R/O. We need to test that it produces the same rank order as LW_RAA/PA beyond these players; a good place to start would be with the cohort of four players who were jumbled in order when looking at RAA/PA and R/O:

Here we get a match in rank order. Of course, looking at fourteen players and finding consistency in their ranking doesn’t prove that there would never be a player for whom the two would not agree. We will need some other approach to demonstrate that.

Let’s leave R+/PA for now and go back to the other approach we were considering, which was making an adjustment to RAA/PA that would allow it to produce meaningful ratio comparisons. The obvious solution is to add the league average R/PA to RAA/PA to get a modified absolute R/PA. After all, RAA/PA  represents an individual’s contribution above average per PA, taking the impact of PA generation into account. Adding back in the league average R/PA will return this to an absolute R/PA basis, that will equal raw R/PA at the league level by definition, and that unlike LW_RC/PA will capture the value of extra PA generated by the batter. We know that by definition it will match RAA, since the difference between an individual and the league average will be equal to RAA/PA (RAA/PA + LgR/PA – LgR/PA).

Here are the top and bottom 5, with “mod R/PA” being the metric we’re discussing (LW_RAA/PA + LgR/PA):

If these figures look familiar, it’s because (with minor rounding discrepancies), they are equal to the R+/PA figures we just calculated. That’s right – all of that algebra to calculate extra plate appearances or outs avoided, and multiply by the league average R/PA or R/O, add back to LW_RC – we could have dispensed with all of it, and just added LgR/PA to LW_RAA/PA.

This is our “proof” that R+/PA as we built it from adjusted absolute runs created meets our criteria for a proper rate stat in a linear weights framework – it’s equivalent to using RAA/PA. The two parallel approaches are not in fact parallel – they are alternative ways of getting to the same place.

To demonstrate why the math works out this way, let’s return to our two linear weights equations from part 1:

LW_RC = .5069S + .8382D + 1.1695T + 1.4970HR + .3495W - .1076(outs)

LW_RAA = .5069S + .8382D + 1.1695T + 1.4970HR + .3495W - .3150(outs)

If we think about everything on a per plate appearance basis and building R+/PA using the RAA/PA + LgR/PA path, what we’re doing is taking the second equation and adding the league average R/PA for every event that corresponds to a plate appearance, which is all of them (in a full blown implementation, where we would be considering non-PA events both in the linear weights formula and the calculation of the PA generation impact, this wouldn’t be as clean). Since the league R/PA is .1363, this results in:

R+ = .6432S + .9745D + 1.3058T + 1.6332HR + .4858W - .1788(outs)

Alternatively, from the Sibelius approach, the plus component can be manipulated to read:

(OBA – LgOBA)*PA*LgR/O = ((H+W)/PA – LgOBA)*PA*LgR/O = (H + W – LgOBA*PA)*LgR/O

Continuing to treat everything on a per PA basis, this is (H + W – LgOBA)*LgR/O

Thus any hit or walk contributes (1 – LgOBA)*LgR/O = LgR/PA = .1363, and each out “contributes” (0 – LgOBA)*LgR/O = -LgOBA*LgR/O = -.3433*.2075 = -.0712. Adding .1363 to the LW_RC value for each event results in the same weights we just found (i.e. .6432 for a single), as the weights for on base events are the same between the LW_RC and LW_RAA equations. The out is now worth -.1076 + -.0712 = -.1788 runs, also the same value. We knew from the results that the two approaches must be equivalent, but I always prefer a “proof” when possible.

At this juncture we reach a philosophical question: should we dispense with the notion of absolute runs created for individuals altogether, and replace it with a R+ approach? In other words, instead of our default way of discussing an individual’s contribution being something like: “Frank Thomas created 131 runs in 508 plate appearances while making 258 outs”, and presenting RC, the rate stat, and some metric compared to a baseline, we could cut to the chase and say “Frank Thomas contributed 147 runs in 508 plate appearances”, with the rate and baselined metric. This approach does away with the need to explicitly think about outs, since their special impact beyond plate appearances has been built in to the 147 R+. It also prevents the awkward situation of having a runs created figure of 131 but no handy denominator with which to convert it to a rate stat without doing a R+ or RAA calculation first. 

There is no right answer, although there certainly is a good case to be made for just reporting Thomas’ runs created as 147, since it limits the likelihood of user-initiated bad math. Call it inertial reasoning if you must, but I still like the idea that the runs created figure is the batter’s direct contribution, and the secondary impact of PA generation is captured when looking at the rate or the baselined stat. 

Rate Stat Series, pt. 6: Rate Stats for Linear Weights I

Last time, I attempted to demonstrate that linear weights producing a result in terms of runs above average properly account for the value of extra plate appearances created by a batter. From here on, I will talk about this contention in the same way as I would any scientifically-demonstrated fact, even though I admit that I have not provided a “proof” in the mathematical sense. The casual use of language is intended to prevent wasting space repeating myself rather than an attempt to claim a more robust result than is appropriate.

Since we have demonstrated that LW_RAA captures the player’s direct and indirect contributions to team offense (at least within the linear framework of metrics), I would contend that it follows that any linear weights-based rate stat we should propose must return the same RAA as we would get from eschewing a rate stat altogether and just applying our linear weights formula with the “-.3 type out value”.  One very simple way to do that is to simply use RAA, rather than a measure of absolute runs created, as the numerator for the rate stat. In this case, the obvious denominator is plate appearances.

Using outs in the denominator would be appropriate for a team metric, but for an individual would overstate the value of his PA generation. For a simple demonstration of this, recall from part three the equation:

PA/G = (O/G)/(1 – OBA)

and the equivalent PA = O*(1 – OBA), given our simple definitions which as a refresher are PA = AB + W, O = AB – H, and OBA = (H + W)/(AB + W)

This direct relationship between plate appearances, outs, and OBA means that if we were to use outs as the denominator rather than PA, we would be inflating the rates for players with higher OBAs, even though we’ve already demonstrated that the PA generation impact of higher OBAs is captured in the numerator (RAA).

Here were the top and bottom 5 performers in terms of RAA/PA, which for ease of use I have restated as RAA/650 PA (This works out to 152.5 full games for a player getting 1/9 of an average 1994 AL team’s PA; sadly, these teams didn’t get anywhere close to playing a 162 game season. There’s no special significance to 650 other than that it is a round number that is reasonably close to the number of PA a full-season batter might accumulate):

You may have noticed that the identity and order of these players did not change from when we used a fully dynamic approach that treated them as if they were their own teams (BsR/O). This recalls a simple truth that I am burying beneath thousands of words of minutia – any reasonable approach will reach similar conclusions for the majority of situations. Even a poorly designed, indefensible metric like OPS will get you 98% of the way there. This series is about the tiny differences that lie beyond that point and the larger differences that arise in values as opposed to rank order.

To wit, while the rank order remains the same, the RAA values are different using linear weights, and with the exception of Griffey, less extreme. All of the other players have moved closer to average, with Frank Thomas losing a whopping 7.7 RAA due to using a linear approach rather than imagining an entire lineup of Big Hurts.

At this point, we could stop, and simply use RAA/PA or RAA/650 PA as our final linear weights-grounded rate stat. However, it lacks the very useful trait of ratio comparability that would be desirable in the ideal rate stat. While linear differences in RAA/PA can be compared (e.g. Thomas contributed an additional .081 RAA/PA beyond what Chili Davis did), the ratios are not particularly useful. Consider two players who each had 600 PA, one contributing +1 RAA and the other -1 RAA. If you can explain the practical baseball interpretation of the resulting ratio of -1, be my guest.

This happens because we have applied the high baseline of average to the metric. However, we have another version of linear weights that is based on absolute runs from which we could build a rate stat. Remember that in order to qualify for consideration, the RAA that results from that rate stat must be equivalent to the RAA from simply applying the linear weights formula and not comparing a rate to the league average.

The obvious first choice is Runs/PA, using our formula for linear weights runs created. In this case, I am not showing the leaders, but rather the same ten players in the same order. RAA in this case is (LW_RC/PA – LgR/PA)*PA:

The results are not even close to what we need, and the reason is simple: we have not accounted in any way for each batter’s PA generation. There is an easy alternative that might correct this: using outs in the denominator. This just takes us to the correct team rate stat, although the numerator is linear rather than dynamic (either in the case of using Base Runs or using actual runs scored on the team level). As such it does implicitly consider PA generation; might it produce satisfactory results for individuals? Here RAA = (LW_RC/O – LgR/O)*O:

A perfect match. Absolute runs per out produces the same RAA as the direct application of LW_RAA. R/O also has the advantages of being meaningfully comparable as both a difference and a ratio and is the same as the correct team rate stat. Everything is great, except we haven’t answered the question: Does it actually work?

Remember, I said that matching LW_RAA was a necessary condition for our proposed alternative linear weight rate stat to meet – it is not a sufficient condition. We’ve already concluded that RAA/PA is a proper rate stat for a linear framework; in order for an alternative to be acceptable, it must produce results that are consistent with RAA/PA. How do we determine this consistency, other than matching the RAA result, when the numerators and the denominators each start from a different basis? 

One simple but obvious way to determine if they are consistent is to confirm that they result in the same rank order of players. They did for our most extreme hitters, but does that hold for all hitters? 

Apparently not. I didn’t have to go to far to find this little cluster of hitters as they rank 7-10 in RAA/PA. None of them rank in the same spot as Vaughn is first in RAA/650 but second in RC/O; Lofton is second/third; Mack is third/fourth; and Clark is fourth/first. 

I slipped OBA onto the chart because it helps to explain what is going on. Will Clark’s .433 OBA ranked fifth among AL hitters with 200 PA; using outs in the denominator helps him as it implicitly assumes that he represents an entire team with a .433 OBA. While all of these hitters had excellent OBAs relative to the league, R/O goes a bit too far in valuing this. Given that R/O still produces the right RAA, any distortions have to be somewhat limited for normal players.

One can come up with extreme thought experiments, like a player with a .999 OBA all from walks versus a player with a .600 OBA, all from home runs. A team made up of the former would score what would certainly feel to the opposing pitching coach like a nearly infinite number of runs; but as a single player in a lineup, his impact would be muted. It’s not necessary to answer the thought experiment as to which would be more productive to see that R/O applied to extreme individual players would break down. Incidentally, it is exactly this type of scenario that I got into trouble trying to “prove” in my last attempt at this series – while I do think that the theoretical team methods I’ll discuss later provide reasonable estimates for this situation, relying too heavily on them for proofs is begging the question. 

So, we have a rate stat (LW_RAA/PA) that works, but it lacks ratio comparability. There are (at least) two ways we could go about solving this problem:

1. We could adjust LW_RC/PA in some way to take into account the value of PA generation

2. We could manipulate LW_RAA/PA so that it’s no longer a measure of runs above average, but instead on an absolute runs basis

Next time I’ll explore these two parallel questions...if in fact they are parallel at all.

Rate Stat Series, pt. 5: Linear Weights Background

Linear methods sidestep the issues that arise from applying dynamic run estimators to players by simply ignoring any non-linearity in the run scoring process altogether. While this is clearly technically incorrect, it is closer to reality than pretending that a player’s performance interacts with itself. Since an individual makes up only 1/9 of a lineup, it is much closer to reality to pretend that his performance has no impact on the run environment of his team than to pretend that it defines the run environment of his team. Linear weights also have the advantage of being easy to work with, easy to adapt to different baselines, and easy to understand and build. Their major drawback is that the weights are subject to variation due to changes in the macro run environment (as distinguished from the marginal change to the run environment attributable to an individual player). 

Linear methods were pioneered by FC Lane and George Lindsey, but it was Pete Palmer who used them to develop an entire player evaluation system, publish historical results, and bring them into the position of the chief rival to Runs Created in the 1980s. Curiously (especially since Palmer is a prolific and brilliant sabermetrician whose pioneering work includes park factors, variable runs per win, using the negative binomial distribution to model team runs per game, and more), Palmer’s player evaluation system as laid out in The Hidden Game of Baseball and later Total Baseball and the ESPN Baseball Encyclopedia never bothered to convert its offensive centerpiece Linear Weights into a rate statistic.

This gap contributed to two developments that I personally consider unfortunate. First, confusion about how to convert linear weights to a rate may have hampered the adoption of the entire family of metrics, and this confusion generally persisted until the publication of The Book by Tom Tango, Mitchel Lichtman, and Andy Dolphin. Second, Palmer did offer up a rate stat, but he did not tie it to linear weights, or in its crudest form to any meaningful units at all. That’s because Normalized OPS (later called Production), which you may know as OPS+, was the rate stat coupled with linear weights batting runs.

To my knowledge, Palmer has never really explained why he didn’t derive a rate stat from linear weights to use; the explanations have instead focused on the ease and reasonable accuracy of OPS. In The Hidden Game, the discussion of linear weights transitions to OPS with “For those to whom calculation is anathema, or at least no pleasure, Batter Runs, or Linear Weights, has a ‘shadow stat’ which tracks its accuracy to a remarkable degree and is a breeze to calculate: OPS, or On Base Average Plus Slugging Percentage.”

Coincidentally, Palmer recently published an article in the Fall 2019 Baseball Research Journal titled “Why OPS Works”, which covers a lot of the history of his development of linear weights and OPS, but still doesn’t explain exactly why a linear weights rate wasn’t part of the presentation.

Without the brilliant mind of Palmer to guide us, where should we turn for a proper linear weights-based rate stat? To answer that question, I think it’s necessary to briefly examine how linear weights work. For this discussion, I am taking for granted that the empirical derivation of linear weights is representative of all linear weight formulas. This is not literally true, as belied by the fact that the linear weights I’m using in this series were derived from Base Runs, not from empirical data. If we were using an optimized Base Runs formula, the resulting weights would be very close to empirical weights derived for a similar offensive environment, but other approaches to calculating linear coefficients like multiple regression can deviate significantly from the empirical weights. Even so, the final results are similar enough that the principles hold for reasonable alternative linear weight approaches.

What follows will be elementary for those of you familiar with linear weights, but let’s walk through a sample inning featuring the star of our series, Frank Thomas. I want to use this example to illustrate two properties of linear weights when using the “-.3 type out value” (i.e. when the result is runs above average): the conservation of runs, and the constant negative value of outs. This example will simplify things slightly, as in reality not every event in the inning cleanly maps to a batting event that is included in a given linear weights formula (e.g. wild pitches, balks, extra bases on errors, etc.) It also will presume that the run expectancy table we use for the example corresponds perfectly to our linear weights, which it does not. Still, the principles are generally applicable to properly constructed linear weights methods, even if the weights were derived from other run expectancy tables or, as is the case for us in this series, by another means altogether (I’m using the intrinsic weights derived from Base Runs for the 1994 AL totals).

Baseball Prospectus has annual run expectancy tables; their table for the 1994 majors is:

On July 18, Chicago came to bat in the bottom of the seventh trailing Detroit 9-5. Their run expectancy for the inning was .5545 as Mike LaValliere stood in against Greg Cadaret. He drew a walk, which raised the Sox RE to .9543, and thus was worth .3998 runs. The rest of the inning played out as follows:

1. If we were going to develop empirical LW coefficients based on this inning, we would conclude that a home run was worth 2.658 runs on average, and thus our linear weight coefficient for a home run would be 2.658. The other events would be valued:

This is in fact how empirical LW are developed, but of course a much larger sample size (typically at least an entire league-season) is used.

2. The team’s runs above average for the inning is always conserved. We started the inning with the bases empty and nobody out for a RE of .5545. This is the same as saying that the average for an inning is .5545 runs scored. The White Sox actually scored 4 runs, and the total of the linear weight values of the plays was 3.4455 runs, which is 4 - .5545. They scored 3.4455 runs more than an average team would be expected to in an inning. The sum of the linear weight values will always match this.

Because of this, we can be assured that the run value of additional plate appearances created by the positive events of the batters has been taken into account in the linear weight values. If this were not the case, runs would not be conserved.

3. Since that is true, it is also true that the sum of the LW values of the positive events (which is 4.8128 runs) plus the sum of the LW values of the outs (-1.3673) must be equal to the runs above average for the inning (3.4455). The sum of the values of the outs will be higher in innings in which more potential runs were “undone” by outs, as was the case here. On the other hand, an inning in which three outs are recorded in order will result in -.5545 runs.

We can use this fact to isolate the run value of the out between the portion that is due to ending the inning (what Tom Tango has called the “inning killer” effect of the out; this is the -.5545 that is the minimum out value for an inning), and that which is due to wasting the run potential of the positive events (what’s left over, in this case, -.8128 runs). 

If we wish to convert our linear weights from an estimator of runs above average to an estimator of absolute runs, we need to back out the inning killer value of the out (since it will be present for every inning equally and serves to conserve total RAA) from the overall value of the out, leaving the remainder which we do not need to worry about as it would have to be debited from the value of the positive events in order to conserve runs.

So we can take -.5545/3 = .1848 and add it back to the linear weight RAA out value, which for our example was -.3150. This results in an absolute out run value of -.1302, In our example we’re using -.1076; these don’t reconcile because:

1. our linear weights don’t consider all events (we’re ignoring hit batters, sacrifices, all manner of baserunning outs, etc.)

2. our linear weights weren’t empirically derived from the 1994 RE table as the .1848 adjustment was

While the numbers don’t (and shouldn’t!) balance perfectly in this case, this is the theoretical bridge for converting empirical linear weights from a RAA basis to an absolute runs basis. I would also contend is serves as a demonstration by inductive reasoning that absolute linear weights do not capture the PA generation impact of avoiding outs, but RAA linear weights do.

Note that converting to the “-.1 type out value” does not eliminate the result of negative runs altogether. An offensive player who is bad enough will be credited with negative runs created (if it helps you to imagine what this level of production might look like, consider that the total offensive contributions of pitchers has hovered near zero absolute runs created in the last decade). For real major league position players, this will not happen except due to sample size. If you’d like an interpretation, I have found this helpful (I stole it from someone, probably Tom Tango, and have badly paraphrased): Since linear weights fix the values of each event for all members of the team, the level at which runs created are negative is the level at which in order to conserve team runs, the weights of positive events cannot be reduced – the poor batter essentially undoes some of the positive contributions of his teammates.

As an aside, the first paper I’m aware of that made the connection between the two linear weight approaches in this manner (rather than simply solving algebraically for the difference between the two without providing theoretical underpinning) was published by Gary Skoog in a guest article in the 1987 Baseball Abstract. This article, titled “Measuring Runs Created: The Value Added Approach” is available at Baseball Think Factory.

Rate Stat Series, pt. 4: Players as Teams

 A dynamic run estimator is a run estimator that allows offensive events to interact with each other, such that the value of a given event is not fixed as would be the case in a linear weights formula (e.g. a single is worth .50 runs), but rather is dependent upon all of the other components of the batting line. Dynamic run estimators are great in theory, since the run scoring process for a team is obviously dynamic and not linear. However, there are two issues:

1. They are harder to design than linear estimators. Any idiot with a spreadsheet and a dataset can run a linear regression on runs scored and have a linear estimator when they are done. It may not be a good one, but it will be functional and will probably have a low RMSE when estimating team runs scored. To develop a dynamic model, one must consider the run scoring process and produce a simplified model, but not so simplified as to not produce reasonably accurate estimates.

This is not a series about run estimators, but the most commonly used dynamic run estimator, Bill James’ Runs Created, suffers from flaws that make it unable to handle extreme offenses. A much better model, David Smyth’s Base Runs, is powerful and will be used here.

2. They are not appropriate to apply to individual offensive statistics. Dynamic estimators always involve multiplying base runners by some factor representing advancement of baserunners (in Runs Created that’s the end of the story, Base Runs accounts for the unique nature of home runs). This multiplication is inappropriate when applied to an individual player, as now Frank Thomas’ high OBA is multiplied directly with his high power which advances runners. In reality, there is some interaction, but Thomas’ impact is diluted by being just 1/9th of the lineup. Inputting his statistics into a dynamic run estimator produces an estimate of how many runs a team would score if each batter hit like Thomas.

Due to this issue, I do not advocate applying dynamic run estimators directly to individuals, but this post will still address the rate stat implications of such applications. Later we will discuss theoretical team methods that allow the use of a dynamic run estimator while still accounting for the fact that the player is just one of nine in the lineup.

This series will now discuss what I believe to the be the proper rate stats for a particular framework for evaluating individual offense. One of my objectives is that for each option of a framework for building a rate stat presented, there be at least one variation that is linearly comparable and one that is ratio comparable. I’ve defined those terms as I use them at length before, so here I will be brief:

* A statistic is linearly comparable if the difference between two figures is meaningful. A hitter with a .400 OBA would reach base 100 times more than a hitter with a .300 OBA over 1000 PA.

* A statistic is ratio comparable if the ratio between two figures is meaningful. Our .400 OBA player reached base 33.3% more frequently than the .300 OBA player

Ideally, our metric will facilitate both types of comparison, but if not, I will endeavor to present an alternative formulation that fills the gap. I will not propose any metrics that are neither linearly comparable or ratio comparable because they are the scourge of sabermetrics (hello OPS).

The underlying principle of the discussion that follows for the three frameworks (treating the player as a team, a full linear model, and a theoretical team model) is that the rate stat should be consistent with the run estimator used. If the run estimator treats the player as if he is a team, then the corresponding rate stat should treat the player as if he is a team.

In this case, that makes it very simple. The proper denominator for a team rate stat is outs. If you apply Runs Created, Base Runs, or some other run estimator directly to an individual player, the proper denominator is outs.

At this point in the discussion, this may ring as a somewhat hollow declaration, as I have only indirectly made the case for why we might want to use a denominator other than outs for an individual when it is so clearly the proper choice for a team. Since I’m suggesting that outs are the proper choice for this framework, I’ll defer that case for later.

In this case, I advocate for using outs when applying a dynamic run estimator to a team because it is the only consistent treatment. The only justification for going down this path (other than needing something quick and dirty) is a theoretical exercise – how many runs would a team that hit like Frank Thomas score? While I don’t think this theoretical result is appropriate for attempting to value Thomas’ contribution the 1994 White Sox, it at least does have an interpretation. If you start mixing frameworks, you really have a mess on your hands. There’s no good reason (other than crude estimation) to apply a dynamic run estimator directly to an individual; there’s no sense in deviating from the corresponding rate stat in order to try to make the results more comparable to a better approach to evaluating individual offensive contribution. Just use the better approach, and if you insist on misapplying a dynamic run estimator to individual players, make outs the denominator so that at least you have a theoretically coherent suite of metrics.

I should note that Bill James in the 1980s took this entire process to its logical conclusion. After applying Runs Created to individuals, dividing by outs, and multiplying by a constant that was close to the league outs/game for the definition of outs chosen, he went a step further and used the Pythagorean theorem to estimate the winning percentage that this team would have if it allowed an average number of runs. He then converted it to wins and losses by using the number of outs the player made to define games, which caused all kinds of problems, but at least he was committed.

This will be the first of several times that I’ll run a leaderboard for the 1994 AL using a particular framework. Here we have the top 5 and bottom 5 performers with at least 200 PA in Base Runs/Out. RAA is “Runs Above Average” and is calculated simply as (BsR/O – LgR/O) * Outs. Spoiler alert: No matter how we slice it, Frank Thomas is going to come out as the leading hitter in this league, as he raked .353/.492/.729 on his way to a second consecutive MVP award.

I am showing at least one more decimal place on each metric than I usually would just to allow for a little more precise calculation if you’re following along; it is no way a statement about the significance of the ten-thousandths of runs per out. 

Runs per out can of course be scaled; Bill James multiplied it by the league average outs/game appropriate given the categories be considered in the computation of outs. For instance, in this case, since we’re defining outs as AB – H, the average outs/game will be around 25.2 (for the 1994 AL it was 25.19). A more complete accounting of outs, like AB – H + CS + SH + SF + DP, would get close to 27 outs/game. While putting individual contribution on a team games basis is nonsensical on some level, since it is just a scalar multiplier it causes no real distortion and provides a scale that is easily understandable, in the same manner that ERA or K/9 are understood by everyone other than Matt Underwood and Harold Reynolds. 

Rate Stat Series, pt. 3: Teams

If I tell you that three teams in the same league-season played the same number of games (113), and that one of them scored 679 runs, another scored 670, and the third scored 633, how confident would you be in using this limited data to rank the productivity of their offenses? As usual in this series, we are ignoring park factors and other contextual factors (like quality of opposition/not having to face one’s own pitching staff); since they are from the same league-season, you don’t need to worry about whether the win value of each team’s runs was the same. Assume also that runs will be distributed across games by a known distribution like Enby, so the distribution is also not a differentiator. Assume that we don’t care about any “luck”; the actual total is what matters, not what a run estimator came up with. What else do you need to know?

I would contend that given the (admittedly restrictive) parameters I’ve placed on the exercise, you now know almost everything you need to know. In a small number of cases, and to a small extent, you are missing valuable information – but for most situations, you should need no additional information.

Now suppose I told you something similar about three players: same league season, same number of games played (111), and three runs created estimates: one player created 106 runs, one 92, and one 88. Do you feel like you need any additional information to put these players in the proper order of offensive productivity?

I hope that your answer here is yes, and a lot of it. I’ve told you how many games each have played, but that doesn’t tell you how many opportunities they’ve had at the plate. Sure enough, in this case one of the players had substantially fewer plate appearances than the others (489, 490, 451 respectively). Given that the player who created 90 runs had 39 more plate appearances than the player who created 86, it seems likely that the latter player was actually more productive on a rate basis.

I did not tell you how many plate appearances each of the three teams had in their 113 games; I don’t think it’s relevant to the question at hand, but the answer is 4493, 4611, and 4556 respectively. Why do we need to know plate appearances (or something) in the case of players, but not in the case of teams? Understanding this gets to the heart of the reason this series needs to exist at all, why applying the same rate stat to team offenses and player offense may not work as intended.

In the previous installment, I asked the question: “Where do plate appearances come from?” The answer is that every inning (excluding walkoff situations) starts with three PAs guaranteed, and only by avoiding outs (reaching base and not being subsequently retired on the bases) can a team generate additional plate appearances.

From a team perspective, then, plate appearances are not an appropriate denominator for a rate stat, because differences in team plate appearances are the result of differences in performance between the teams. To return to the three teams discussed above, they are the 1994 Indians, Yankees, and White Sox respectively. The Indians had the fewest PA of the three yet scored the most runs. Does this mean that their offense, which already scored more runs than the other two clubs, was even more superior than the raw numbers would suggest?

An offense does not set out to maximize its plate appearances, nor does it set out to score the maximum number of runs it can in the minimum number of plate appearances. An offense sets out to maximize its total runs scored. Plate appearances are a function of the rate at which a team makes outs. At this point it might be helpful to consider the three teams:

New York’s OBA was 22 points higher than Cleveland’s and thus they generated an extra plate appearance per game. When ranking team offenses, it wouldn’t make sense to penalize the Yankees for this, which would be the case if we used R/PA. The difference in plate appearances simply reflects the different manner in which New York and Cleveland went about creating runs. For a team, plate appearances are inextricably linked with their OBA. Each inning, a team attempts to score as many runs as it possibly can before making three outs. It’s possible to score one run in a complete inning with as few as four or as many as seven plate appearances. Whether a team uses four, five, six, or seven plate appearances to score a single run is irrelevant in terms of that run’s impact on them winning or losing the game (*). Thus outs or an equivalent like innings are the correct choice for the denominator of a team rate stat.

(*) I am speaking here simply about the direct impact of the runs scored and not any downstream effects or the predictive value of team performance. Perhaps the team that uses seven PA to score one run benefits by wearing down the opposing pitcher or is more likely to have success in the future because they had four of seven batters reach base compared to one in four for the team that only needed four PA. Here we’re just focused on the win value directly attributable to the run scored and not any secondary or predictive effects.

The fact that outs are fixed for each team each inning (ignoring walkoffs) means that outs are also fixed for each team each game (ignoring walkoffs, rainouts, extra innings, and foregone bottom of the ninths). Which means that outs are also fixed for each team each season (ignoring those factors and cases in which teams don’t play out their full schedules, or have to play tiebreakers), which means that R/G and raw seasonal runs scored total are essentially equivalent to looking at R/O for a team. So for the question I asked at the beginning of the article, just knowing that the three teams had played an equal number of games, we had a pretty good idea how they would “truly” rank using R/O.

For players, this is not at all the case, since even in an equal number of games, players will get different numbers of plate appearances for a variety of reason (batting order position, the team’s OBA (remember, higher OBA teams will generate more PA), whether or not they play the full game), a fact that is intuitive to most baseball fans. What is less intuitive, though, is that even in the same number of plate appearances, players can make very different numbers of outs. Since we’ve already accepted that team OBA defines how many plate appearances a team will generate, it isn’t much of a leap to conclude that if we have two players who create the same number of runs (using a formula that doesn’t explicitly account for their impact on the team’s OBA) in the same number of plate appearances, the player who makes fewer outs was more productive when we consider the totality of their offensive contribution. Even though the two players were equally productive in their plate appearances, the player who made fewer outs generated more plate appearances for his teammates, a second-order effect that needs to be considered when evaluating individual offensive contribution. For teams, the runs scored total already reflects this effect.

This would be an appropriate time to note that this series is focused on evaluating offenses, but of course every offensive metric can be reviewed in reverse as a defensive metric. However, since the obvious denominator for teams is outs, it is also the obvious denominator for individual pitchers. We don’t need to worry about a pitcher’s impact on his team’s plate appearances – when he is in the game, he is solely responsible (setting aside the question of how the team’s performance should be allocated between the pitcher and his fielders) for the number of plate appearances the opponent generates, and his goal is to record three outs while minimizing the number of runs he allows, regardless of how many opponents come to the plate. Outs are clearly the correct denominator for the rate stat, and innings pitched are nothing more than outs/3 (and even better, IP account for all outs, including many that don’t show up in the standard statistical categories).

In thinking about the development of early baseball statistics and the legacy of those standard statistics on how the overwhelming majority of fans thought about baseball before the sabermetric revolution took hold, it is striking that the early statisticians understood these concepts as they applied to pitchers. When pitchers were completing almost all their starts, simple averages of earned runs allowed sufficed, for the same reason that team R/G tells you most everything you need to do. As complete games became rarer, ERA took hold, properly using innings in the denominator. For most of the twentieth century, and even post-sabermetric revolution, baseball fans are conditioned to think about innings pitched as the denominator for all manner of pitching metrics – even those like strikeout and walk frequency for which plate appearances would make a much more logical denominator. (Of course, present day sabermetrics has embraced metrics like K% and W% for pitchers, but the per inning versions remain in use as well).

The parallel development of offensive statistics resulted in the opposite phenomenon. While early box scores tracked “hands out” (essentially outs made) for individual batters, batting average eventually became the dominant statistic. Setting aside the issues with “at bats” and how they distort people’s thinking and saddled us with the mouthful of “plate appearances” to describe the more fundamental quantity of the two, the standard batting statistics have conditioned fans to think about batting rates (walk rate, home run rate, etc.) in the correct manner (or one adjacent to being correct, depending on whether at bats or plate appearances are the denominator), but leave people struggling with how to properly express a batter’s overall productivity. Again, this is the opposite problem of how pitching statistics were traditionally constructed. One can imagine that it all might be very different had the Batting Average taken the form of a hit/out ratio rather than hits/at bats.

Rate Stat Series, pt. 2: PA Generation

This is a little bit of a detour and certainly nothing new (I don’t know who originally laid out this logic/math – the earliest use I’m aware of was in 1960 by D’Esopo & Lefkowitz as part of their Scoring Index model), but I think a discussion of it is appropriate in the context of this series, and I will later make use of these formulas.  It’s also ground I covered in the original series, but I think my explanation this time is slightly more coherent.

Each batting team starts each inning (excluding scenarios where a walkoff is possible) with three plate appearances guaranteed. Thus each team starts each game with twenty-seven plate appearances guaranteed (excluding scenarios where the home team forgoes batting the bottom of the ninth, rainouts, post-2020 doubleheaders, etc.). Any plate appearances beyond that must be earned by batters avoiding outs. Since it’s more natural to think of a positive outcome rather than the avoidance of a negative outcome, I will simplify and say that each extra plate appearance must be earned by a batter reaching base (and not being subsequently retired on the basepaths).

For the sake of discussion (and keeping with the simple set of statistics being used in the metrics in this series), I’m going to ignore the existence of baserunning outs, including caught stealing, pickoffs, outs stretching, outs advancing, and runners retired on double/triple plays (although not on fielder’s choices, since the batter is charged with an out in that case). I’m going to assume that the out rate is the complement of on base average, which in this series will be defined simply as (H + W)/(AB + W). In reality, considering all the ways in which outs can be made, it would be a more involved equation (I’ve used the acronym NOA for Not Out Average and OA for the complement, Out Average) which would look something like this, although it still doesn’t think I’ve accounted for every possible event (you try incorporating fielders’ choices without complicating the equation significantly):

NOA = (H + W + HB + CI + ROE – CS – DP – Outs Stretching – Outs Advancing – Pickoffs – 2*TP)/(AB + W + HB + SF + SH + CI)

Alternatively, for a team when LOB data is available (and ignoring the walkoff situation), you could have OA = (Plate Appearances – Runs Scored – Left On Base)/Plate Appearances. All of this is just an attempt to calculate, as best we can from the available statistics we have restricted ourselves to, Outs/Plate Appearances. NOA or OA as appropriate could be substituted for OBA in the equations that follow as long as the appropriate corresponding adjustments are made to the numerator.

Let’s assume for the purpose of developing an equation for team plate appearances that the OBA is constant across each of the nine batters in the lineup and doesn’t vary for any other reason (this is obviously never true, but it is a fine simplifying assumption for modeling PA generation). Then a team will start an inning with three plate appearances. For each of those three guaranteed PAs, there is a probability (equal to OBA, given our assumption) that the batter avoids an out (reaches base, given that there are no baserunning outs). This increases the expected number of plate appearances by OBA.

It doesn’t stop there, though. Each additional PA that is generated also has an OBA chance of creating an additional PA, which itself has an OBA chance of creating an additional PA. Thus, for each of the guaranteed PA, the expected final number of team PA is:

OBA + OBA*OBA + OBA*OBA*OBA + … = OBA + OBA^2 + OBA^3 + … OBA^n

which when n is infinity and OBA is between 0 and 1 (which it must be by definition) resolves to:

OBA/(1 – OBA)

The 1994 AL had an OBA of .343. Thus, each guaranteed plate appearance should have generated .343/(1 - .343) = .522 additional plate appearances. In an average inning, starting with three guaranteed PA, we would expect 3 + 3*.522 = 3*(1 + .522) = 4.566 PA, and thus in a game we would expect 9*4.566 = 41.09 PA. Note that instead of calculating the .522 additional PA, we can simplify this to 3/(1 – OBA) for an inning or 27/(1 – OBA) for a game. In reality there were 39.24 PA, so we have an unacceptable 4.7% error. What went wrong?

I’m mixing definitions of plate appearances and definitions of OBA incorrectly, and also ignored that the three guaranteed PA are equal to the number of outs permitted in the inning. In order to estimate the number of plate appearances per inning or game consistently, we need to divide the average number of outs/game by 1 – OBA:

PA/G = (O/G)/(1 – OBA)

The definition of outs that corresponds to our simple (H + W)/(AB + W) complement of out average is AB – H. In the 1994 AL there were 25.19 outs/game using this definition, so our expected PA/G is:

25.19/(1 - .353) = 38.34

The actual average was 38.35; we’re off due to rounding as this is now just a mathematical truism since by our simplified definitions plate appearances = outs + times on base. Using this equation to estimate team PA/G from their OBA for the 1994 AL, the RMSE is .259, which is about .7% of the average PA/G. We shouldn’t expect perfect accuracy at the team level since team PA will be affected by different quantities of all the statistical categories we’re ignoring that have an impact on the actual number of PA a team generates, as well as differences in number of extra inning games, foregone bottom of the ninths, and walkoff-shortened innings.

The key points to keep in mind as we move forward in discussing rate stats are:

1.      The number of plate appearances a team will get is a function of their out rate, and simplifying terms we can very accurately estimate team PA as a function of on base average

2.      Since players have an impact on the number of plate appearances their team gets, and thus the number of plate appearances they get, a proper rate stat for measuring overall offensive productivity must account for that impact

Almost Perfect

In my earlier days as a baseball fan, I was really interested in no-hitters, and outside of the Indians winning the World Series, my most fervent desire as a fan was to witness one even if only on the radio. Eventually this faded, due to some combination of growing jaded about the extent to which baseball fans sometimes elevate trivial events above game outcomes, the pernicious influence of Voros McCracken on how I thought about the hits column for pitchers, and after fifteen years of intense baseball-watching finally witnessing one (I'm now up to five).

Perfect games retain a bit more of their mystique for me, due to being much more rare (someone who has watched as many games over the years as I have is bound to have seen a no-hitter, but one can't really expect to see a perfect game) and not relying on any arbitrary distinction between hits and errors (which of course doesn't affect all no-hitters). The three closest games I have taken in to being perfect games prior to last night were Mike Mussina against the Indians in 1997 and Armando Galarraga's should-have been perfect game against the Indians in 2010. The latter game is case in point of what I meant about fans sometimes being more interested in trivial events than game outcomes - there was more outcry in favor of replay as a result of that game then there was cumulatively from many calls that much more directly influenced which team won a given game.

Last night's effort by Carlos Rodon combined elements of both of the ninth innings of these games in the way that people who believe in hocus pocus should embrace. From Galarraga's, we took the extremely close play at first base, with Josh Naylor playing the role of Jason Donald, desperately trying to reach first after making weak contract towards first base. In this case, the play was actually much closer, but no replay was required as the call on the field was that Jose Abreu beat him to the bag by a narrow margin. 

From the Mussina game, we borrowed the man, lineup slot, and fielding position to break it up. With one out in the ninth, the Indians catcher. Sandy Alomar singled off Mussina, while Roberto Perez was only hit in the back foot with a slider, but history repeated itself in who ended it. Of course, if Rodon had to lose the perfect game, he got the better outcome than the other two, as he at least got to keep the no-hitter.

Naturally, all of the near perfect games I've seen have been pitched against the Indians. In addition to the infinitely more important distinction of now having the longest World Series drought, after Joe Musgrove's no-hitter for the Padres, the Indians now have the longest drought between no-hitters, it having been nearly forty years since Len Barker's perfect game.

I was keeping score of the Mussina game and Rodon's effort last night, but not the Galarraga game, which I listened to on the radio while I watched some other game on TV.