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Linear Weights and Dividing Offensive Credit

Last week, I discussed the two questions you should ask when you are trying to evaluate any problem with numbers, whether in the world of baseball or the so-called "real world" (like there's a real world outside of baseball). Here is what I said about evaluating a player's total performance.

As a a result, the takeaway here is the following: if you want to measure how productive a player is (and this is generally what people mean when they ask about how "good" a player is), you need to find a measure that converts all of his offensive and defensive performances into runs, which can then be converted into wins.

I then said that we would tackle how to approach turning a player's production into runs this week, and that is exactly what we'll be doing here. But why do we need a "sabermetric" method for dividing up runs? Why can we not just use the classic staple of runs and RBI in this case? After all, they have the right units, right?

In order to clarify this question, we need to go back to our previous discussion and ask those two pertinent questions again: 

1. Always ask yourself what question am I trying to answer and what stat can I use to answer that question before discussing.

Let's apply these to a basic baseball situation.

Side Note: An Example of Why RBI and Runs Are Limited

Consider this set of plate appearances occurring in the first inning of a Marlins game.

1. Emilio Bonifacio leads off with a single up the middle.

2. Omar Infante doubles down the left field line. The ball hangs up in the air a bit, so Bonifacio had to hold just long enough that he ends up at third base instead of scoring.

3. Hanley Ramirez hits a sacrifice fly, scoring Bonifacio from third.

The question is, who gets credit for this run and how much credit?

If you go by runs and RBI, you would presumably give half the credit to Bonifacio and the other half to Ramirez, since those are the only two receiving run stats for this set of circumstances. But does that tell you everything that happened? Bonifacio ended up on base with a single, but Infante was critical to move him to third base on the double. However, when you use runs and RBI, Infante receives no credit for that run, despite the fact that he was very important in moving the runner into position to score.

This gets to the point of what runs and RBI measure. Yes, they measure runs scored and driven in, but there are more factors to those numbers than meets the eye. Runs scored encompass not only a player's ability to get on base, but also his team's ability to drive him in. Similarly, RBI is dependent not only on a player's ability to move runners down the bases with hits, but also his team's ability to provide runners for him to move. In other words, there is little separation between the individual and team aspects of runs scored and RBI. If we want to evaluate the individual, we cannot be using stats that encompass the team's performance as well.

Note that this does not mean that runs and RBI are "useless." Like all statistics, they do tell you something, and they can tell a story. As is the case with many stats, it is not the stat's fault, but rather the misuse of the stat that makes it look bad. These stats can tell you a lot about the type of player you are dealing with and the quality of the team surrounding him in the lineup, but they cannot determine how much that player has produced.

Enter Linear Weights

So our goal in this case is to isolate each player's performance from his team and give that performance a run value. I do not want to give Infante extra credit for hitting a double just because his teammate is on base, and similarly I do not want to credit Ramirez more for making an out just because his teammate was on third; neither of these situations were in an individual player's control.

How can we resolve that? One way to handle this is to give the player an average amount of credit for each play. Every base-out state (for example, runners on second and third with no outs, or runner on first with no outs, or runner on second with one out) has a run expectancy, a value of runs that is expected to score based on empirical evidence (for example, how many runs have scored after a second and third with no outs situation). One can do this with play-by-play data from any era. For example, let us use this table by Tom Tango for the 1999 to 2002 era.


RE 99-02 0 1 2
Empty 0.555 0.297 0.117
1st 0.953 0.573 0.251
2nd 1.189 0.725 0.344
3rd 1.482 0.983 0.387
1st_2nd 1.573 0.971 0.466
1st_3rd 1.904 1.243 0.538
2nd_3rd 2.052 1.467 0.634
Loaded 2.417 1.65 0.815

Note: The columns are the number of outs, while the rows indicate the base situation.

Let's go back to our previous situation. Going from the start of the inning to after Bonifacio's single, we see that the Marlins would have increased their expected runs scored from 0.55 runs to 0.95 runs. This means that, in this situation, Bonifacio's single was worth 0.4 runs. Then, Infante's double changed the run expectancy from 0.95 to 2.05, meaning in this case that Infante's double added 1.1 runs. Finally, Ramirez's sacrifice fly scored a run, meaning we automatically add one run to his total, but it changed the state to runners on second with one out (0.73 run expectancy). This means that in total, the change from runners on second and third with no outs and runner on second with one out and a run scored was worth -0.3 runs (yes, that means Ramirez hurt the team's chances of scoring a run by driving in that run via a sacrifice fly).

But remember, we wanted to give credit isolated from teammates. So what we can do is average all of the changes in run expectancy resulting from singles, doubles, home runs, walks, and other basic events. That is, take all the changes in base-out state that resulted from a single, measure the run expectancy change from each single, and take the average. This gives a player credit for the average event, rather than credit him for what his team provided in terms of runners on base or teammate ability to drive runners in. It eliminates the effect of the teammates surrounding the player in the lineup and isolates only that player's production.

What do linear weights look like in terms of run values for each event? You can see an example for the 1992 to 2002 era at the end of this table by Tom Tango. Here it is summarized:


Event Runs Above Average
Non-intentional BB 0.33
Hit-by-pitch 0.39
Single 0.47
Double 0.76
Triple 1.06
Home Run 1.41
Stolen Base 0.20
Caught Stealing -0.46
Strikeout -0.31
Other Outs -0.30

These values, measured for the current run environment, are the basis for the weights found in wOBA. Therefore, wOBA more properly weighs the value of different hits and steals than OPS does, making it a more accurate offensive stat. As you can see, there are numerous interesting questions that are resolved by this table:

- One can see the actual relative value of an average home runs versus an average single. SLG measures a 4:1 ratio in difference, while linear weights shows that the difference is more like 3:1.

- A triple is essentially worth a run when you take into account the average number of baserunners aboard when a triple is hit. A home run is always worth at least one run, but the average number of baserunners aboard when a home run is hit adds 0.4 runs to that value.

- The value of a walk is very similar to the value of an out, which makes total sense seeing as though they are polar opposites of each other (a walk is the least valuable non-out event at the plate).

- The value of a stolen base is significantly less than the value of a caught stealing. Outs on the bases are significantly more damaging than base advances.

These little nuances help in a lot of different analyses, but one thing this allows us to do is properly evaluate, in terms of runs above average, the performance of a given hitter, not only in quality but in quantity. For example, in 2011 Jose Bautista led the majors in wOBA with a .436 mark. He was the best hitter in terms of rate, but because he missed some playing time, we can see that he was actually just outproduced by Miguel Cabrera by about one run above average.

This was a quick and dirty explanation about the methodology and the reasoning behind why we use linear weights. Any questions, leave them in the comments section and I'd be happy to try and address them. Getting this concept really helps to understand valuing offensive performance.