SoSH Glossary: Linear Weights

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Linear weights is a broad term for systems of run estimators that assign run values to offensive events. These values, or weights, are based on the connections between events and the scoring of runs during a large number of actual games, and can be seen in the equations of popular statistics – wOBA and FIP for example.

Why Linear Weights?

In the 1980s, Pete Palmer expanded on the work of sabermetric pioneer George Lindsay and created the batting runs statistic, which was the first form of linear weights. Palmer wrote in The Hidden Game of Baseball:

The relationship of individual performance to team play is stated poorly or not at all in conventional baseball statistics. In Linear Weights it is crystal clear: The linear progression, the sum of the various offensive events, when weighted by their accurately predicted run values, will total the runs contributed by that batter or that team beyond the league average.

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The values calculated through linear weights and assigned to each offensive event are representative of the real world value of the event. A double is not really twice as good as a single, for example, but in the calculation for slugging percentage, it is given that much weight. Linear weights values a double around .75 runs and a single around .45, depending on the run environment.

How Are Linear Weights Calculated?

The first thing one needs to do is figure out the time frame that one wishes to use as the basis for the calculations; over the decades, as the dominance of pitchers has waxed and waned, the relative value of plays (singles versus home runs, for example) has changed slightly.  Next, one must create a run expectancy matrix for that time frame, because linear weights are based on the difference in run expectancy from the beginning to the end of a play:

Linear WeightsThe results for each play type are then added together and then divided by the total number of such events. So the run value for all the walks would be added and then divided by the total number of walks in the chosen time frame. The result would be the coefficient, and run value, for a walk in that linear weights model. This process is repeated for each event to find the run value for each event.

What Can Linear Weights Tell Us About the Game?

By awarding a player with the average number of runs his actions SHOULD have created, based on the linear weights model, we eliminate the effect of the lineup, providing a clearer picture of the player being evaluated. Looking at this chart of the average values of the linear weights taken from the bottom of this chart from Tom Tango’s site, we can learn some interesting things:

Event Runs Above Average
Put Out on Error -0.953
Outfield Assist -0.572
Caught Stealing -0.456
Strikeout -0.310
Out -0.299
Pickoff -0.255
Bunt -0.001
Foul Error 0.026
Defensive Indifference 0.063
Intentional Walk 0.102
Stolen Base 0.195
Balk 0.237
Wild Pitch 0.284
Passed Ball 0.285
Non-intentional Walk 0.330
Hit by Pitch 0.385
Single 0.474
Reach Base on Error 0.546
Double 0.764
Triple 1.063
Home Run 1.409

The origins of some sabermetric idioms come to light here. For instance, a stolen base is only worth 0.195 runs, while a runner being caught stealing costs an offense 0.456 runs. It’s often stated that a runner needs to have a stolen base rate over 70% to be successful, and using linear weights we can demonstrate why. A baserunner who steals 71 bases in 100 attempts will gain 71*0.195 = 13.845 runs, but will also lose his team 29*0.456 = 13.224 runs, and so the net result is positive – 0.621 runs – but just barely.

Here are some other observations from the table:

  • A home run is worth nearly three times as much as a single, not the four times that slugging percentage and ISO value it at.
  • A walk is nearly the inverse of a strikeout.
  • Bunts have a negative run value.
  • Intentional walks generally help the offense.

Follow Pete on Twitter @PeterWHodges.