Relief pitchers can enter the game in very different situations, and that should be accounted for when evaluating their performance. Coming in to clean up a 12-0 victory (or defeat) is different than coming in during a close, tense game with a lot on the line. We measure this disparity using Leverage Index (LI).
Intuitively, we want LI to measure how the game “hangs in the balance”: how likely is it that the game could change for the worse if the new pitcher does a poor job?
Luckily we’ve already discussed Win Expectancy, which tangentially addresses this question. When a new pitcher enters the game, we want to compute all the possible changes in Win Expectancy as a result of the first batter they face. Not only that, we need to weight those changes by how likely they are to happen. So, we find the probability that each event could happen, and the resulting change in Win Expectancy were the event to happen, and multiply them together; if we add up all these values, we get a measure of how much win expectancy change we’d get on average in this exact situation.1For those of you with any background in probability, we’re just computing the expected value of the change in win expectancy.
However, we would like this normalized so we know what an “average” situation is. So, we find the average change in Win Expectancy across all situations in the league, and that becomes our divisor.
\text{LI} = \frac{\sum\limits_{\text{all events}} \text{P}(\text{event})\times \text{WE Change}(\text{event})}{\text{avg. WE Change}}As such, an average LI value is 1; a high leverage situation is greater than 1, while a low-leverage situation is between 0 and 1.
When reporting on LI for a pitcher, we typically look at the average LI across their appearances.2Note that there are multiple ways of calculating LI. We don’t have to just look at the LI when a pitcher enters the game; we can instead consider their average LI across all at-bats. Similarly, we could measure LI for pinch-hit appearances. For example, Zack Britton had an average game-entering LI of 1.66 in 2016.
Leverage Index is an interesting statistic for relievers in its own right, but it will mainly come into play tomorrow when we discuss pitcher WAR, where we want to accurately reflect the relative value of starters and relievers.
Let’s go back to a few statistics.
- Mariano Rivera had a tense start to the 21st century. In 2001 and 2002, his average game-entering LI was 2.02 and 2.06 respectively.
- In 2008, Francisco Rodriguez set the MLB record with 62 saves in 1 season. Across 76 appearances, his average LI3Baseball Reference doesn’t make it clear if they’re using game-entering LI, or average across all batters faced. was 2.55. That’s the third-highest all time in a season since 1969.
Continue to Day 20 – Pitcher WAR
- 1For those of you with any background in probability, we’re just computing the expected value of the change in win expectancy.
- 2Note that there are multiple ways of calculating LI. We don’t have to just look at the LI when a pitcher enters the game; we can instead consider their average LI across all at-bats. Similarly, we could measure LI for pinch-hit appearances.
- 3Baseball Reference doesn’t make it clear if they’re using game-entering LI, or average across all batters faced.