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Leverage measures how important a given situation is during a game (relative to the start of the game). The start of a game is defined as a Leverage of 1.0, a neutral situation. As the game progresses, the Leverage can fluctuate based on the inning, the number of outs, the runners on base, and the difference between the two teams' runs scored so far.
Leverage is the ratio between how much a single run scored changes the expected probability of winning in the current situation and how much a run would have changed the expected probability of winning at the very beginning of the game.
For example, if a run scored in the eighth inning increases the probability of winning by 20%, while a run at the start of the game increases it by 10%, then the Leverage of the situation in the eighth inning is 20%/10% = 2.00 Leverage.
APR, or Adjusted Pitching Runs, measures the number of runs a pitcher prevented as compared to a league-average pitcher in a neutral park in the same number of innings. Derived from the pitching component of linear weights, or pitching runs, APR includes an adjustment for ballparks and is based on all runs allowed, not just the earned runs.
APR = L x IP - (R/PF)
Where "L" is the league average of runs per inning and "PF" is the park factor for the pitcher's home park.