Leverage Index by inning

Here’s a little something you may not have known: on average, games get less critical as the game progresses. In other words, on average for all games, the ninth inning is actually less critical and tense than the first when measured correctly.

To illustrate the point, here is a table of the average Leverage Index of a play from the first through the ninth innings in the 2013 season, broken out by the top and bottom of each inning. Remember, all games go through the top of the ninth (barring a rainout), but games progress to the bottom of the ninth only when the home team is tied or behind. So you need to examine just the top of the inning to be sure you are capturing the true trend of a game:

          1       2       3       4       5       6       7       8       9
Top     0.96    0.96    0.94    0.95    0.95    0.95    0.97    0.98    0.92
Bottom  0.97    0.94    0.94    0.96    0.93    0.95    0.95    0.96    1.74

If you look at just the top of the inning, the average LI of a game actually creeps up a little in the seventh and eighth innings but then drops in the top of the ninth. This was true in 2013 and 2012, and I’m pretty sure you’ll find it’s true in all seasons. Close games raise the LI of a play in later innings, but runaway games lower it and, on average, as measured by LI, the runaway games have a bigger impact than the close ones.

If you combine the top and bottom of all innings, you get this:

           1      2      3      4      5      6      7      8      9     10
All     0.97   0.95   0.94   0.95   0.94   0.95   0.96   0.97   1.21   2.44

(I threw in the 10th inning for the heck of it.) When you look at it this way, yes, games get more critical as they progress, but now you can see that this occurs only because of the selection bias of including the bottom of the ninth.

Why is there a selection bias when including the bottom of the ninth? First of all, the highest LIs tend to come in the bottom of the ninth, when the game is more clearly coming to an end from which there is no return. Said differently, the home team always has a chance to come back from a key play in the top of the ninth, but the visiting team has no chance of coming back against a key play in the bottom of the ninth. Hence, the bottom of the inning is naturally more critical.

Secondly, while there may be a relatively even balance between close games and blowaway games for the home and visiting teams, tie games (which tend to have higher LIs) are included in both innings. This will raise the average LI of the ninth in the very first play of the inning, everything else being equal.*

*Note: I looked it up. The average LI of the very first play of the bottom of the ninth inning was 1.48 in 2013, compared to 0.87 for the very first play in the top of the ninth.

Last fun fact: a majority (56 percent) of games go into the bottom of the ninth, even though home teams win 54 percent of the time. If I’m interpreting that correctly, it means that a little less than 20 percent (10/54) of home team wins are decided in the bottom of the ninth or later. Is my math correct?

Tom M. Tango asked for some follow-up information, so I’m going to post it here and on his blog. In 90% of all 2013 games, the Leverage Index rose about 2 at least once. Here is the percent of times it rose above two, by half innings

           1      2      3      4      5      6      7      8      9     10
Top        9%    13%    12%    14%    18%    21%    23%    26%    29%   100%
Bottom    10%    12%    14%    18%    19%    23%    23%    26%    51%    90%

Dave Studeman was called a "national treasure" by Rob Neyer. Seriously. Follow his sporadic tweets @dastudes.
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9 years ago

Interesting.  How are you measuring the LI for each half inning?  Do you take an average LI for the entire half inning?  Do you take the max LI for the entire half inning?  The initial LI for the entire half inning?  Or something else?

vr, Xei

9 years ago

I took the average initial LI of all plays in an inning.

9 years ago

Point of clarification request: when you say you’re averaging the LI of all plays, what is the definition of “play”?

If a runner tries to steal while a batter is up, would the steal attempt be a first play, and the result of the plate apperance a second play?

What about pickoff attempts? Are those all counted as plays, are none counted, or are only those resulting in an out counted as distinct plays?

9 years ago

A play is an event with an outcome. So a stolen base is a play, and the outcome of the resulting plate appearance is a separate outcome. A pickoff attempt is counted as a play if there is an out or the runner advances.

You can take a look at the game logs at Fangraphs or Baseball Reference to see how this works.

8 years ago

(Comment nearly 1 year after the post)

The bottom of the 9th LI can be “fixed” by including a 0 for all the games that end in the top of the 9th. Based on what’s in your post, if 56% of games get to the bottom of the 9th, the LI for that half-inning should be multiplied by 0.56 (since the other 0.44 has an LI of 0). So the “fixed” bottom of the 9th LI is: 0.974. Very much in line with the rest.

You could similarly “fix” the 10th if you have the percentage of games that actually get to that inning.