# What’s the most important at-bat in an inning?

The first hitter in the inning sets the tone for the rest of the frame.  Right?  I love sportscasters for one reason.  Whenever I need an idea on what to write here at StatSpeak, I listen to a baseball game on MLB’s Gameday Audio.  Before too long, somebody says something of dubious validity that I feel the need to test against the evidence.  The other day, I heard a whopper.  “The most important hitter in the inning is the leadoff hitter.”  Surely, the leadoff hitter in the inning can do something to help the team (he could walk, hit a home run, or just walk slowly to the plate to allow the team’s radio network 10 more seconds of advertising time), but is his at-bat the most important?
Well, first let’s define important.  The job of the team at bat is to score runs, so the leadoff situation (nobody out, nobody on), if it is that important, should be vital in determining whether or not a run scores in the inning or not.  The leadoff hitter could ensure that a run scores by hitting a home run, but in the absence of that, he could do such useful things as not making an out.  In other words, he should ideally do something that increases, rather than decreases the chances that his team will score in the inning, like lacing a single to right field.  Come to think of it, all hitters are charged with the same task.  Some succeed, some fail.  How important the situation is depends on how big a swing in those chances might happen.
Those of you familiar with Sabermetrics can probably see where I’m going with this.  For those of you who aren’t (or are just confused by my writing), I’m suggesting a modification of what is commonly known as the leverage index.  Leverage is the idea that during a game, some points are more influential on the outcome, in terms of winning and losing than other points.  The leverage index is a way of telling us how important it is.  I’m proposing something along the same lines, but instead looking at the construct within an inning.
There are two factors that determine the general run-scoring chances of a team during an inning: outs and runners.  There are eight possible configurations of runners and three possible numbers of outs, for a total of 24 states.  We can look back through a season (in my case, 2006) of data and see what the chances were that, given a particular state of the inning, a team would succeed in its primary goal of scoring more runs.  For example, from the state of a runner on first and no one out, batting team scored at least one run in the inning 44% of the time.  This is up from the “leadoff” state of no runners and no outs, where the batting team has a 29% chance of scoring.  (Note that not all “leadoff” at bats actually lead off the inning.  If the first hitter in an inning hits a home run, the next batter also hits with no runners on and no outs.)  So, by drawing a walk or singling or getting hit by a pitch, the leadoff hitter has increased the chances of his team scoring a run by 15%.  We can calculate the change in probability for all at-bats over a season, and see which situation, on average, had the most at stake in determining the probability of whether a run would score in the inning.
Gritty details: I used the 2006 PBP data from Retrosheet, and calculated from each at bat whether there were any more runs to be scored in the inning.  For example, a leadoff home run followed by three outs would be counted as one at-bat in the 0 outs, no runners with a run scored (on the home run), followed by one at-bat in the same state with no more runs scoring.  I calculated the average probability of any more runs scoring, whether one more or twelve more, from that state, both before the plate appearance and after.  All home halves of the ninth or later innings were excluded, as were baserunning events such as stolen bases or caught stealings.  The probability of scoring further runs after a plate appearance in which the third out of the inning was made was set to zero, for obvious reasons.  Run-scoring probability after a plate appearance for plate appearances in which a run actually scored were set to 100%.  That is, if a batter singled in a runner from second with no one out, he saw his team’s chances of scoring more runs jump from 63% to 100% in his at-bat.  However, the next batter, batting in the state of runner on first, no one out, will start with a team run chance of 44%.  I took the difference in probability from before the plate appearance to after and used a root mean squared of the difference method to generate how much, on average, probability changed from each state.  Then, I divided by the weighted mean RMS to generate a standardized (mean = 1.00)  leverage value for each state.  A value over 1.00 means that the state is more important than the average plate appearance in the inning and a value of 1.50 means the plate appearance is 1.5 times as important.
The most important at-bat in an inning?  The one that takes place with 2 outs and the bases loaded, with an inning run leverage value of 1.99.  Part of this is due to my methodology.  In that particular case, the only possible outcomes of any plate appearance are that either a run will score (in which case the probability afterward will become 1.00) or the third out will be made (in which case the probability will become 0).  But still, the top three spaces on the chart featured situations in which there were two outs and a runner on third.  The leadoff situation?  A leverage value of .82.  Not only is the leadoff hitter not the inning’s most important hitter, he’s not even above average in importance.  Even if he makes an out, the team’s run scoring probability only drops eleven percentage points from 29% to 18%.  Sure, it’s a drop, but there are other situations in which the stakes are much higher.  On looking through the list, which I’ve appended to the end of this post, one thing becomes very clear.  The most important plate appearances in the inning are the ones that take place with at least one out and runners in scoring position.
Like anything else, this is a work in progress and I welcome your critiques.  Full chart is behind the cut.

Leverage Value Situatuion
1.74  2 outs, runners on 1st & 3rd
1.68  2 outs, runner on 3rd
1.62  1 outs, runners on 1st & 3rd
1.61  2 outs, runners on 2nd & 3rd
1.60  2 outs, runners on 1st & 2nd
1.56  2 outs, runner on 2nd
1.45  1 outs, runner on 3rd
1.39  1 outs, runners on 1st & 2nd
1.34  1 outs, runners on 2nd & 3rd
1.14  1 outs, runner on 2nd
1.06  0 outs, runner on 1st
1.04  1 outs, runner on 1st
1.04  0 outs, runners on 1st & 2nd
1.00  2 outs, runner on 1st
0.87  0 outs, runner on 2nd
0.82  0 outs, no runners
0.76  1 outs, no runners
0.76  0 outs, runners on 1st & 3rd
0.72  2 outs, no runners
0.71  0 outs, runner on 3rd
0.65  0 outs, runners on 2nd & 3rd

### 4 Responses to What’s the most important at-bat in an inning?

1. Cory Humes says:

Yesterday, Tim McCarver said “more often than not, when multiple runs are scored in an inning, a walk is involved.” I just nodded and smiled.

2. Pizza Cutter says:

I do that a lot when Tim McCarver is involved. Maybe I’ll take a quick look at that when I need to procrastinate on actually working on my dissertation. (Which is really all I do on this site.)

3. Sean Smith says:

There’s some truth to the statement. If nobody is on base, the leadoff batter has the highest leverage index.
The more baserunners, the greater the LI, obviously, but first you’ve got to get those baserunners.

4. Kyle J says:

You’d probably really need to look at the weighted average of different situations for the first, second, and third hitters in an inning, right?
The statement relates to which hitter is most important, not which stitution is most important. I suspect you’ll arrive at the conlusion Sean does through a simpler methodology–all things being equal, the first hitter is most important.