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March 8, 2007

Schrodinger's Bat

Using Your Leverage

by Dan Fox

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"Pressure is a word that is misused in our vocabulary. When you start thinking of pressure, it's because you've started to think of failure."

- former Dodgers manager Tommy Lasorda on what are today known as "high-leverage" situations

Last week's column detailing the leaders and trailers in Win Expectancy (WX) and Win Expectancy Clutch (WXC) generated plenty of feedback. This week, in an effort to sate the curiosity of our readers, we'll delve a little deeper into the subject by looking at WX from an aggregate perspective.

Leveraging Your Opportunities

Perhaps the most frequently asked question concerned the concept of Leverage.

As mentioned last time, Leverage is calculated as the ratio of the impact of one additional run at a point in time to the impact of one additional run at the beginning of the game averaged over all plate appearances.

So for example, on June 21st of last season the Mets held a 5-4 lead over the Reds at Shea Stadium with two outs in the top of the ninth inning. The bases were loaded as Brandon Phillips stepped in against the Mets closer Billy Wagner. At the point, given the score, inning, base state and run environment, the Reds theoretically had just a 23% of winning the game. If Phillips were to come through and the Reds were to score an additional run, their chances would improve to 53.6% for a net gain of 30.6%. To calculate the Leverage of the situation under the definition given above, we then take that 30.6% or .306 and divide it by the change in WX that would have occurred had the Reds scored an additional run at the beginning of the game. That turns out to be 10.2%, or .102 with the resulting Leverage score coming out almost exactly at 3.0.

As several of our more sabermetrically-oriented readers noted, there are alternate ways to calculate leverage, with a good explanation of one technique given by Tom Tango last year. Alas, a more robust discussion of the pros and cons will have to wait for another day.

Incidentally, Phillips lined a single to center field scoring two runs which held up in the Reds 6-5 win.

Given this definition, other readers wanted to know if Leverage could throw any insight on the perennially thorny issue of lineup construction. The following table and associated graph shows the average Leverage by lineup position across all games in 2006.

Lineup Pos   Avg Lev
1              1.016
2              1.018
3              1.018
4              1.012
5              1.009
6              1.002
7              1.011
8              1.010
9              1.013

image 1

Here we see that hitting in the third spot in the order affords the highest average leverage across all plate appearances. Intuitively, this makes a certain amount of sense. In the first inning, Leverage is actually relatively high since the score is often tied (making the impact of a single run greater) and so one would expect the first three lineup positions to score highly. As runs are scored before the fifth and sixth hitters take their first turns at bat and as the score differential increases in the latter parts of the game, their ability to influence the game with an additional run, and hence their Leverage, goes down.

There are two points to keep in mind, however. First, the difference between the third spot in the order at 1.018 and the sixth spot at 1.002 represents a change in WX of a well under 1% drop, as is reinforced when you notice that the scale of the y-axis increases in increments of .002. Second, since the first spot in the lineup actually garners 4.4% more plate appearances than the third spot, the weighted Leverage of the leadoff position--as well as the second spot in the order--is greater than that of the third spot. Similarly, even though the ninth spot has the fourth highest Leverage, when weighted by plate appearances it ranks last. The end result is that Leverage is simply not a useful tool in lineup construction because the contingencies of the game tend to even out the opportunities. This result was hinted at last week when we noted that players with lots of plate appearances have Leverage scores that are very closely clustered. And as many analysts have pointed out in the past, short of batting your very weakest hitters at the top of the order, the impact of lineup construction is, for the most part, minimal.

The above discussion, though, begs the question of how the inning influences Leverage. Perhaps counter-intuitively (see the table and graph below), as the game progresses Leverage scores do not generally rise until you reach the ninth inning. In fact, Leverage scores steadily decrease from innings one through five before rising again in innings six and seven, dipping in the eighth inning and then skyrocketing in the tenth inning and beyond.

Inning       Avg Lev
1              0.961
2              0.960
3              0.954
4              0.943
5              0.941
6              0.945
7              0.948
8              0.937
9              1.213
10             3.273
11             3.208
12             3.294
13             3.323
14             3.438
15             3.841
16             3.855

image 1

Leverage scores tend to increase in the eighth and ninth innings because an additional run at that point pushes a team much closer to winning on account of the scarcity of outs remaining. Obviously in extra innings the score is tied during the vast of the plate appearances and so an additional run almost always ensures a victory.

And as you might have guessed, that leads us to the question of what Leverage looks like as a function of run differential. This time your intuition would be correct if you guessed that as the difference in the score increases the Leverage decreases exponentially.

Diff         Avg Lev
0              1.496
1              1.439
2              0.981
3              0.669
4              0.434
5              0.265
6              0.157
7              0.094
8              0.048
9              0.029
10             0.018
11             0.006
12             0.003
13             0.001
14             0.000
15             0.001
image 1

By the time the difference in score reaches five runs, the Leverage is less than a fifth of what it was when the differential was at one. At its core this is a reflection of the fact that while moving from a tie to a one-run lead is obviously important, the impact (in terms of the probability of winning a game) of moving from a one-run lead to a two-run lead (and to a lesser extent from a two-run to a three-run lead, and a three-run to a four-run lead) is much greater.

Perhaps an internalization of this fact was exemplified by new Cubs manager Lou Piniella last weekend when he told Cubs broadcasters Len Casper and Bob Brenly that while he doesn't like bunting in the early innings, he will bunt when up by one or more runs in order to push across that additional run that makes it more difficult for the opposing team to mount a comeback.

The dichotomy between the graph of Leverage by inning and the one above showing it by run differential is what underlies the gospel preached by many analysts (and our own Keith Woolner in Baseball Between the Numbers) that closer usage patterns could certainly be better optimized, since it isn't so much the inning that matters but the other variables of the game situation.

What Does it Take to Win?

After last week's column I proceeded to post all of the WX1, WX and WXC scores for 2006 on my blog. A particularly attentive reader pointed out that when WX is summed over all the players you end up with a total of…123 runs. At first glance this appears incorrect since theoretically all of the WX values assigned to individuals when summed should total to zero since a change in WX credited to one player should be debited from another.

What I neglected to mention last week is that in this calculation of WX all non-batter events were excluded. Specifically, this means stolen base attempts, passed balls, balks, other advancement, errors on foul balls, other fielding errors, defensive indifference, pickoffs and wild pitches. In all those make up approximately 7,500, or less than 4%, of the over 193,000 total events recorded for the 2006 season. When these are added into the mix the aggregate WX comes in at -0.91, or less than one win.

This observation leads one to a question. Just how impactful are the various categories of events in terms of winning and losing? To answer that question (at least for 2006 in total, since the WX for any individual event varies according to the score, base state, and run environment--as shown above) the following table and graph show the various events that are tracked in our play by play database, the number of those events, their total impact on winning and losing, the average change in WX per event, and the average Leverage at the time of the event.

Event         Number    Wins   WX/Ev Avg Lev
Home Run        5383   679.7   12.6%    0.99
Triple           951    88.6    9.3%    0.98
Double          9129   602.1    6.6%    0.98
Error           1716    75.7    4.4%    1.00
Single         29580  1207.5    4.1%    1.01
HBP             1815    50.8    2.8%    1.00
Balk             145     4.0    2.7%    0.94
BB             14428   384.4    2.7%    1.03
WP              1376    36.1    2.6%    1.00
PB               278     7.3    2.6%    0.96
INT               19     0.5    2.4%    0.89
SB              2417    43.5    1.8%    1.10
IBB             1409    23.9    1.7%    1.25
DI               207     0.4    0.2%    0.49
Foul Error        44     0.0    0.0%    0.83
FC               512    -6.8   -1.3%    0.98
Pick Off         471   -10.4   -2.2%    1.12
Other Out      91379 -2311.1   -2.5%    1.00
K              31635  -842.6   -2.7%    1.03
CS               758   -31.8   -4.2%    1.08
Other Advance     54    -2.7   -5.0%    1.10

image 1

It should come as no surprise that home runs have the highest impact, with triples and doubles coming in second and third. A home run on average pushes the beneficiary over 12% towards a win. Another way of looking at it is that every 100 home runs are worth 12 ˝ wins. The fact that extra-base hits are at the top is not surprising, although errors ranking before singles may at first be confusing. It turns out this is the case since singles are typically limited to the batter only reaching first base, while errors can result in the batter advancing to second and beyond and also occur somewhat disproportionately with runners on base. In a similar fashion, unintentional walks have a greater impact than intentional walks since the former often advances a runner while the latter does not.

On the opposite side of the ledger, a fielder's choice typically results in an out on a forced runner and so it is slightly negative overall (-1.3%), while strikeouts (-2.7%) are slightly more costly than other outs (-2.5%), and caught stealing (-4.2%) are much more costly than either. Pickoffs, on the other hand, are less costly than caught stealings since they include both pickoffs that are successful and the 30% that result in errors by the defense. The category of "other advancement" ends up being the most costly, since the vast majority record runners thrown out attempting to advance after a batted ball, which is especially costly when runners are thrown out attempting to score.

From a Leverage perspective it is not surprising that defensive indifference ranks the lowest (0.49) since it occurs only in non-crucial situations. Nor is it a shock that the intentional walk (1.25) has the highest, since it is used when runners are on base, and the score is tight.

In a Pinch

Finally, one reader wondered how Leverage and WX can be applied to pinch-hitting. The following table summarizes the number of pinch-hitting plate appearances, the average Leverage for those appearances, the total change in WX as a result, and the change in WX per PA broken down by team for 2006.

Team              PA Avg Lev      WX   WX/PA
CIN              273    1.32    1.12    0.4%
WAS              314    1.15    1.08    0.3%
KCA               91    1.40    0.29    0.3%
SDN              258    1.20    0.16    0.1%
LAN              286    1.13   -0.30   -0.1%
MIL              235    1.22   -0.34   -0.1%
FLO              247    1.10   -0.57   -0.2%
SLN              271    1.27   -0.80   -0.3%
DET               81    1.21   -0.24   -0.3%
BOS               93    1.36   -0.29   -0.3%
SFN              210    1.33   -0.70   -0.3%
CHN              268    1.18   -0.90   -0.3%
CHA              135    1.18   -0.52   -0.4%
LAA              101    1.37   -0.43   -0.4%
ATL              300    1.16   -1.29   -0.4%
COL              258    1.28   -1.21   -0.5%
NYA              107    1.26   -0.70   -0.7%
TOR              113    1.00   -0.78   -0.7%
HOU              286    1.34   -2.16   -0.8%
ARI              275    1.23   -2.12   -0.8%
NYN              246    1.17   -1.96   -0.8%
CLE               96    1.25   -0.79   -0.8%
MIN               92    1.23   -0.79   -0.9%
SEA              117    1.31   -1.17   -1.0%
PIT              262    1.44   -2.64   -1.0%
TBA               80    1.17   -1.01   -1.3%
PHI              297    1.22   -4.78   -1.6%
BAL               72    1.17   -1.22   -1.7%
TEX               39    1.09   -0.80   -2.1%
OAK               62    1.26   -1.53   -2.5%

From this. two things are evident. First, most teams don't come out real well in pinch hitting, although the Reds, Nationals, Royals and Padres at least registered a net positive impact. The Reds took the top spot by registering three of the five largest impact pinch-hitting plate appearances in 2006:

Overall, the Reds recorded an astounding 13 pinch-hit home runs and 65 pinch-hits in 2006. Even so, teams don't do very well pinch-hitting, generally speaking, for two reasons. Pinch-hitters are typically not very good hitters to begin with and there is a "penalty" of sorts built into pinch-hitting where hitters perform more poorly in those situations than they otherwise would (even correcting for the quality of the pitching).

In addition, the average Leverage in pinch-hitting situations is higher than other plate appearances averaging out at 1.23 (the average over all plate appearances is 1.01), which is slightly higher than the average Leverage in all ninth inning plate appearances in 2006.

In the end, this is yet another confirmation of the Earl Weaver adage that when given the choice of populating the bench with an additional productive hitter or another (12th in the modern game) pitcher, go with the hitter. As a Cubs fan, this provides another small ray of hope. After several years of enduring a bench that couldn't hit their way out of a paper bag, Jim Hendry has added Daryle Ward to the roster, who had 22 pinch hits (four of them homers) in 2006 in 75 plate appearances.

Note: Several readers have asked whether the baserunning numbers I've written about are available on the site. I'm happy to report that subscribers can download the spreadsheet here and that it includes not only the individual and team totals for 2006 but all of the individual totals for 2000 through 2006. Enjoy.

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