One of the truly under-studied areas of baseball theory is pitch sequencing, the idea that pitches earlier in the at-bat influence hitter behavior against pitches later in the count. Effective Velocity lays out a basic theory behind a potential pitch sequencing strategy. The core tenet of EV is that hitters swing naturally late against the fastball up-and-in, and naturally early against the off-speed pitch down-and-away, due to the bat path required to make quality barrel contact against pitches in these locations. Hitters realize this, and time their swing earlier or later based on the speed and location of the previous pitch. The pitcher can hence maximize his effectiveness by mixing up the effective velocity of his pitches such that the hitter will stay off balance. For example, if the last pitch was low and slow, the hitter will adjust to swing later on the following pitch - which makes them vulnerable to getting jammed against the fastball up-and-in.

To see if this theory is valid as it applies to sequencing, we can look at how the up-and-in fastball performs, based on the characteristics of the current pitch and the previous pitch. If the EV theory is correct we'd expect this pitch to perform better if the last pitch was slow, down, or away, as such pitches would influence the hitter's approach, causing them to get jammed more often.

We start our study by defining what "performance" is. We use Statcast wOBA, but rather than using the wOBA that resulted from the current pitch - which would exclude those pitches that did not end the plate appearance - we use a modified wOBA metric that measures the change in wOBA expectation from pitches that are taken for balls and strikes, as well as those that ended the plate appearance on balls in play. We define this change by finding the average wOBA value of the plate appearance for all plate appearances that began in a given count. For example, the average wOBA value starting from 0-0, according to our table below and using the Statcast wOBA scale, is 0.329. If the hitter takes ball one to go to 1-0, all plate appearances that start from 1-0 have an eventual wOBA value of 0.370. The 0-0 pitch would then result in a 41-point wOBA gain to the hitter as their expectation moves from 0.329 to 0.370. And if the hitter takes a strike to drop to 0-1, their expectation would fall from 0.329 to 0.279, a 50-point loss. We show the average wOBA value for all counts in the below table:

The value of each pitch is then calculated as in the below example, which started in a 0-1 count:

If the pitch ended the plate appearance, such as the single in the above example, we use the wOBA of the outcome minus the starting wOBA of that count to find the value of the pitch. Otherwise, we use the change in average wOBA by count. Note that this metric is centered around 0 - since all plate appearances that start in a 0-1 count result in an average wOBA of 0.279, the combination of all outcomes will have an average of 0.279, resulting in an average net change of 0. The benefit of this metric is that we can consider the gains/losses to the pitcher and hitter from getting ahead or behind in the count in addition to what happens when a PA-ending event takes place.

With our metric defined we then must define what an "up and in fastball" is. We define "up" as anything within one foot below the top of the official strike zone, as defined by PitchFx (the sz_top column for those who have used this data) and "in" as anything inside of the dead-center of the plate, but over the plate. We define "fastball" as anything considered a two-seam or four-seam fastball by the Statcast data. Finally, we also only include pitches that are inside the official strike zone, we exclude any pitch from any plate appearance that had a bunt attempt on any pitch, we only include pitches involving hitters who had at least 200 plate appearances in other games and pitchers who had at least 150 plate appearances, and we exclude pitches that took place in 0-2, 1-2, and 2-2 counts (as well as 0-0 counts where there is no last pitch), where the pitcher's goals are far different than they should be earlier in the count.

With our sample defined, we now measure the impact of the last pitch on the value of the current pitch, using a linear regression model. There is one adjustment we must make to the regression, which is to control for the fact that throwing a fastball for a strike is much better for the pitcher in some counts, specifically hitter's counts, compared to others. We use separate intercepts for each count (not shown) to control for this.

In the above table we have marked the variables dependent on the last pitch in yellow, while the variables dependent on the current pitch are in white. Keep in mind that negative is better for the pitcher/worse for the hitter.

Since we are limiting our pitches to those that are in the official zone, obviously, pitches closer to the edge are better, and the numbers reflect that. For every 1 foot inside from the center of the plate, we would expect the hitter to average 0.0265 wOBA worse, meaning a pitch 0.8 feet inside the center on the inside corner would have an expected wOBA 0.020 lower than a pitch down the middle. The impact of pitch height, at 0.0355 wOBA less per foot higher, is even more significant. Finally, with the up-and-in fastball, the pitch is 0.0037 wOBA more effective per MPH of velocity when thrown up-and-in. So for example, a 95-MPH fastball would be expected to perform 0.0185 wOBA or 18.5 wOBA points better than a 90-MPH fastball.

The yellow box variables, related to the previous pitch, are the key subject of this study. Only the vertical height of the last pitch has a statistically significant influence on the wOBA value of the next pitch. For each foot of height the last pitch was, the up-and-in pitch will perform 0.0085 wOBA points better for the pitcher. The strike zone is around two feet high, so if the last pitch was at the bottom of the zone, the following pitch up-and-in will perform about 0.017 wOBA or 17 wOBA points worse for the pitcher than if the last pitch was at the top of the zone. The speed or inside-outside location of the last pitch have no statistically significant impacts.

It is helpful to visualize the influence of the previous pitch location through a heatmap plotting the value of the current pitch as a function of the location of the previous pitch:

The above heatmap splits the strike zone into 0.3 x 0.3-foot boxes, with the black box representing roughly the official strike zone. Left-right coordinates have been flipped for left-handed hitters meaning we are comparing inside to inside pitches across handedness.

While there is volatility across these boxes, anytime a pitcher locates his fastball high and tight in the zone in a non-two-strike count, they have thrown an above-average pitch, and the negative wOBA value across almost all last-pitch locations reflects this. But we can see a general trend where, when the last pitch was down in the zone, the up-and-in fastball is less effective, as is shown by the red and pink zones which indicate a relatively hitter-favoring situation, while if the last pitch was above the zone or in the same area, the subsequent pitch is more pitcher-favoring.

The EV theory suggests that it is better to go from low-to-high, the idea being that the low pitch will slow down the hitter's bat, but here, we see that is better to go from higher-to-high, or perhaps more common, it is worse to go from low-to-high. It is worth investigating why this is the case, through performing the same regression study as above on four major components of hitting - swing rate, launch angle, launch speed, and pull tendency. Rather than show the individual regressions, we have combined them into a single table, listing only the coefficients tied to the influence of the last pitch:

Although they do not all influence how well the hitter performs, all three of our previous pitch-related variables influence the hitter in various ways. For every foot outside the last pitch was, the hitter swings at the up-and-in pitch 0.62% less often. Perhaps the last pitch being outside causes an inside pitch to appear relatively more inside than it otherwise would. The hitter also hits the up-and-in pitch following an outside pitch at a higher average launch angle, at 1.09 degrees higher per foot outside.

When the last pitch is high, hitters tend to get under the ball (1.12 degrees of launch angle per foot) and go to the opposite field (1.1% more likely to go to the opposite field per foot) more often on the next pitch. This results in slightly lower launch speeds (0.343 MPH per foot) and consequently, worse performance. While higher launch angles are usually good, the average launch angle on balls in play against these up-and-in pitches is already very high - at 21.4 degrees - meaning a higher launch angle tends to hurt the hitter here as it leads to more pop-ups.

Finally, a faster previous pitch causes the hitter to swing less often and get more on top of the ball, other things being equal.

Overall, the underlying metrics following high pitches support the idea that a high-high sequence performs better than a low-high sequence. Following a high pitch, the hitter tends to swing too far towards the bottom of the ball and too late, and this hurts them on the next pitch. But we'd also expect the out-in sequence to perform better, although our first model suggests that it does not. After all, the hitter swings less often at pitches that are defined as strikes, and they also hit at a higher launch angle, against pitches where hitters already struggle to avoid popups. It turns out there is one other factor we have not accounted for - while all these pitches are "strikes" according to the official zone, the probability these pitches are actually *called* a strike by the umpire is also heavily influenced by the location of the previous pitch.

The influence of pitch sequencing on the umpire's strike zone is a rich, complex subject worthy of several future articles. But for now it is enough to know that for every foot high the last pitch was, the umpire is 1.93% more likely to call the up-and-in pitch a strike, and for every foot outside the last pitch was, the umpire is 1.88% less likely to call the pitch a strike. This is after accounting for the count (which also has a ridiculous influence on how often the pitcher gets the call) as well as the location and velocity of the pitch. These differences drive the out-to-in sequence, which should be pitcher-favoring, to neutral, as the pitcher doesn't get the call on marginal strikes as often as they otherwise would, and they make the high-to-high sequence more favorable than would be expected by launch speed alone.

My theory is that the strongest pitch sequencing impacts stem from the idea that both the hitter's and umpire's reference frame of pitch location is influenced by the location of the last pitch, rather than the hitter's timing. If the last pitch was higher in the zone, the pitch that follows will look relatively lower. This will cause the hitter to swing slightly late (aligning with the EV concept) and perhaps more towards the bottom of the ball than they otherwise would, a poor approach against the up-and-in pitch, and the umpire to give the pitcher more calls on marginal high pitches. The velocity of the last pitch, while having a small impact on the hitter's timing, is not as influential.

The impact of pitch sequencing on pitches in the zone is modest - we have cherry-picked the location and pitch type where it is most important in this article - and furthermore, most pitchers are not accurate enough to exploit it anyway. Unless the pitcher is accurate enough to locate their fastball in a target slightly larger than the size of the catcher's glove, the pitcher's expectation in most counts is better if they simply aim near the middle of the plate, because the gains from painting the edge are not enough to offset the additional balls that result from an off-center target. Where there *are* substantial gains to be had from pitch sequencing is when there are two strikes, and the pitcher is looking to get the hitter to chase. For more on this and other topics, buy my book.