Illustration 1. Tunneling can be partly described by measuring the distance between consecutive pitches along their paths at the tunnel exit (A) and at the plate (B).

For the model sketched above, the next couple paragraphs explain the gritty details computationally and in terms of Baseball Prospectus’s tunneling data. The model was coded using the Bayesian R package rstanarm as

stan_gamm4(whiff.pitch2 ~ s(tunnel.diff, plate.diff, px.pitch2, pz.pitch2), random = ~ (1 | mlbcode), family = binomial(link = “logit”), subset = b_side == “R” & p_hand == “R” & swing.pitch2 == TRUE …)

where whiff.pitch2 indicates the outcome of the latter of consecutive pitches when the batter has swung, and whether the batter whiffed or not; tunnel.diff and plate.diff denote the distance in inches between two consecutive pitches measured at the tunnel exit and the plate, respectively; px.pitch2 and pz.pitch2 denote, from the batter’s perspective, the horizontal and vertical distance in inches from the center of his strike zone[2].

As each batter’s strike zone may differ vertically, the top and bottom for each was estimated with linear regression using batter heights and data from a U.S. Army Anthropometry Survey[3]. As whiff.pitch2 measures a binary outcome (contact or no contact), it was modeled as a probability using the logit link function. The probability of a whiff is neither linear, nor depends on any of one variable independent of the others. Thus, we can model the interactions between variables using a spline (that’s what the s(…) means). “Splines are continuous, piece-wise polynomial functions,”[4] which can better capture the chance of a whiff, as this outcome is not proportional to its causes.

In the model we simultaneously consider each batter’s unmeasured characteristics, which create variation in whiff rates across the league of batters even when all measured characteristics in an observed pitch have the same value. If we ignored specific batters (assigning the same whiff probability to each for the same measured values) we risk ignoring important variation in baseline whiffs. The variation could hide association between characteristics. Conversely, if we estimated a unique rate for each batter, ignoring the league, we ignore that each batter helps us estimate the whiff rates of other batters. In short, our multilevel model takes the best of both worlds, in one level modeling the league of batters and, in the other, modeling specific batters (mlbcode).

Further, this analysis focused on matchups between righties (b_side, p_hand) to avoid the complexity of modeling platoon advantages and other characteristics that depend on matchup handedness. Righty-righty matchups are also the most common, giving us plenty of data for an analysis. Finally, as mentioned, Bayesian modeling software—Stan (specifically, stan_gamm4)—was used for modelling so that we can describe uncertainty in our estimates and predictions.

Figure 1. Whiff probabilities when swinging are shown as contour lines at 10 percent intervals. In this example, the distances between consecutive pitches is held constant at 8.2 inches at the tunnel exit and 15.5 inches at the plate. Low-and-in, from this perspective, is the location where a swinging batter is most likely to make contact.

Of course, these contour lines represent mean probabilities of a whiff. Depending on location of the second pitch, our uncertainty in the probability varies. The posterior distribution of mean probabilities of a whiff when balls cross the upper- inside corner (blue dot in Fig. 1), the middle of the strike zone (gray), and the lower-outside corner (red) differentiate themselves below in Figure 2.

Figure 2. The expected probabilities of a whiff at any point along the contour lines in Figure 1 follow distributions. Here are distributions at three locations in relation to the strike zone.

The results here agree with intuition: the probability distribution of a whiff in the upper-inside corner (blue) is markedly higher than for pitches crossing center (gray). Less obviously: the probability distribution of a whiff in the upper-inside corner reflects more uncertainty in outcome than does the distribution in the lower-outside corner (red).

Figure 3. These small-multiples represent the change in expected whiff probabilities (30 percent shown blue) along nine discrete combinations of distances between pitch pairs at the tunnel exit (rows, inches) and at the plate (columns, inches). The center plot is also Figure 1.

Tunneling distances were chosen to show how whiff probabilities change across the middle 80 percent of the data. Thus, read from left to right, the predicted whiff probabilities change as the distance between consecutive pitches at the plate increases from 5.9 inches (10th percentile) through a league (righty-righty) mean of 15.5 inches and finally 29.3 inches (90th percentile). Similarly, read from top to bottom, the predicted whiff probabilities shift as distance between pitches at the tunnel exit increases from 3.1 inches (10th percentile) through the league average of 8.2 inches and finally to 15.5 inches (90th percentile). The looser the circles in each plot, the easier it is for the batter to make contact. To aid comparison, an estimated whiff probability of 30 percent is shown in blue.

As a group, the plots provide several insights. First, the top row of plots—representing probabilities where the pitches are proximal at the tunnel exit and range in distance at the plate—show relative stability across that range of adjustments. That stability stands in contrast to the predictions represented along the bottom row of the same range where the distance between consecutive pitches at the tunnel exit increases. Probabilities in this progression—unlike those of the top row— change markedly across the range of distances between pitches at the plate. When the distance between pitches at the tunnel exit is large and at the plate is small, for example, the contours of whiff probabilities spread and flatten. The second of consecutive pitches may include roundhouse curves that cross the plate near the first pitch: “the roundhouse curve is not usually effective . . . because the slow break is easy to follow.” (Kahn, 1997).

But even where the general contours seem more stable across the range, real differences between the contours of whiff probabilities exist, and suggest that moderate distances between pitch pairs at the plate relate to higher whiff rates more than pairs with either low (e.g., 5.9 inches) or high (e.g., 29.3 inches) distances between pitch pairs at the plate. With the distance between pitch pairs at the tunnel exit held constant at 3.1 inches, for example, the model estimates that the league average distance between pitch pairs at the plate sees higher whiff rates than those with a large distance 97 percent of the time and more than those with a small distance 91 percent of the time. Distributions of whiff probabilities in these scenarios follow:

Figure 4. Holding the distance between pitch pairs at the tunnel exit constant at 3.1 inches (Figure 3, top row), the whiff rate is higher when distance between pitch pairs at the plate is league average compared with either the shorter (differences in blue) or larger distance (differences in gray).

Simply put: throwing consecutive pitches that look the same are more likely to earn swinging strikes than pitches that look vastly different. These results agree with anecdotal evidence of results by former Yankee closer Mariano Rivera who achieved greatness using a moderate break and avoiding the middle of the plate.

Figure 5. Pitch combos coincide with the various distances between pitches. Here are the top fifteen pairs (top three, blue) for each of the nine discrete distances between pitches, again aligned by distance at tunnel exit (rows, increasing top-to- bottom) and distance at the plate (columns, increasing left-to-right).

Each histogram shows the fifteen most frequent pitch-pair combos corresponding with the given distances between pitch pairs at the tunnel exit and plate. The pitch pairs are listed alphabetically top to bottom by the first two letters of their names[5]. The fastball/fastball (FA | FA) appears most frequently along the middle diagonal of histograms above, while, fastball-curveball (FA | CU, CU | FA) combinations arise most where the difference between pitch pairs is low at the tunnel exit and high at the plate. More generally, basic physics presents itself: high distances between pitch pairs at the plate generally follow greater distances between pitch pairs at the tunnel exit. That’s because pitch paths form a parabolic curve.

Figure 6. Here, we overlay home runs (blue) and all pitches (gray) onto the contour plots we’ve been studying (Figure 3). Roughly, the frequency of home runs seems to co-vary with whiff rate.

Figure 7. The second of consecutive pitches faced by Carter crossed the plate at markedly different locations than did those faced by Iglesias. Contours here, unlike elsewhere, demarcate the density or relative frequency of pitches.

The two batters have also faced comparatively different frequencies of pitch pairs, especially when it comes to slider combinations. The six most frequent pitch pairs are shown below.

Figure 8. Pitch pairings seen by Carter and Iglesias, 2016.

Carter has seen more pitches overall, but the combinations are a little surprising. The less-dangerous Iglesias has seen a greater degree of fastballs, as pitchers are more likely to challenge the shortstop. But he’s also more likely to see sinker/sinker, perhaps counterintuitive for a batter already hovering around a 50 percent groundball rate.

Figure 9. The distribution of distances between pitches at the tunnel exit (gray) and at the plate (blue) for Carter (left) and Iglesias (right): the former is more likely to see different looks, given the greater number of off speed pitches heading his way.

After controlling for these differences, Carter’s overall estimated chance of a whiff is about 9.6 percentage points lower than Iglesias’s! So, when the two batters face consecutive pitches in the upper, inside corner where the difference in pitch paths are, say, 8.2 inches at the tunnel exit and 15.5 inches at the plate, Carter’s estimated chance of a whiff is around 45 percent. Iglesias’s estimated chance—facing that same pitch sequence—is around 55 percent. This example is represented at the blue point in Figure 10, which generally shows Carter’s whiff probabilities minus Iglesias’s at all locations for the nine discrete distances at the tunnel exit and the plate.

Figure 10. These small-multiple contour plots represent Iglesias’s expected whiff probabilities minus Carter’s.

More importantly, the model provides a tool that allows different kinds of questions to be asked: which player is less likely to whiff when facing similarly tunneled pitches? How might a batter fare against a new pitcher? The model examined here can be improved, too, by considering additional characteristics of pitches.