Research Article Kyle Burris* and Jacob Coleman Out of gas: quantifying fatigue in MLB relievers Abstract: As relief pitcher usage in Major League Baseball has spiked in recent years, optimal bullpen decision-making has become increasingly vital for team managers. Throughout the season, managers must be mindful to avoid overusing their most talented relievers, due to the risks of injury and ineffectiveness. De- spite the substantial amount of attention given to pitcher arm health and injury prevention, the effect of workload on pitcher fatigue is poorly understood. Asa result, many of these overuse decisions are driven by feel and intuition. In this paper, we borrow ideas from toxicology to provide a framework for estimating the effect of recent workload on short-term reliever effectiveness, as measured by fastball velocity. Treating a thrown pitch as a fatigue-inducing “toxin" adminis- tered to a player’s arm, we develop a Bayesian hierarchical model to estimate the pitcher-level dose-response relationship, the rate of recovery, and the relationship between pitch count and fatigue. Based on the model, we find that the rate of reliever fatigue rises with increasing pitch count. When relief pitchers throw more than 15 pitches in an appearance, they are expected to suffer small, short-term velocity decreases in future games; upon crossing the 20 pitch threshold, this dip is further amplified. For each day that passes after the appearance, we estimate that the effect on a player’s velocity is cut roughly in half. Finally, we identify the relievers most affected by fatigue, along with those most resilient to its effects. Keywords: baseball, pitcher, Bayesian, recovery, dose-response 1 Introduction In the 2016 World Series, Chicago Cubs relief ace Aroldis Chapman made three appearances in four days, throwing 42 pitches in Game 5, 20 pitches in Game 6 after a day of rest, and 35 pitches in Game 7. By the final game of the seven-game series, Chapman’s trademark fastball had dropped from his season average of 100.9 miles-per-hour (mph) to an average of 98.9 mph, and he gave up a game-tying *Corresponding author: Kyle Burris, Department of Statistical Science, Duke University Jacob Coleman, Department of Statistical Science, Duke University 2 K. Burris and J. Coleman home run off of a 97.8 mph fastball in the 8th inning. Pundits and experts alike questioned Cubs manager Joe Maddon’s usage of Chapman - including Chapman himself [24]. Though equipped with player feedback, medical expertise, and pitch- tracking data, managers like Maddon are required to make bullpen decisions that are largely judgment calls. Could Maddon have predicted the drop in Chapman’s fastball velocity in the winner-take-all Game 7, based on previous usage? In this paper, we aim to develop a rigorous framework for quantifying velocity loss based on cumulative usage in preceding days. The Rise of the Modern Bullpen 3.4 ● ● 3.2 ● ● ● ● ● ● ● ● ● ● ● ● 3.0 ● ● ● ● ● ● ● ● ● ● ● Bullpen innings pitched per game Bullpen innings pitched ● ● 2.8 ● 1990 1995 2000 2005 2010 2015 Season Fig. 1: Average bullpen workload, 1990-2017. Reliever usage in the MLB has seen a steady increase in the last few decades, including a spike in last few years (see Figure 1). This has coincided with higher velocity fastballs thrown by relievers and a widening gap between reliever and starter effectiveness [14]. Because most relief appearances are one inning orless, relievers can exert near-maximum effort on every pitch, resulting in higher strike- out rates and fewer runs allowed. By 2016, starters were replaced by relievers at never-before-seen rates, especially in the playoffs [5]. Relievers became valued more than ever, as contracts for free-agent relievers rose dramatically during the past two offseasons [6], [21]. Increased usage of relief pitchers throwing at maximum exertion comes at a cost, however. It is well accepted that throwing a baseball at high velocity puts strain on a player’s elbow and shoulder, leading to greater risk of significant injury MLB reliever fatigue 3 [1], [9], [4], [23], particularly when the number of days between consecutive ap- pearances is low [22]. In addition, fatigued pitchers may change the biomechanics of their deliveries [7], which can put even more stress on the arm and further increase injury risk [10]. The difficulty reaching and sustaining peak performance associated with athlete fatigue [16] may manifest itself in pitchers via decreased velocity, spin rates, and/or command. Prior attempts to quantify pitcher response to workload have primarily fo- cused on within-game fatigue and its cumulative contribution over the course of a season. Authors in [3] developed a linear model to estimate the performance effect of cumulative workload, though their response variable, ERA, is a volatile measure of in-game performance for relievers. Sonne and Keir [20] utilized a three-compartment biomechanical model of the arm to estimate muscle fatigue when the time between pitches is varied, arguing that implementing a pitch clock may impair recovery from fatigue. However, the impact of short-term workload on a pitcher’s performance has not been studied extensively in the literature, even though this may have a large impact on a manager’s bullpen decisions. Since fastball velocity is recognized by both scouts and analysts as an impor- tant component of MLB pitcher success [8], we focus on estimating the relationship between short-term reliever workload and fastball velocity. We estimate this re- lationship via a principled, Bayesian hierarchical framework, borrowing modeling ideas from toxicology. By treating pitches as a fatigue-inducing “toxin" adminis- tered to a pitcher, we estimate 1) the relationship between the number of pitches thrown in a game and the toxicity of an outing, 2) the rate at which the toxin exits the pitcher’s system, and 3) the relationship between the dose of the toxin and mean fastball velocity for each pitcher. In Section 2, we motivate our approach and detail the Bayesian model used to assess the relationship between short-term reliever workload and fastball veloc- ity. Section 3 describes and interprets the model results in context. A discussion, including directions for future research, follows in Section 4. 4 K. Burris and J. Coleman 2 Methods 2.1 Data To answer our research question, we scraped pitch-level Statcast data from MLB.com, collecting available information on every regular-season pitch thrown between 2015 and 2017. We choose to focus on pitchers with exclusive relief roles, defined as those without any starts over the three-year period. For each reliever, in order to avoid the confounding influence of injuries, we only included years in which he threw at least 200 pitches and games in which he pitched with fewer than ten days of rest. Figure 2 and Figure 3 display the typical workload of the 324 unique relief pitchers in our study. Most successful relief outings tend to last for an inning (≈ 15 pitches), but can be extended to multiple innings in some situations. Although relievers throw far fewer pitches per outing than do starters, they pitch far more frequently, with over half of appearances coming with fewer than two days of rest. Distribution of Reliever Rest Days Reliever Pitches per Outing 5000 0.3 4000 0.2 3000 2000 Proportion 0.1 Number of Games 1000 0.0 0 0 1 2 3 4 5 6+ 0 10 20 30 40 50 Days of Rest Number of Pitches (a) (b) Fig. 2: Reliever usage patterns between 2015-2017. Relief pitchers have high-frequency, low-volume workloads, often pitching consecutive days or every other day. Appearances with more than 50 pitches (70 in total) are not shown in (b). A reliever’s mean fastball velocity in a game is the response variable of interest in our study. Although Statcast classifies the type of each pitch, not all relievers throw a four-seam fastball. As such, for each reliever, we define his fastball as the fastest pitch among four-seam fastballs, two-seam fastballs, sinkers, cut-fastballs, splitters, or sliders that he throws at least 10% of the time. Since some relief pitchers throw both a four-seam and two-seam fastball, we label both pitches as MLB reliever fatigue 5 fastballs if they average within two mph of each other. Because a pitcher’s fastball velocity is slightly left-skewed, we only consider relief appearances in which at least three fastballs are thrown, so that the sampling distribution of a pitcher’s mean fastball velocity in each game is nearly normal. After final processing, a total of 268,966 fastballs spread over 26,774 unique appearances were included in the three-year study. The Impact of Rest on Blake Treinen 99 ● 98 97 Mean Velocity 96 95 0 1 2 3 4 5 6+ Days of Rest Fig. 3: The relationship between days of rest and average fastball velocity for Blake Treinen, the closer for the Oakland Athletics. From exploratory data analysis, there appears to be a small, but noticeable trend in the relationship between the number of days of rest that a player receives and his average fastball velocity in the next game. Blake Treinen is an illustration of this phenomenon (Figure 4); he steadily gains a few tenths of a mph on his fastball when he is allowed an extra day to recover from an outing. After about two days of rest, there appear to be diminishing returns to increasing days of rest. In addition, preliminary nonparametric additive modeling approaches find a negative relationship between pitch count in the previous outing and mean fast- ball velocity. Affirming intuition, the strength and magnitude of this relationship diminishes with an increasing number of days of rest.