In an effort to prevent fatigue and injury among pitchers, many baseball talent overseers (e.g., managers, coaches, trainers) have suggested limiting the number of pitches that pitchers are allowed to throw. For example, during the 2010 season, the Washington Nationals limited top-prospect rookie pitcher Stephen Strasburg to 100 pitchers per game and to 160 total innings pitched for that year in an attempt to protect his future health and effectiveness (7). This regimen proved unsuccessful because Strasburg would require an elbow ulnar collateral ligament replacement after pitching a total of 123.33 innings between the major and minor leagues.
Strasburg's treatment is not an isolated case. Recent trends in pitcher usage indicate that practitioners have been reducing the number of pitches thrown per game by pitchers. Figure 1 maps the maximum pitches per game thrown by season since 1988 for major-league baseball pitchers, showing a clear downward trend in the number of pitches that managers allowed their starting pitchers to throw. The maximum pitches thrown in a game declined from highs in the 160s and 170s in the 1980s and 1990s to highs in the 130s in the 2000s. Although the maximum number of pitches per game had a declining trend, the average number of pitches per game thrown by starters did not change. Figure 2 shows that median pitches per game remained stable from 1988 to 2009. However, over this same period, the lower bound of pitches per game increased. Although the managers reduced the maximum number of pitches that they allowed their pitchers to throw per game, they also increased the minimum number of pitches thrown.
Despite the recent growth in the popularity of using pitch-count limits to protect pitchers, there has been scant study of the effectiveness of setting pitch limits to regulate effectiveness and to prevent injuries among adult pitchers. Nearly all past analysis of pitches thrown on injuries has focused on adolescent pitchers. Several studies (6,8,9,12,13) have found evidence to show that pitches thrown and overuse are associated with injuries and pain, and limiting pitches thrown can reduce injuries among young pitchers. However, given the rapid development among this age cohort, the results may not translate to adult major-league pitchers.
To our knowledge, there have been no peer-reviewed studies of pitches thrown and the days of rest on the performance of adult baseball pitchers. This is largely because previously such data were not widely available. Using newly available pitch-count data, we estimate the impact of past pitches thrown and the days of rest on future performance among major-league baseball pitchers.
Experimental Approach to the Problem
We used the past individual-game pitching performances of major-league baseball pitchers to estimate the impact of pitches thrown and the days of rest on pitching performance. The use of real-world data outside of a controlled laboratory setting required the use of statistical methods to estimate the impact of these factors on performance while controlling for other outside factors.
We used individual-game performance values of starting pitchers in Major-League Baseball in the regular-season games from 1988 to 2009 who pitched after <15 days of rest. The sample included games started by 1,058 pitchers over 22 seasons, for a total of 77,131 observations. The year 1988 is the earliest that pitches thrown data are available with any reliability. Data were from the open-access website Baseball-Reference.com, and we included all available data from games during that time period, with some data not being available in the 1990s. Baseball-Reference compiles its data from several sources and is a partner of the National Baseball Hall of Fame. The analysis was conducted using publicly available data from outside a laboratory setting, and therefore, Institutional Review Board permission was not required.
We collected and organized a dataset of all major-league baseball pitchers from 1988 through 2009, who started games after <15 days of rest. The cutoff for rest days was chosen for 2 reasons. First, pitching rotations typically include 5 pitchers who receive between 4 and 5 rest days between starts. When off-days permit, weak or tired pitchers often have their turn skipped to give them 8–10 rest days between starts. Pitchers who have more-than-normal rest are typically inferior pitchers who switch between starting and relieving roles or bounce between the minor- and major-league levels. Second, injured pitchers are placed on the disabled list, which requires them to spend a minimum of 15 days without playing before returning to the lineup. Including only pitchers with <15 rest days excludes inferior and recently injured pitchers who may perform poorly for reasons other than days of rests. We also collected data on several factors hypothesized to impact pitcher performance (detailed in the Statistical Analyses section), and then we analyzed the data using the statistical procedures given below.
Although the statistical technique employed to estimate the relationship between pitches thrown and performance is complicated, the general method is quite simple: observe how pitching performances typically changed based on how many pitches a pitcher threw in the previous games. The multiple regression estimation method employed uses the past performances of pitchers to estimate the average impact of many factors that potentially affected performance. In addition, the sample is partitioned by age to observe if younger and older pitchers respond differently to pitches thrown.
The procedure produces an estimate of performance as function of factors included in the regression model (equation 1). To measure potential nonlinear impacts of marginal pitches thrown—each pitch beyond a certain threshold may have a greater or lesser effect than the preceding pitches—we used multiple-variable fractional polynomial regression to estimate the model. We selected this particular estimation technique because it does not impose a predetermined functional form on the relationship between variables and permits controlling for multiple factors that ought to affect pitcher performance. The fractional polynomial estimation procedure used an iterative process to select a transformation of the explanatory variables and a coefficient (β) to generate a functional approximation of the observed relationship between pitcher performance and each explanatory variable, while holding all other explanatory variables constant. Royston and Altman (15) demonstrated that fractional polynomial estimation is good at measuring curved relationships concisely and accurately. We tested the null hypothesis that the past pitches thrown and the days of rest were not associated with performance. Estimates of βs with p ≤ 0.05 were considered to be statistically significant and result in hypothesis rejection. The equation was estimated using Stata 10 statistical software using the “mfp regress” command.
P is the performance of the pitcher in game g using one of several measures of performance: earned run average (more commonly referred to as ERA), strikeout rate, home run rate, and walk rate (all measured per 9 innings pitched). Earned run average is a cumulative measure of performance. The other metrics are components of pitching performance that do not require the help of fielders, which McCracken (10) and Bradbury (3) demonstrated to measure individual pitching performance better than the ERA.
In equation 1, PT is the number of pitches thrown in the preceding game (g−1), the average number of pitches thrown in the previous 5 games (g−5), or the average number of pitches thrown in the previous 10 games (g−10). The measures proxy the immediate and cumulative effects of pitches thrown on performance, estimated in separate equations. The DR is the number of rest days the pitcher had before game g. Average performance P in the year of analysis t is included to serve as a proxy to control for the ability of the pitcher (e.g., seasonal ERA was used when estimating the dependent variable ERA per game, and the seasonal strikeout rate was used when estimating the dependent variable strikeouts per game), which should be positively associated with game performance. Age is the age of the pitcher as of the game day measured continuously in years, which is included to capture any effects of durability due to aging. To further capture aging effects, separate estimations by age cohorts were conducted. Here Y is a vector of the year indicator variables that equals 1 for games played in the year of analysis and 0 for all other games. The indicators control for factors unique to individual seasons (e.g., run scoring, rule changes) that impact performance in games played in each season. The α is a constant term, and ε is a standard error term. Table 1 gives the summary statistics for the included variables.
Figure 3 maps the estimated impact of pitches thrown on pitcher performance in the game, 5 games, and 10 games preceding the present game on ERA performance. Figures 4–6 map the estimated impact of pitches thrown on strikeouts, home runs, and walks, respectively. The figures graphically depict the estimated relationships between pitches per game and the performance for each performance metric. The graphs are easier to interpret than the raw regression estimates of the transformed variables; however, the regression estimates are available from the authors upon request.
For ERA and pitches thrown, all estimates were statistically significant. Each pitch in the preceding game was associated with an increase of a pitcher's ERA by approximately 0.007 in the following game. Although the relationship is nonlinear, the graph reveals that the curvature of the function is so slight that a linear approximation is appropriate for practical purposes. Each pitch averaged in the previous 5 games was associated with an increase of a pitcher's ERA by 0.014, and each pitch averaged in the preceding 10 games was associated with an increase of a pitcher's ERA by 0.022.
For strikeouts and pitches thrown, the estimates were linear, small, and not statistically significant. Each pitch in the preceding game was associated with a decrease of a pitcher's strikeout rate by 0.0008. Each 1-pitch increase in the 5- and 10-game averages was associated with a strikeout rate lowered by 0.0011 and 0.0027, respectively. At the average strikeout rate for the sample of 6.1 strikeouts per 9 innings pitched, a 1-pitch increase in the preceding game, 5-game average, and 10-game average lowered the strikeout rate by 0.13, 0.18, and 0.44%, respectively.
For home runs and pitches thrown, all the estimates were statistically significant. A 1-pitch increase in the preceding game was associated with a 0.0013 increase of home runs allowed (a 1% change at the average). A 1-pitch increase in the 5- and 10-game averages was associated with an increase of the home run rate by 0.002 (1.6%, estimated at the 101st pitch) and 0.0025 (2%), respectively.
For walks and pitches thrown, the estimated impact was nonlinear, statistically significant, and was the opposite of the expected effect. Each pitch in the preceding game was associated with a walk rate decreased by 0.0024 (0.66%) at the 101st pitch. The 101st pitch for the preceding 5- and 10-game average pitches thrown was associated with a lowered walk rate by 0.0038 (1%) and 0.006 (1.67%).
Table 2 reports the overall impact of previous pitches thrown on ERA and by 3 age cohorts: 25–34 years (10 years centered on the estimated peak age for pitchers as estimated by Bradbury ), <25 years, and >34 years. The top half of the table lists the impact of each additional pitch thrown on ERA, and the bottom half lists the number of pitches needed to raise a pitcher's ERA by 0.25. Younger pitchers were no more sensitive to high pitch performances than those of the middle age cohort. Older pitchers suffered much less than younger pitchers did from pitches thrown in the previous game; however, older pitchers suffered more from increased cumulative pitching loads than their younger counterparts did.
The estimated impact of the days of rest on ERA was small and insignificant, with each rest day associated with an improvement of 0.015. Based on this estimate, skipping a pitcher in a 5-man rotation—giving him 4 additional days of rest—lowered his ERA by 0.06. Also, rest days were not strongly correlated with performance components. The relationship with strikeouts was not statistically significant. Each rest day was associated with a home run rate lowered by 0.012 (0.98%), and the estimate that included pitches thrown in the previous 10 games was not statistically significant. The estimated impact of rest days on walks was to increase the walk rate by 0.032, approximately 0.08% at the average walk rate. As with pitches thrown, the estimated effect was statistically significant and counterintuitive.
The finding that pitches thrown were negatively correlated with future performance should be interpreted with caution. Although the estimated effect was statistically significant, it was small. The findings are consistent with the results of Escamilla et al. (5), who examined the change in pitching mechanics over the course of simulated games using a sample of collegiate baseball pitchers. The researchers found that the pitching mechanics of pitchers who threw between 105 and 135 pitches for 7–9 innings were “remarkably similar,” and the results did not support the idea that pitching more increased shoulder and elbow forces and torques, which Anz et al. (1) found to be positively correlated with injury. The value is within the upper range of pitches thrown that major-league starting pitchers are typically allowed to do. According to the estimates reported in this study, the ERA difference in a game following 105 pitches vs. 135 pitches was approximately 0.19—a small effect of 0.33% at the sample average.
Murray et al. (11) also looked at the performances of major-league baseball pitchers; however, the focus of the study was to examine the impact of pitching on fatigue within a single game. The researchers used video observations of several physical markers to compare pitchers in their first and last innings of play in a game and identified kinematic and kinetic changes that were consistent with fatigue. However, alternative explanations for the changes could not be ruled out, and there was no follow-up on the future impact on performance. The findings of this study do not support or contradict those of Murray et al.
Although the estimates reported in this study indicate that there is a clear relationship between pitches thrown and overall performance, the relationships between pitches thrown and the performance components differ. The strongest effect occurred with home runs—each pitch was associated with an increased home run rate of between 1 and 2%. The weakest effect occurred with strikeouts—each pitch was associated with a decreased strikeout rate of between 0.13 and 0.44%, and the estimates were not statistically significant. The counterintuitive relationship between pitches thrown and walks is difficult to explain. In summary, the analysis of the performance components indicates that high pitch counts are likely to impact pitchers' run prevention ability through giving up home runs, as opposed to reducing strikeouts or increasing walks.
As a regressor, age was not associated with changes in performance after controlling for the other factors in the regression equations. However, when the sample was separated into age cohorts, there was a clear difference in responses to pitches thrown, among age groups. Older pitchers were more sensitive to cumulative pitches thrown than younger pitchers were; however, that older pitchers were less sensitive to pitches thrown in the preceding game is interesting. This response is consistent with the results of the past studies that found older athletes using experience to counteract diminishing physical skills. For example, Baker et al. (2) found evidence of golfers using strategy to substitute for decreased driving distance to remain competitive. Among baseball players, Bradbury (4) identified differences in aging functions across skills that were consistent with players improving strike-zone judgment to compensate for diminished hitting and pitching skills. If such compensation is occurring, then veteran pitchers may be able to exploit their knowledge of the game to pitch effectively following a high pitch game even though their physical stamina has decreased. However, further research is needed to examine this hypothesis.
Alhough days of rest did not appear to affect the performance of pitchers, it is likely that rest days are important for maintaining performance; otherwise, teams would not give pitchers any rest days. Less than 0.5% of the pitchers in the sample pitched with <3 days of rest; therefore, it would be unwise to extrapolate the estimates to predict the impact of rest days below that threshold. The results of this study indicate that additional days of rest beyond the normal do not appear to have a strong impact on performance. This finding is consistent with that of Potteiger et al. (14), which found that after 3 days of rest, markers of muscle damage returned to baseline levels.
It is our hope that future researchers will examine the usefulness of pitch counts as a predictor of performance and injury more precisely than we have identified here. Researchers should draw upon the vast amounts of sports data that are becoming increasingly available to them to examine factors relating to performance and injury. In addition, future studies should examine the direct impact of pitching loads on injury.
This analysis of 1,058 pitchers across 22 seasons found that pitching performance is affected by the number pitches thrown in the previous games; however, the magnitude of the effect is small. It takes a rather large change in pitches thrown to have even a modest effect on performance; therefore, the guidance offered by pitch counts and pitch-count restrictions may be limited. Pitchers and coaches should be mindful of potential overuse, but occasional high- or low-pitch games likely have only a minor effect on future performance. The longer the high- or low-pitch counts are maintained, the greater the dampening or improvement will be. Furthermore, additional days of rest beyond the ordinary appear to have little effect on performance.
One interesting finding of the study is that the estimated relationship between pitches thrown and performance was virtually linear. Even in cases wherein nonlinear estimates were found, the curvature was small. Therefore, the estimates provide simple rules of thumb that can be used to estimate the future performance consequences from pitches thrown in a game. For example, this study found that each pitch thrown in a game increased a pitcher's ERA in the following game by 0.007. Managers can use such rules to quickly weight the strategic risk of leaving a pitcher in a game versus taking him out.
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