ࡱ>     g ybjbj B{z\{z\L, DBBBLBY( CYEYEYEYEYEYEY$[n^~iY iY ~YLLL   CYL CYLLL 2<>L/YY0YL^?^LL^+L L iYiYE Y ^ > : Chapter Six Offensive Issues The goal of this chapter is to explore some areas relevant to offense other than its evaluation that have garnered research attention. Age Overall Offensive Trajectories As I grew up watching games on television, I heard on several occasions that players generally peak between ages 28 and 32. Statistically, this guesstimate is both inaccurate and vague. In general, it is earlier, and it is dependent on specific skill. Further, there are methodological problems that invalidate any easy analytic method for determining the peak. Phil Birnbaum (2009a) addresses two of these problems, both of which are relevant to what statisticians call selection bias, defined on Wikipedia as the selection of individuals, groups or data for analysis in such a way that proper randomization is not achieved, thereby ensuring that the sample obtained is not representative of the population intended to be analyzed. Suppose you just average the performance of players at different ages, and then use that average at each age as your estimate of offensive production. If you do, you will overestimate the productivity of the average older player, and here is why: The very best players may have lost some of their skills when they reach their mid and late 30s, but they are likely to still be performing at a fairly high level. Weaker players may have lost no more skill than the best ones, but that loss will be enough to end their careers. Phil used the example of Mike Schmidt, whose runs created per 27 outs was 8.9 runs at age 30, and Damaso Garcia, whose RC/27 at the same age was 3.7. At age 35, Schmidt had fallen to 7.3, but Garcia was gone. A nave attempt to use average performance of players still in the game would result in a higher performance average (Schmidts 7.3) at 35 than at 30 (6.3 combining the two). The analogous problem can occur with very young players. Others noting this issue have been Schulz, Musa, Staszewski, and Siegler (1994), Hakes and Turner (2011), and, in his second book, Michael Schell (2005). As Phil noted, a different path would be the paired season method, which is to take all players at each age cohort, compute the average change there is in their performance from one year to the next, and then use that change as an estimate for the change for everybodys estimate. So, for example, just suppose that the average decrease in RC/27 from age 32 to 33 is one run per game. But, as Fair (2008) pointed out, players are far less likely to retire or be released after a good year than a bad year. If a player has a season in which he performs above his actual ability level due to random processes, than it is more likely that some team will want his services for another season than if he plays below his norm. But if random processes are involved, it is unlikely that he will repeat that seasons performance, so there will probably be a large dip between that and the next year, hastening his departure at that point. As a consequence, players are likely to perform unusually well in their next-to-last season. This leads to a statistical bias in which the next-to-last season should result in greater error in estimation; Fair demonstrated that the error is relatively large. Phil Birnbaum also showed that this error would be a bias toward overestimating declines as a whole, as the soon-to-be-released player is not really as bad as his performance implies. There is a third problem with methods such as this. As will be described later, different skills increase and decrease at different rates. Just to name two obvious examples, speed is greatest with young players whereas plate discipline often improves such that walks increase with experience. As a consequence, all things being equal, a player whose overall performance relies on the former will probably peak earlier and leave the game before a player who relies on the latter. It took a long time for the sabermetric community to understand the implications of these issues. In particular, early on there was a tendency to look at overall performance rather than categorize players according to differing specific skills. As with so much else, Bill James led the way in his first conventionally-published Baseball Abstract (1982). Bills method was relatively informal, but his findings have been fairly well corroborated since then. He used an informal method he called Value Approximation, which assigns points for various annual achievements, e.g., 1 point for a batting average of .250 or more, and one additional point for each BA increase of 25 points, such that a .400 average is worth 7 points. Using all 977 players who appeared in the majors that were born in the 1930s, Bill found that the total approximate value of all players in the data set increases in an inverted-U curve that is flatter for older players than for younger, which implies that older players lose ability slower than younger players gain it (more on this below). The curve was highest at ages 26 and 27, with sharp increases before than after; the total was half as much at ages 23 (3 or 4 years before peak) and 33 (6 or 7 years after peak). This method is problematic in that it makes no correction for the number of players in the majors at each age, and a greater number of players as a particular age will increase total approximate value over ages with fewer players. But for what Bill was attempting to show, this is not a problem, because it means that there are more players capable of playing in the majors at those ages; in other words, fringe players peak around 26/27 and so are good enough to be in the majors only around that point. Even so, a better method examines individual player trajectories, and Bill did some of that, with analogous findings. Sticking with Bills relevant research for the time being, he later (1985a) looked at all players with at least 200 plate appearances at age 37 (chosen because only about five percent of major leaguers are still playing at that point) who had accumulated at least 1000 PA in their 25-29 and 35-39 age ranges, and compared performance between those ranges. Every index he examined except for BB/PA decreased substantially for those players over time (BA down .20, SA down .51, OBA down .17 despite the increase in walks, RC/game down 1.05, also SB% and HR/AB). Keep in mind that there is selection bias here, such that this data set mostly included players who were unusually good at maintaining their skills (still getting significant playing time at age 37). The implication is that a more general set would decrease even faster. Other work on composite indices has also noted that overall performance peaks around age 27. Bob Boynton (2004) examined Win Shares totals for every hitter listed in the 8th edition of Total Baseball, and uncovered a peak at age 26. Mark Armour (2018a) contributed a study to the SABR Statistical Committee blog that reveals age 27 as associated with the highest overall bWAR between 1876 and 2017, 26 right behind, and 25 to 29 as the overall peak. Fairs (2008) study included all players between 1921 and 2004 with at least 10 full-time seasons, defined as at least 100 games for position players. For batters, the peak was 27.6 for OPS and 28.3 for OBA. His work provides equations for the trajectories, with that for OPS declining more rapidly than that for OBA, which means power skills fade faster than on-base abilities. Fairs method assumed the same trajectory for everyone but allowed for estimations of prediction errors for each player for each year, providing the opportunity to look for unusual performance. He noted a sample of players who tended to have positive errors toward the end of their careers, in other words, consistently playing better than what would have been expected given normal rates of decline. There were 17 who had at least four seasons of this sort from age 28 on, including Barry Bonds, Ken Caminiti, Luis Gonzalez, Mark McGwire, Rafael Palmeiro, and Sammy Sosa; prime suspects for drug-fueled performance enhancement. Most of the others were also recent (the only pre-1970s name was Charlie Gehringer), such as Andres Galarraga and Larry Walker, both almost certainly due to their stints as Colorado Rockies. As Bill Jamess earliest work (1982) mentioned above shows, it is important to note that the data distributions of performance by age are not symmetrical, but rather are what statisticians call positively skewed. In plain English, they go up fairly quickly as player performance increases in their early 20s, but go down more slowly, as player skill erodes relatively slowly through the early 30s. As such, although the peak is usually around 27, players will usually create more total offense after than before 27. In his Win Shares book (2002), Bill James explored aging patterns for 148 great position players, those with at least 280 career Win Shares (see the Overall Evaluation chapter for definition) or more with fewer than 10 as pitchers. Grouped together, total Win Shares among these players increased very rapidly through the early 20s until its peak at age 26, and decreased slowly (still at 90 percent at age 32), and then at the more rapid clip of about 10 percent a year thereafter. Nate Silver (2006b) presented the following 1946 to 2004 data concerning mean percentage change in Equivalent Runs from present season to the next at different ages. Age212223242526272829Change26.915.116.46.18.24.0-1.1-2.5-3.1Age303132333435363738Change-4.6-5.9-8.0-8.1-10.2-11.1-11.6-13.0-19.3 According to Mark Armours (2018a) work discussed above, bWAR on average is a bit greater at age 32 (five years after 27) than at 23 (four years before 27), about the same at age 34 (seven after) as 22 (five before), and a bit more at 36 (nine after) than 21 (six before). This is important analytically, as it means that one must judge a given players peak performance year independently of the (so to speak) midpoint of his productivity, as the former usually occurs earlier than the latter. In follow-up work (2018b) also discussed, Mark restricted the sample to 1988-2017 and divided it into bWAR categories. He noted the greatest drop in average age of best season, from about 29 to 27, to have occurred between 2005 and 2009 for players whose peak was a bWAR of 3 or greater. The drop was from about 29 to 28 for 1 to 3 bWAR as their peak, and 28 to 27 for those less than 1. In other words, the best and worst position players currently tend to peak a bit earlier than those in the middle. If playing time corresponds at all to ability, than Dallas Adamss (1982c) study of the former across ages provides additional support to the age 27 estimate. He tallied the number of position players games at each age that has been represented (16 through 46) from 1901 to 1968. Here are the percentage of total career games played at each age, separately for all position players and Hall of Famers excepting catchers, and the percentage of players of that age appearing in at least 95 games: All Position Players (except catchers)Hall of Famers (except catchers)CatchersAgePercentage>95Percentage>95Percentage>95258.44341.26.30589.27.72122.4269.29345.46.47192.09.19324.1279.55349.56.70697.49.17227.7289.21950.46.58592.210.11026.0298.67152.56.44486.89.30629.2307.95253.66.41295.68.48025.5316.69753.06.17786.77.23728.5325.77151.66.15287.86.36025.1 Note that for all non-catchers, overall and HOFers, the peak percentage of games occurred at age 27. However, the greatest proportion of 95-game players was later. I would speculate that this is because once a player wins a regular spot, it has historically taken a lot for him to lose it. The trajectory for catchers is displaced a year, with the peak at 28. Individual performance across the years can be erratic. Bendtsen (2017) defined a regime as a phase in a position players career within which offensive performance is relatively consistent for a significant period of time, but distinctly different than beforehand and after wards. The author evaluated a model for determining regimes and the boundaries between them using 30 seemingly randomly-chosen players whose careers began no earlier than 2005 and who had at least 2000 entries in Retrosheet, the source of study data. The number of regimes for the chosen players ranged from 3 (with one exceptional 2) to 6 and averaged 4.36; and the sample includes quite a few who were still playing when the data ended, meaning this average is almost certainly an underestimate of the number of regimes the sample will accumulate in their careers. Only forty percent of the boundaries between regimes could be accounted for by reported injuries, changes in teams, or a new season; the other sixty percent occurred within-season for no discernible reason. In addition, all but two had separate regimes that were statistically analogous. A detailed examination of two of the sample (Nyjer Morgan and Kendrys Morales) shows that differing regimes generally reflect obviously different OPS values for substantial periods of time. Some economists have examined the role of experience in number of years as major leaguers by taking a given season and examining batting averages for different age cohorts for this year. Experience is a less useful index than age, because players who come into the league at an earlier age will probably peak later in their MLB careers than those entering at a relatively advanced age; and batting average measures one skill only. But having said this, Scully (1989), based on players active in 1986, found the usual inverted-U curve for batting average, peaking at the sixth or seventh year. Sommers (2008), using players with at least 100 at bats per included season for 1966, 1976, 1986, 1996, and 2006, noted a progression over time, with peaks at year six in 1966 increasing to years nine or ten in 1996. As the American League regressed to year eight in 2006, Sommers insinuated that the increase is steroid-fueled and the regression is due to baseballs tougher strategy to fight them; to me, as the increase over the decades was gradual, it looks more like responsibility for the general increase could also be attributed to more attention to diet and conditioning, allowing players to maintain their ability later in their careers. Differences among Specific Skills Most of the studies described thus far did not seriously consider the issue of differences among specific skills. The earliest appears to be Schulz, Musa, Staszewski, and Siegler (1994, see also Schulz and Curnow, 1988), who examined all 235 position players with careers of at least 10 seasons who were active in 1965. They noted that most of the included indices (batting and slugging averages, home runs per at bat, and stolen bases per times on base, along with analogous counting indices) peaked between 27 and 28 years, with strikeouts and walks per at bat at about 29 and fielding average at 30. Scott Berry, with associates Shane Reese and Patrick Larkey (1999a), in a study to be discussed in detail below, found that batting averages tend to peak at age 27 with little variation between 25 and 31, whereas home run percentages were highest at 29 and basically at their top between 26 and 32. Consistently with those for total performance, trajectories for both skills went up to their highest ranges relatively quickly and dipped down fairly slowly. John Charles Bradbury (2009) took on this issue head-on. Based on a sample of every position player with 10 years experience and at least 5000 plate appearances between 1921 and 2006, Bradbury wisely adjusted the data for ballpark and normalized for year, the latter by transforming player performance on each index into z-scores. As a consequence, he in essence was working with performance relative to the sample mean for each season. However, Bradbury erred by in effect weighting each players impact on his estimates by the length of their careers. Given that career trajectories are positively skewed rather than symmetrical, the findings are biased toward higher estimates than they ought to be. As a consequence, the following numbers are probably a year or two too high, but their relevant ordering is probably correct, and the study does make a contribution for that reason: Doubles plus triples per at bat 28.26 Batting average 28.35 Slugging average 28.58 OPS 29.13 Linear weights with stolen bases removed due to absence of caught stealing data pre-1950 29.41 Home runs per at bat 29.89 On-base average 30.04 Walks per plate appearance 32.30 Those most impacted by speed (doubles plus triples per at bat, BA as compared to other averages) peaked at the earliest, followed by those directly indicating power (particularly HR/AB), and finally those most affected by plate discipline (walks/PA, OBA as compared to the averages that do not include walks). Bradbury also looked but uncovered no evidence that peak would be later in more recent decades than earlier due to better conditioning and diet. Michael Schell (2005) used career at bat milestones (or plate appearances in the case of the indices including walks), but included the mean age for when those milestones were reached. The ordering is consistent with Bradburys. Based on 1140 players with at least 4000 at bats: Strikeouts in the first 1000 MLB PAs, with declines until around 5000 and steady afterward Triples alone in the range of at bats 1000 to 2000, with steady declines afterward Stolen bases at bats 2000 to 3000, with declines that accelerate after 6000 Batting average at bats 2000 to 3000, but there is almost no drop-off until beyond at bat number 5000 Doubles plus triples at bats 2000 to 3000, with declines that accelerate after 7000 On-base plus slugging plate appearances 3000 to 4000 Slugging average at bats 3000 to 4000 On-base average plate appearances 4000 to 6000 The last three steadily rise beforehand and drop afterward Home runs interestingly, these peaked at ABs 1000 to 2000 before 1920 as most HRs then were inside the park and so speed-dependent; currently a steady rise until 5000 to 7000 (age 34) Walks a rise to plate appearances 6000 to 7000 In addition, Krohn (1983) looked at 66 players between 1955 and 1960 and noted batting average to peak at a mean of 28 and to decline an average of 4 points by age 31, 13 points by age 34, and 20 points by age 37. As part of a study of shirking to be discussed later in this chapter, Maxcy (1997) estimated the peak for slugging average to be age 27. Differences among Players With the exception of Bendtsen (2017), all the research described so far has presumed that although player skill differs, career trajectories are the same across them. This is not true. Along with the overall pattern described above, Bill James in Win Shares (2002) presented figures for the age in which 148 great players had their best season. The number was negligible in the early 20s, at about 15 per year from ages 24 through 26, 20 per year at 27 and 28, and then dipped quickly to about 10 per year from 29 to 32, thereafter decreasing rapidly. This is a far less smooth pattern than the grouped-together data I described earlier. Jim Albert (2002b, 2017) modeled career trajectories for batting runs for all 463 players born after 1910 with 5000 or more plate appearances. Although the average trajectory was the normal inverted-U, there were significant individual differences across players, from Roger Mariss fast rise and fall to Hank Aarons long-term relative consistency across the years. Interestingly, estimated career peaks for each birth decade were later than in most analyses, usually over 28 years and as high as 29.8 for players born in the 1960s. Unfortunately, Jims method included the same assumption as Bradbury, that increase and decrease around career peaks is symmetrical, which of course is incorrect. Hofmann, Jacobs, and Gerras (1992), basing their work on 128 position players who entered the majors between 1970 and 1980 and remained in the league for seven to ten years (the analysis only included the first ten for those lasting longer), uncovered the usual inverted-U curve for batting averages for 107 players but an overall-increasing pattern for 11 and decreasing for 10. A possible explanation is that the former group began in the big leagues at a younger age than average, such that they were just reaching their peak at the ten year mark, whereas the latter group entered late, with their peak toward the beginning. However, Hofmann et al., specifically made such a comparison, and no reliable age difference between the three groups existed in their data. Of particular interest are the discovery of interpretable patterns among trajectories. Nate Silver (2006b), in a continuation of the study on overall career trajectories mentioned earlier, presented evidence that players with two or more skills relevant to offense (high batting average, power, batting eye, and speed) tend to age better than those with only one, probably because if one fails they can fall back on the other. In addition, great players tend to peak later and lose their skills more slowly than do the average players. Returning to a previously mentioned study, Schulz, Musa, Staszewski, and Siegler (1994) divided players into performance-level groups for comparison. They did not find many differences in peak batting average among the groups, but Hall of Famers maxed out significantly later than the norm for home run average (almost 30) and walk average (33). Fielding average was interesting, with peaks of 28.44 for the bottom third, 30.02 for the middle third, 32.93 for the top third, and 35.59 for HOFers. The authors do however admit the selection bias against the very young and old described earlier with this analysis. Hakes and Turner (2011) addressed the issue of calculating differences among career trajectories based on performance data from 1985 through 2005 for hitters with a minimum of 130 at bats in given seasons. As a consequence of the selection bias against very old and young players described earlier, Hakes and Turner divided their sample into quintiles based on OPS adjusted both for season and for position (as replacement level is far lower for shortstops than for first basemen). The data demonstrated that career trajectories do differ substantially based on skill. First, the mean peak (estimated as the third highest seasonal OPS, assuming that the first and second highest represent randomly playing above ones real ability) ranged from 25.6 years of age for the lowest quartile up to 28.2 for the highest, additional evidence that career peak is later for the higher skilled. Second, not only did the lower skilled enter the majors at ages just a bit older than the highly skilled, but young highly skilled players improved their performance more quickly than did the equally-young lowly skilled. This resulted in differences in performance among the five OPS levels beginning immediately upon entering MLB and remaining constant throughout careers, resulting in longer careers for the better hitters. The lowest quintile never rose much above replacement level during their shorter tenures. Finally, given that skill differences matter more for how long careers last than for when they begin, Hakes and Turner determined that the impact of overestimation due to the career-length selection bias is greater for old major leaguers than for young ones. In attempting to address the skill-difference bias, Hakes and Turner may have fallen victim to a couple of biases of their own. Using the third highest adjusted OPS to represent peak performance makes it hard to compare their findings to other studies that have used the very best season. In addition, there is a problem that may be endemic to all this research but that Hakes and Turners approach makes particularly salient. Suppose a player falls below replacement level for a season at a relatively young age and is banished to the minors. Then suppose that player not only reverts to greater than replacement level but performs better than ever, yet has been saddled with the reputation of minor leaguer and is not given the opportunity to show his stuff in the majors during his peak (think Esteban German as a possibility here). If this were the case for a substantial number of lower skilled players, then Hakes and Turner would be underestimating the age of that skill groups peak. Chris Constancio (2007) described a forecast model based on research concerning the development of power in young players. Not surprisingly, it is correlated with player weight, such that heavier players begin with more power and develop it at a quicker rate as they age. Constancio calculated that 175 pounders in the major leagues would expect an isolated power figure of about .090 at age 21 which would increase to about .105 at age 25. In contrast, 215 pounders would increase from .110 to .140 over that span. Defensive position appeared to have no impact beyond player size. Interestingly, upon making the majors, players beginning their careers right out of high school and international signings have typically had more power than those going to college, but the latters power has increased at a faster rate thereafter, closing much of the gap. This may tell us more about team strategy in player scouting and drafting than about the players themselves. Career Length An issue related to age and performance is career length for position players. For most players, it is not long. Rosenberg (1980) examined the issue for players reaching the majors between 1960 through 1964 and computed a mean of 5.67 years but a median of 4.92, denoting that the majority of players have below-mean career lengths and the mean was dragged upward by a relatively few unusually long careers. Rosenbergs data concerning withdrawal rate is instructive. In that cohort, 17.6 percent only appeared in games during one season. Of the remaining players (the hazard rate; the proportion of players who never played another season divided by the number of players still playing during that season), 15.5 disappeared after their second season. The hazard rate dipped to 8.8 in year five, implying that careers lasting this long tend to continue, but then increased to over twenty percent for the ninth and tenth years, after which Rosenbergs data become less useful due to careers still in progress at the time of his analysis, 1975. Witnauer, Rogers, and Saint Onge (2007) performed an analogous but far more inclusive study, including all players with more than a month of major league time whose careers began no earlier than 1902 and no later than 1993. Their computed mean was basically the same as Rosenbergs, but the hazard rate was larger early on 20 percent after one season, 18 percent after the second, and about 12 percent for years 4 through 8, but then back up to 20 percent at year 11 and increasing steadily thereafter. The probable reason for the discrepancy was that Rosenberg chose expansion years for his sample, which likely increase MLB service time. Career length was, not surprisingly, associated with the age of entry, with half-lives of 20-year-old rookies at 10 years, 22 year olds 7 years, 24 year olds 5 years, and 26 year olds 3 years. Note that these data implies that the typical MLBer is done at 29 or 30. Finally, career length increased markedly through the 20th century; the 20 percent dropout rate after the first year hides a decrease from 28 percent between 1902 and 1945 to 14 percent between 1946 and 1968 to 10 percent between 1969 and 2003, with similar decreases through year 7. Krautmann, Clecka, and Skoog (2010) performed a Markov analysis for players between 1997 and 2007, based on the odds of playing in the majors in one year given playing in the majors the previous year. Their method differed from Witnauer et al. in that it allowed for players who had not completed their careers after 2007. They presented tables for both then-active and inactive players listing the mean, median, and modal number of additional expected major league seasons for age cohorts from 20 through 45, along with other data. Schall and Smith (2000a) examined normalized, i.e. z-scores, for batting averages for all twentieth century players in the context of career length, and noted the following: a career length average 5.6 for position players, with hazard rates of 23 percent after the first year going down to a minimum of 13 percent in year seven and then beginning to rise. They did the same analysis for pitchers, who probably due to health had shorter careers but the same general trends; see the Pitching Issues chapter for details. The problem with these studies is that they made no allowance for player quality or type. First and foremost, the better player, the longer the career. For the 1980s, Bill James (1981 Abstract, pages 167-170) uncovered a correlation of .48 between career length and offensive won-lost record for 112 players during the year in which they turned 28. Schall and Smith (2000a) computed a .23 correlation between first-year BA and career length, and Ohkusa (2001) noted that higher slugging average was associated with longer careers for Japanese players between 1977 and 1990. Groothuis and Hill (2008) noted that factors with the largest statistical contribution to the prediction of career length were home runs, RBI, and runs scored per game, stolen base average, and (negatively) strikeouts per game. In addition, performance is associated with age of entry. In an informal but interesting study in the 1987 Baseball Abstract (pages 55 to 73), Bill formed pairs of rookies who had statistically similar seasons but differed on other variables to see if total career performance was affected by those other variables. Age had a huge impact. For example, as compared to those in the data set entering the majors at age 20, those debuting at age 21 hit only 62.3 percent as many home runs, those at 22 46 percent as many, those at 23 36.9 percent as many, with further decrements as rookie age increased. Analogous although less extreme effects occurred for games played, stolen bases, hits, and walks. The players position is also significant, because a player with a given offensive performance can maintain a career longer if he plays a more demanding defensive position. Complicating the matter further is that players often move to less demanding positions as they age, with greater offensive requirements. Groothuis and Hill observed catchers and infielders to have had longer careers than outfielders and first basemen, indicative of the fact that poorer hitting is more likely overlooked for the more challenging defensive positions. Bill James, in the 1987 Baseball Abstract (pages 55-73), in a study just discussed, also noted that the more defensive responsibility, the longer the career, given equivalent offense. Catching was a partial exception, as overall careers were relatively shorter, but only if the catcher was a good offensive player. (I wonder how much of the latter point is due to the stereotype that catchers who are good hitters must be poor fielders.) Second basemen were also at a relative disadvantage compared to shortstops, probably because of the extra susceptibility to injury on double plays (remember that shortstops are facing the baserunner coming from first on double play but second sackers are facing away and thus susceptible to surprise take-outs by baserunners, plus the latter are just in the vulnerable middle-man DP position more often). Longer careers were also associated with beginning ones career with a better team (Bill believed that this leads to better instruction and less force-feeding) and with young player (i.e., speed) rather than old player (i.e., power/walks) skills. Finally, Blacks had far more successful careers than matched Whites, an issue that will come up again later in this book (in the Organizational Issue chapter, I summarize evidence that Blacks tend to hit better than both Whites and Latins). This leads to the question of whether there have been racial differences in career length independently of performance. Jiobu (1988) examined this issue, using position players with careers of at least 50 games between 1971 and 1985. Jiobu first noted that Blacks tended to have significantly longer careers than Whites, but after adjusting for their performance advantage (based on BA, SA, and OBA), Blacks careers turned out to be shorter, all else being equal. Responding to this first attempt, Groothuis and Hill (2008) considered year-by-year performance rather than career totals. They included all 3185 players who appeared in a game between 1990 and 2004; I will describe their findings for hitters here and for pitchers in Pitching Issues chapter. In general, the hazard rate was relatively high for the first year (for whites, 18 percent), went down to a minimum around the sixth and seventh (for whites, 7 percent), and then started increasing, topping 20 percent at year eleven and 30 percent at year fourteen. In contrast with Jiobu, after taking these indices into account, career length differences between whites and blacks failed to reach statistical significance; Hispanic career length was shorter for the same performance during the first ten years of the data set but not the next five. Sophomore Slump It is true that outstanding rookie seasons are usually followed by poorer second ones, but the reason is likely the fact that outstanding rookie seasons are often due to players randomly hitting above their then-current ability level and then returning toward that level afterward. This is an example of regression toward the mean (Bill Jamess Plexiglass Principle) that occurs for both players throughout their careers and teams over the years after unusually good or bad seasons. Taylor and Cuave (1994) put it to the test, based on an evaluation of 82 rookie hitters between 1945 and 1983 who produced at least one of the following: the number of home runs (at least 26), RBI (at least 99), or batting average (at least .298) one standard deviation better than the average rookie. Comparing the first, second, and combined third through fifth seasons, their data show no real differences across time for HR and RBI and regression toward the mean for BA (.300 in Year 1, .276 in Year 2, and .269 in Years 3-5). In their 1993 Baseball Analyst (page 28), Siwoff, Hirdt, Hirdt, and Hirdt examined the first and second year batting averages of hitters from 1942 through 1991. Of 228 batting .250 or above in their rookie season, 63 percent declined the next season, whereas the exact same 63 percent of 217 players hitting less than .250 as freshmen increased those averages. The implication is that rookies hitting over their head should be expected to decline and those under their heads to improve the next season. And Robert Murden (1990) claimed that there is indeed such a thing as a career year in the sense of one season that was at least 20 percent higher than any other. He noted that 18 percent of 996 position players with at least five years of 300 at bats or four years of 400 at bats had career years as just defined in three of the following five indices: total extra base hits, batting average, stolen bases, runs scored, and runs batted in. I certainly grant that the career year exists, but Murdens analysis cannot distinguish between differences due to chance and differences due to actual peak skill during one season. Clutch Hitting What Counts as Evidence for the Existence of Clutch Hitters? In the Situation chapter, I covered the extent to which offensive performance is affected by the extent to which the game is on the line. The issue at hand now is whether or not there is a specific skill involved in clutch hitting, such that players are characteristically good or bad at it. Unenlightened commentary assumes as such, but without any good evidence. The idea of a good RBI man as a clutch player is unhelpful; runs batted in can be amassed in all sorts of situations, clutch and otherwise. Further, players often identified as good RBI men/clutch hitters may just be reaping the benefits of batting with an inordinate number of baserunners; Tommy Henrich and Tony Perez come to mind as examples. The game-winning RBI index was supposed to measure clutch hitting, but, for reasons already discussed, it died a quick and well-deserved death. Earnshaw Cook (1971, p. 92) believed that clutch hitting, as everything else in the game, is related only to the batsmans general level of skill inevitably controlled by the laws of chance. Unfortunately, his method for demonstrating this, showing that the 1969 batting averages of a group of players as pinch-hitters were not significantly different than the overall batting averages of those players, is not convincing; clutch hitting and pinch hitting are in no way synonymous. Somewhat more usefully, Cook defined a clutch hitting factor (CFI) as a players batting average with runners on base divided by the players overall batting average, and estimated through simulation methods that an increase in a teams CFI of 10 points would result in about 36 additional runs per season. Confusing the matter, an accompanying diagram includes a definition of CFI as number of hits with runners on base divided by total number of hits. These two measurement procedures are not equivalent except in the unlikely case that the odds of base-runners is identical in the cases when players both do and do not get hits. The claim that there is such a thing as a clutch hitter presupposes two potentially-testable assumptions: first, that clutch hitters hit better when the game is on the line than when it isnt, and second, that this occurs year after year. The first of these presuppositions relates to the fact that hitters often identified as good in the clutch may just be good hitters overall, in which case the clutch label means nothing over and above good hitter. The second distinguishes players who might hit well when the game is in doubt in a given year through luck from those with a real skill. The first serious effort at meeting these requirements was in Richard Cramers (1977) article discussed last chapter in which he proposed Batter Win Average. It turns out that the relationship between BWA and the Mills PWA in Dicks data set could be expressed in the following simple regression equation PWA = .484 + (1.37 x BWA) which accounts for eighty percent of the variation in the data. In other words, eighty percent of PWA is due to general batter skill irrespective of clutch hitting. The remaining twenty percent, however, could be a function of clutch hitting, and can be exploited to examine whether or not it is a skill independent of general hitting prowess. The way to do this is simple. First, compute the difference between the actual PWA for a player in a given year and what his PWA would be as predicted by the equation above. If actual PWA is greater than predicted PWA, then the player has performed better in the clutch during that season than overall; if actual PWA is lower, then the player has wilted under the pressure. The larger (smaller) the actual/predicted discrepancy, the better (worse) the player has performed with the game on the line relative to his normal output in a given season. Second, correlate the discrepancy across players over two seasons. Dick did just this (which I will henceforth call the Cramer method) across 1969 and 1970. He did not report correlations but rather variance shared across the two years, which is the correlation squared. The results: for 60 NL players, .038; for 62 AL players, .055. His conclusion; good hitters are good hitters and weak hitters are weak hitters regardless of the game situation (page 79). This is an important preliminary finding, but the conclusion is premature. First, Dick did not state how the players in the sample were chosen. Second, a squared correlation of .038 means the original correlation was .195 for the NL; of .055 signals a correlation of .234. These are small but not zero, and the latter is statistically significant at .10. Further, the study ran into the issue of statistical power, a measure of the odds that if there is a real phenomenon, the researcher will find it given the size of the researchers sample and the effect size (the strength of the phenomenons impact). With a sample size of only 60, if the real-life effect size would be a correlation of, for example, .1, then Dick had only a .19 chance of finding it. The jury was clearly not out on the clutch-hitting-as-a-skill question. In an essay from 2004, Bill James claimed that Dicks conclusion was not merely premature, but founded on a flawed method. Basically, Bills argument is that a comparison of differences between measures compounds measurement error in those measures to the point that, if no difference exists, there is no way of telling whether that absence of difference really means no difference or is only a function of accumulated measurement error (Jim Albert [2005b] made the same point in a response to this essay). This argument is absolutely correct in principle. Let us suppose that PWA has a measurement reliability of .8, which (given that 1.0 is perfect measurement and 0.0 is absolutely no reliability at all) is fairly error-free. The prediction of PWA would then have the same reliability, such that the difference score between the two has a reliability of the two multiplied by one another (.64). Further, Cramer compared those differences across two years, so we again have to square the reliability figure. As .64 times .64 equals .41, reliability is indeed poor. Thus a reliable index treated in this way results in an analysis that cannot be trusted. The problem is that Bill then went on to make the following statement (page 31), quoted both here and in Phil Birnbaums (2008a) response to Bills essay: Random data proves nothing and it cannot be used as proof of nothingness. Why? Because whenever you do a study, if your study completely fails, you will get random data. Therefore, when you get random data, all you may conclude is that your study has failed. This quotation is fundamentally wrong in two ways. First, the quotation suffers from an obvious logical error, as failed study means random data (A causes B) does not at all imply random data means failed study (B causes A). As Phil pointed out, random data make it possible that the study has failed. This leads to the second error. Although it is true that random data cannot be proof of nothingness, this is irrelevant, as one cannot prove anything with numerical data. One can, however, use random data as evidence of nothingness in a circumstance in which if there is something going on in the real world statistical power is sufficiently high to have found that something. The implication of these two points is that when statistical power is high random data are strong evidence (although not proof) that the phenomenon under investigation is indeed a random process. Incidentally, Tom Conlon (1990) made the same criticism of some work by Rob Wood to be discussed below, correctly attributing the issue to one of statistical power, and challenging researchers to insure its sufficiency. In the empirical part of his response, Phil first used Retrosheet data to compute correlations between the differences between clutch and non-clutch batting averages (defined as Elias LIP) for players with at least 50 clutch ABs in every pairing of two seasons from 1974-1975 to 1989-1990 (excluding the two pairings including the 1981 strike season). Interestingly, 12 of the 14 correlations were positive, but all of these positives were less than .1, and the overall average correlation was .021. Second, Phil simulated what the distribution of these clutch-non clutch differences would have been if clutch hitting is a randomly distributed skill, such that about 68% of the players had a difference between 1 and -1 standard deviations from the mean, 28% had a difference either between 1 & 2 s.d.s or -1 and -2 s.d.s from mean, and 5% more extreme than either 2 or -2 s.d.s. In this case, the mean correlation across two-season pairings was .239 and was likely to occur by chance less than five percent of the time for 11 of the 14 seasonal comparisons. Thus it was likely that if clutch hitting was a randomly distributed skill, Cramers method would have uncovered evidence of it. Third, Phil computed the statistical power for such correlations, and noted that if clutch hitting was a skill but weak enough such that the season-by season correlation was only .2, Cramers method would still have a 77 percent chance of finding it. Statistical power for a correlation of .15 would still be slightly in Cramers favor (.55), although it finally drops below that (.32) with a correlation of .10. The conclusion we must reach is that if clutch hitting actually exists, its impact on performance must be relatively small, because if there was any appreciable difference between clutch and choking players it would have been revealed in these tests. Phils response was followed by Bills (2008) defense of his claims. Here, he brought up an important issue: that clutch hitting could be a real skill but only possessed by a minority of players, and the effect of their skill is buried when we evaluate the existence of the skill among all players. Bill simulated two seasons of a situation in which 100 players accumulating 600 at bats each in a season include 80 whose performance is normally distributed around a mean batting average of .270 and 20 whose performance in 150 of those at bats (representing clutch opportunities, however defined) can range between either their normal batting average to as much as 50 points better or worse than that norm, with the mean of the range 25 points. Over two seasons, 62.5% of these 20 players were either above or below their overall BAs in both simulated seasons and 37.5% were above on one and below in the other, which is what one would expect by chance. The other 80 players were analogously consistent across seasons 50% of the time and not 50%, again as chance would dictate. However, when one combines 62.5% for 20 players and 50% for 80 players, the overall consistency-across-seasons percentage would be 52.5%. Bill then argued that this is too close to 50% for an analyst to tease out the presence of consistent clutch players. Oh yes the analyst can, Phil (2008b) demonstrated in the final statement in this dialogue, if she uses more sensitive statistical than did Bill. Bill basically stacked the deck in his favor by reporting his conclusions as if his data were ordinal (greater or less than, but nothing more) whereas the batting average (or whatever index is used) allows the researcher to treat it as interval (exact numbers greater or less than). Interval data allow for more sensitive analysis than does ordinal. Phil duplicated Bills simulation over 56 seasons and noted what would be taken as significant deviation from randomness in 11 of them, which would alert any analyst that there is something going on in the data. The entire colloquy ends with an essay by Dick Cramer and Pete Palmer (2008). Here, they presented more evidence that clutch hitters probably do not exist, using the Cramer method with a Retrosheet-based sample of 857 players with at least 3000 plate appearances between 1957 and 2007. The difference between clutch situations (defined according to the top 10 percent as defined by the Mills brothers method) and non-clutch situations in consecutive 250+ PA seasons correlated something in the order of a nonexistent .05. To conclude, there is some disconnect between the two, as Bill was chiefly concerned with whether the sort of comparison that Dick did is valid in general (he mentioned quite a few other relevant circumstances in the original article, and this issue will crop again when I cover streakiness later in this chapter) whereas Phil and Dick/Pete were testing the very existence of clutch hitting as a real phenomenon. But the point remains that while one cannot prove that there is no such thing as a clutch hitter, one can show evidence that if there is such a thing, its impact very small. Other Evidence That Clutch Hitters Do Not Exist Other studies exist that uncovered no evidence that there is such a thing as a clutch hitter, but some of these studies were not performed as well as one would like. One must define what counts as a clutch situation. I would define a clutch situation as existing when the outcome of the plate appearance has a relatively large impact on which team wins the game; in other words, there is high leverage. Some researchers have used the runners in and not in scoring position distinction as distinguishing clutch and not clutch, but that has little to do with leverage. A better definition is that offered by STATS; a close and late circumstance occurs in the seventh inning or later with a one-run lead, a tie score, or the tying run on base, at bat, or on deck. Using this definition, Dan Levitt (2003b), based on 40 players with at least 350 at bats for every year between 1992 and 1998 (280 player-seasons), first calculated an overall BA of .276 in clutch situations and .291 in non-clutch. Using the .015 spread as a correction when applicable, Dan used a number of non-parametric procedures to show that players were better or worse in clutch than non-clutch in random patterns, and the correlation for the difference in player BA in clutch minus non-clutch circumstances between 1992-1995 and 1996-1998 was a scant .13. However, the sample size was low, just the 40 players, limiting the studys value. Using Retrosheet data from 1989 to 1992, J. C. Bradbury (2011) adopted runners in scoring position position versus none, as noted a poor measure of clutch ability, and noted only tiny differences over and above overall differences in offensive ability. With a decent sample size of 245 and clutch defined the STATS way, David Grabiner (1993) noted a correlation of .01 in OBAs for clutch situations between 1991 and 1992. In the most comprehensive study to date, Rob Mains and Pete Palmer (2018) compared Batting Runs with and without correction for situational leverage for all 8963 players since World War II (ending I suppose in 2017) who had at least 500 PA, and uncovered little consistency across seasons in whether players did better or worse with the correction than without. The player who did better the most was Bert Campaneris, and the player who did worse the most was Manny Ramirez; six times for each, and keep in mind that both had relatively long careers to help provide the opportunity to be best or worst. Flawed Claims that Clutch Hitters Exist A well-publicized paper by a University of Pennsylvania student named Elan Fuld that is easy to access online (search for Elan Fuld clutch) claims that clutch hitters really do exist. Fuld defined the importance of clutch situations according to his computation of their leverage, and then compared through regression analysis the batters performance in terms of bases gained per plate appearance (0 to 4) on the plate appearances specific leverage. If a player did substantially better (worse) in high leverage situations than in low during a given season, then Fuld labeled the player as clutch (choke) in that season. The real issue was whether a player was consistently clutch or choke across their entire career. He used Retrosheet data for 1974 through 1992 for 1075 player with at least two seasons with 100 PAs, including each season reaching that threshold of play (6784 player-seasons in all). He then computed a measure of clutch tendencies across seasons with a threshold defined such that only 1 percent (11 of 1075) of players would be considered clutch and another 1 percent (another 11) choke by chance. When Fuld treated sacrifice flies under the very strange assumption that they are analogous in value to walks, as many as 24 players met the criteria of consistent clutchness across seasons, although never more than 7 reached that for chokeness. As Phil Birnbaum noted (2005c), this assumption inflates the value of a fly ball with a runner on third over fly balls in other situations, as SFs are more likely to be clutch in clutch situations than the average base-out configuration, while at the same time treating them as walks credits the batter an extra base they did not really earn, artificially inflating their bases gained in clutch situations. When Fuld excluded SFs from the data set, no more than 8 hitters met his criteria for clutchness. Therefore, despite a U. Penn press release claiming that the existence of clutch hitters had been proven along with the media sources that accepted that claim, Fulds study failed to find the existence of clutch hitters. As usual attempting to defend the status quo they represented against the onslaught of sabermetrics, the Elias people were absolutely convinced that clutch and choke hitters were a reality, and presented data that they believed proved the point. In the 1985 Baseball Analyst (pages 383-385), they listed the ten hitters with the biggest positive and ten with the biggest negative differential between their batting average in late inning pressure situations and their overall BA, and noted that a large majority of the former again hit better in LIP situations in 1984 than overall whereas most of the latter again hit worse. Analogous lists for 1984 resulted in the same tendency for 1983. They then claimed that this proved that clutch hitting is a true skill. In the 1987 Baseball Analyst (pages 54-55), they looked at the 30 players with the biggest positive and negative differentials for 1975 through 1984, and again the majority remained consistently above or below respectively. In the 1988 edition (pages 61-62), they took four three-year periods and reported that players whose LIP differentials were positive in the first two years had a better differential than players whose differentials were positive in one season and negative in the other, whose third-year differential was in turn better than players whose differential first two years were both negative. They claimed to have proven the point, and unfortunately badly marred their attempt with consistent unnecessary insults of the competence of researchers such as Cramer who had concluded otherwise. Harold Brooks (1989) demonstrated that they were wrong about their own data. The players chosen in 1985 and 1987 were cherry picked to be consistent with their point. Although they did look at the entire relevant distribution in 1988, they did not conduct any true statistical tests, and when Brooks did, he uncovered the fact that the actual number of players who were either always above or always below average was fewer than expected by chance. In addition, Brooks did the following: 1 for all six pairings of seasons between 1984 and 1987, he calculated correlations for late inning pressure situations batting averages separately for each league for all players with 25, 50, or 75 such at bats. Of 36 total correlations, 20 were positive and 16 negative. He did not do so, so I calculated the mean correlation across the 36 as .015, showing absolutely no overall relationship across seasons. For comparison, Brooks included the correlations across overall batting averages for the same group of players; these averaged to .413, in the ballpark of the customary BA correlations noted in the Offensive Evaluation chapter. 2 Among those 135 players who had 25 or more late inning pressure at bats across all four seasons, the distribution of batters above average across leagues in zero (11), one (26), two (56), three (34), and all four (8) seasons was pretty close to what chance would allow for (1/16th, or 8 for zero and all four seasons; 1/4th, or 34 for one and three; 3/8th, or 51 for two). The distributions for 50 and particularly 75 at bats are admittedly not as close to chance, although here sample size problems become evident. 3 The total number of players with LIP BAs one standard deviation either above or below for two of the four seasons (15) is the same as the number of players with one season one s.d. above and one season one s.d. below (14). Other researchers have used Eliass own data against them. Gary Gillette (1986), in an essay that comprised most of the very first issue of the short-lived Sabermetric Review monthly, employed the Cramer method based on late inning pressure situations and overall BAs between 1984 and 1985 for all 238 players with relevant data in the Elias 1985 and 1986 books. Pete Palmer (1990) used 330 players listed by Elias with at least 250 at bats in late inning pressure situations over the previous 10 years. In both cases, the resulting distributions of difference scores were basically normal, implying random differences. Rob Wood (1989) also used Elias data (exactly which is not clear) and concluded that the number of players clearly exhibiting clutch tendencies were what would have been expected by chance. As noted above, Tom Conlon (1990) questioned the statistical power of Robs analysis, and Keith Karcher (1991) provided evidence that Robs study was indeed underpowered for claiming that clutch hitters do not exist. Nonetheless, in summary, a competent analysis of Eliass own data reveals absolutely no evidence that clutch hitting is a skill. Good Evidence That Clutch Hitters Exist Some of our more skilled researchers have uncovered evidence that clutch hitting may be a skill, but if it is, its effect on the outcome of games is much smaller than baseball people believe. Nate Silver (2006c) examined players between 1972 and 2005. For each player season, he computed their overall Marginal Lineup Value (see the Offensive Evaluation chapter on that) translated into wins, weighted it by the average leverage scores for their at bats (which does not vary very much across players), and then compared that figure to their Win Expectancy (as mentioned in that chapter, a version of Player Win Average). A regression equation indicated that the two numbers shared seventy percent of their variance, implying that the remaining variance, the extent to which players Win Expectancy was better or worse than their weighted MLV, consists either of random variation or actual differences in clutch hitting ability. He then broke the career of the players in his sample in half, to see if the difference between Win Expectancy and MLV was consistent. The two career halves shared about 10 percent of their variance, implying that this may be the contribution of actual clutch ability to the difference whereas 90 percent is random luck. Interestingly, this difference correlated .29 with a measure of walks versus strikeouts, suggesting that if there is some skill involved in clutch hitting, it may be at least a bit connected with plate discipline. But adding this difference to the regression equation above added 2 percent more variance accounted for than the previous 70. In summary, Silvers work would mean that performance in clutch situations is 70 percent overall skill, 28 percent luck, and 2 percent clutch ability. An exception to the general finding that clutch hitting is not a skill can be found in work by the TMA group (2006), specifically (according to Lee Panas) in work by Andrew Dolphin. In The Book, Andrew used defined clutch as eighth inning or later with team one, two, or three runs behind, and compared player OBA and wOBA in such situations with their career levels. Based on Retrosheet data from 1960 through 1992 and 2000 through 2004, for 848 players with at least 100 clutch and 400 non-clutch plate appearances, he noted a non-random number of players significantly above or below their norm (at least 8 OBA or 6 wOBA units) during clutch situations. However, regressing to the mean dropped this effect to about 1 measly point. In on-line posts, Andrew continued this work. Dolphin was concerned that previous studies were underpowered, so he guaranteed large sample sizes in two ways: first, a liberal definition of what counts as clutch (6th inning or later, tie score or tying run on base, at bat, or on deck) with Retrosheet data from 1969, 1972 to 1992, plus American League 1963, 1967, and 1968. The result was 612 players with at least 250 plate appearances in clutch situations and 1000 in non-clutch. He used a slightly modified OBA (hits and walks divided by plate appearances minus sacrifice bunts and flies. Correcting for the fact that overall modified OBA was a bit lower in clutch situations than non-clutch (.322 versus .331), Dolphin compared the distribution of actual performance by the 612 relevant players with the results of Monte Carlo trials assuming the same overall modified OBA (.329) and noted significant differences, such that 28% of the variation among player performance could be credited to differences in clutch ability. Having said this, random effects were five times larger than clutch effects, revealing that clutch hitting may be a real skill but it is not particularly important. Assuming that it is real, Dolphin found it to be negatively related with slugging average, leading to the result that some of the best clutch hitters (in terms of getting on base more often in clutch situations than not) were singles hitters, some exceptional (such as Tony Gwynn and Rickey Henderson) and some not (Rafael Ramirez, Alfredo Griffin), and some of the worst were power hitters (Carl Yastrzemski, Jack Clark, and interestingly, Dave Winfield [remember Steinbrenner calling him Mr. April?] and Mr. October himself, Reggie Jackson). This conclusion is analogous to Silvers correlation between his clutch index and plate discipline as described above. Interestingly, Bill James (1985b) presaged this insight while performing a detailed examination of the LIPS data in the 1985 Elias Baseball Analyst. Of 221 regular players included, the 115 above the mean in frequency of walks lost only 3 points in batting average in LIPS compared to other circumstances, with almost half (57, or 49.6%) actually gaining. In contrast, only 41 (39%) of the 106 batters below mean in walk frequency hit better in LIPS, and the average loss was 14 points. In the same piece, Bill also reported a study of player experience, but his results are inconclusive. One problem in his analysis is that it included some very experienced players still in the lineup despite no longer being competent hitters, and he thought that might muddy up the findings. In either case, Cy Morong (2000) looked at 1985 STATS close and late OPS data for 175 players and, after removing the variance in these data from players overall OPS, uncovered a significant positive effect for experienced players (2000 PA or more) over inexperienced. One might question his simple dichotomous division rather than using actual PAs and getting a more powerful test, but if one accepted the washed-up experienced player argument above, one would need to remove such players beforehand. Jim Albert (2007) also reported some evidence suggesting that there might be clutch differences in walk and strikeout rates. Conclusion Recent evidence suggests that there may well an actual skill involved in clutch hitting. If so, its impact is extremely weak, about the same as that for base running, and the skill is most noticeable in disciplined singles hitters rather than sluggers. This is one time in which the received wisdom imparted by television analysts may have a grain of truth, but only a grain. Also, as Pete Palmer pointed out to me in a personal communication, a player who really bears down when the game is on the line is in effect being lazy when the game isnt; shouldnt he be bearing down throughout the game? Great Feat Odds Statisticians have put significant effort into examining the odds of achieving great hitting feats, based on assumptions of differing complexity. Hal Stern (1995, 1997b), for example, calculated the odds of hitting streaks of 10 and 30 games for players with different batting averages, with the number of at bats per game either fixed or variable and whether the odds of a hit each at bat are either fixed or variable (dependent on opposing pitcher skill and whether the platoon advantage is or is not in the hitters favor). Odds are greatest when both at bats per game and odds per at bat are fixed, because when they are not fixed, the chance of either fewer at bats or tougher pitchers than average tends to deflate the odds. Statisticians have been particularly attentive to two feats in particular, hitting .400 in a season (inspired by 1941 Ted Williams) and 56 games in a row (Joe DiMaggio, of course), with several appearing in the relatively non-technical statistics journal Chance. I review these in turn, and then turn to other great feats. Odds of Hitting .400 As I write this, it has been more than seventy five years since Ted Williams hit .400, so that achievement certainly counts for a great feat, and there is interest in how likely it is to happen again. Dallas Adams (1981a, 1983) was the first to consider the issue, using Shoebothams (1976) relative average measure (to be discussed in the next [Historical Changes] section of this chapter), the ratio between a given players BA and the league average BA. Dallas noted that in the 1902 through 1980 period the distribution of the relative average for the league leading BAs was distributed about normally with a mean of 1.36 and a standard deviation of .075. Using this normal distribution allowed Dallas to compute the odds of hitting .400 given differing mean league batting averages. For example, in the 1941 to 1976 stretch the mean batting average was .255, so that it would take a relative average of 1.57 to hit .400, which is about the highest ever achieved. Thus it may not be surprising to note that there was only an 18 percent chance of someone hitting .400 in this time period. It would take a .295 league average for the odds of someone reaching the .400 plateau to be fifty percent. Several years before he took over the Statistician Looks at Sports column from Hal Stern, Scott Berry (1991) took the issue in a different direction. If we assume that a players at bats are independent of one another (i.e., streakiness is random and there is no such thing as a hot hand), then one can compute the probability of hitting .400 through computations based on the binomial distribution given the true batting average and a given number of at bats. For example, for 500 at bats, the odds of a .300 hitter reaching .400 is .00000072; for a .350 hitter, it is .011; for a .400 hitter, it is .50 (Berry included hitting .3995 here and raised the probability to .52, but that is impossible; the next lowest possible average for 500 BA is .396. Thanks to Cliff Blau for raising this issue). Berry concluded that if the mean batting average in a given season were .270, the odds of a .400 hitter over a stretch of 50 years would be three and one-half percent. As Lackritz (1996) pointed out, although at bats within a season may be independent of one another, Berrys assumptions precludes the fact that players do not have the same true batting average throughout their career but generally traverse the U-shaped relationship discussed earlier in this chapter. This means that Berrys .300 hitter would actually be considerably better than .300 at his peak, which implies that Berrys numbers are probably underestimates. Lackritzs basic thinking was actually similar to Berrys, but he added a number of interesting points. First, the odds of hitting .400 are increased with fewer at bats, because random fluctuations in the hitters favor are more likely. Thus the feat is more likely to occur in to shorter seasons (strike seasons come to mind), to players losing time to injury, or a willingness to walk a lot; Lackritz even mentioned the early 1960s increase from a 154 to a 162 game schedule as a detriment. Second, expansion may dilute pitching talent enough to increase batting averages by 10 to 20 points for a couple of years, enough to double or triple the odds. Third, home park effects can be significant. Anyway, Lackritz used data from 1985 and 1986 as the basis for the estimate that, given the general performance level of that time, a .400 hitter was two-fifths of a percent likely in a given year, which works out to a 20 percent chance in 50 years. Bickel (2004) did an insightful analysis of why no one hits .400 anymore. Simply put, batting average can be looked upon as a composite of two parts, the odds of striking out and the odds of getting a hit when not striking out. If the former is higher, then the latter also has to be higher to make up for it. The strikeout rate has pretty much doubled since 1941, and the rate of getting hits on batted balls hasnt changed to make up for it. Using Bickels method, I calculated that with the current mean strikeout rate you would need an average of .494 on hits on batted balls to reach .400, and according to Bickel the best ever has been .478 (the Babe in 1923 and Manny Ramirez in 2000). When Tony Gwynn (.394 in 1994) and George Brett (.390 in 1980) flirted with the achievement, both struck out a bit less than five percent of the time, which is the sort of proportion one probably needs to have a reasonable chance, but their BA on batted balls (.413 and .410 respectively) wasnt quite good enough. Odds of a 56-Game Hitting Streak In the case of the other popular issue, many different approaches have been taken to figuring out the odds of Joe DiMaggio having a 56 game hitting streak, also in 1941. Several of them have multiple flaws. Let me list some of the ways that a researcher can err in this computation: 1 Taking the average number of at bats per game DiMaggio had in 1941 (3.89), and using that as the basis for figuring out the odds of him getting a hit in game. Problems with this include A Neglecting to realize that you have to use plate appearances rather than at bats, because if the player gets a walk or a hit by pitch, he cannot get a hit; a game with one of these every PA gives you a great OBA but ends the streak. DiMaggio averaged 4.44 PA per game. B Assuming that all games are created equal. DiMaggio had three games with only three and one with just two PA. Although the odds of getting a hit in a six or seven PA game would be greater than in one with 4.44, this advantage is more than counterbalanced by the lower odds of a hit in a two or three PA game. 2 Computing the opportunities DiMaggio had to begin a 56-game streak given that he played 139 games that season; i.e. games 1 to 56, 2 to 57, 3 to 58, and so on until games 84 through 139. And, as there are 84 possible opportunities in this sense, then taking the computed odds per game and multiplying it by 84. However, as pointed out by Scott Berry (1991) and Michael Freiman (2003), there is a problem with in essence adding the .000016 84 times, because this only works if the stretches were all independent of one another. In this case, stretches are not at all independent because they all overlap (for example, for the stretches games 1 to 56, 2 to 57, and 3 to 58, games 3 through 56 occur in all three). 3 Using DiMaggios career average of .325 as a basis for computing the odds of hits per PA; not only is this guilty of 1A, but it uses the wrong baseline; you have to use the 1941 figure. To be fair, there has also been research explicitly examining the odds of a 56-game streak throughout his entire career. The guilty parties, with their errors and computed figure, include Short and Wasserman (1989; 1A, 1B, 2, 3; .00000213), Blahous (1994; 1A, 1B, 2; .00134), Coelho, Grunow, Myers, and Rasp (1994; 1A, 1B, 2; who used DiMaggios .325 to estimate the odds for his entire career at .00049), and Warrick (1995; 1A, 2 and 3; .000274). In addition, Coelho et al. used data for 245 players who appeared in 100 games during 1993 to estimate odds of .002697 for any of them achieving a 56 game streak during that season. Warrack also did further calculations, for example concluding that the odds of a 56 game streak in a 3000 game career for a player with a .400 lifetime average is .25, but for a more realistic .350 lifetime average would be .017. Obviously, none of these figures can be trusted. Incidentally, all but Blahous were published in statistics outlets, such that at least some of the mistakes they made should have been caught. Blahouss attempt was in the Baseball Research Journal, which redeemed itself in 2004 by including an article by Brown and Goodrich detailing Blahouss mistakes and demonstrating why it is a far more difficult problem to solve than had been realized by those attempting it. In a somewhat more defensible piece of work, Chance (2009, ironically publishing his work in Chance), was only negligent in the case of 1B, but used a questionable measure of plate appearances. He considered any attempt at a hit as a plate appearance minus hit by pitches, intentional walks, and sacrifice hits, under the assumption that when batters get unintentional walks they are unsuccessfully trying to get a hit. Chance then used DiMaggios total number of such batting opportunities and success rate given that measure of at bats (.370) during the streak to calculate its odds as .00036. Interestingly, estimated the same way over his entire career it was lower, .000295, because his lifetime BA was much lower than during the relevant 56 games. Chance also estimated the odds of the top 100 batting average players with at least 3000 plate appearances (Ty Cobb topped the list at .00489; DiMaggio was in 28th place on the list) and the odds of any of them doing it as .0442. Better yet, Beltrami and Mendelsohn (2010) used the actual distribution of hits and at bats per game during the streak (he hit .408 during it), simulated the theoretical distribution of hits per game given those data and an otherwise random process in order to estimate how many of the 56 games ought to have had a hit (45), and determined that 56 is significantly more than 45 at lower than .01 odds. An analogous study of Pete Roses 44 game streak using Retrosheet data had similar results. Of course, this does not provide us with the actual probability of either streak, but it does allow us to conclude that the odds of each occurring by chance were extremely low. Brown and Goodrich (2004) smartly used DiMaggios performance for his entire 1936-1940 peak, in which he averaged .343 and had 31.4 hits per 100 plate appearances, but unfortunately assumed 4.5 PA each game (error 1B) in their simulation. Anyway, running a million simulated seasons, they ended up an estimate of 1 in 18519 (.000222). Better yet again, Joe DAniello (2004) used DiMaggios actual number of plate appearances for each game during the streak and a random sprinkling of his actual 91 hits during that streak in a million season computer simulation to see the odds of at least one of those 91 hits occurring in each game, and wound up with 1 in 66,667. Michael Freimans (2003) method took into consideration the fact that, for example, if a player had a 56 game streak between for example games 52 and 107, he must not have had a hit in games 51 and 108, so figuring out the odds for that specific streak disqualifies streaks including those two games. Anyway, his method led Freiman to claim DiMaggios 1941 odds at 1 in 9545 (.000105), and as the most unlikely 30+ game hit streak of all time (second place went to Tony Eusebios 2000 24-game streak, at 1 in 7436 given his usual hitting performance). Incidentally, in a relevant discussion Levitt (2004) read Freiman as .000197; I dont see where he got that figure from. Some researchers were more interested in the odds of anyone hitting in 56 consecutive games rather than only DiMaggio. Scott Berry (1991) attempted to compute the odds for various performance levels given four at bats per game, plus the assumption that a players at bats are statistically independent from one another. Through matrix manipulations Berry arrived at a probability of .0000052 for a .300 hitter, .00031 for a .350 hitter (almost what DiMaggio hit that season), and .0057 for a .400 hitter. Gosnell, Keepler, and Turner (1997) also assumed that a player gets four at bats a game and based their calculation on a players career batting average. This method underpredicts the number of actual batting streaks of 25 of more games in a sample of 100 randomly chosen players with at least 3000 at bats active between 1876 and 1987, due to at least the following two reasons: an underestimate of at bats per game and ignoring career trajectories (during the peak of a players career, they are more likely by chance to have a long hitting streak than during the beginning and end phases). Anyway, given their career batting averages, they calculated the odds of Willie Keelers 44-game streak as .0571, Pete Roses 44-game streak as .0097, and Joe DiMaggios 56-game streak as .0002. Arbesman and Strogatz, in an unpublished paper detailing material summarized in the March 30th, 2008 edition of the New York Times, simulated 10,000 baseball histories including the records of all regular position players from 1871 to 2004. Their simplest model represented each players plate appearances per game as the average over their career and no variance in their performance across games; more complicated models included 10% and 20% variation in one or the other of these, but the findings did not change much. They concluded that the odds of someone having a 56 game hit streak across all the included years were 49 percent. The nineteenth century saw some highly inflated batting statistics; for the 1905 through 2004 stretch, it was a still-large 20 percent. Thomas (2010) explicitly followed up on the Arbesman and Strogatz work by figuring out a model that would predict the number of 30+ game streaks and used that model to predict DiMaggio streak odds. This model estimated plate appearances per game as 1/9th the total for a team per game, a very problematic move the authors as it assumes the same plate appearances per game for all lineup positions). It also added a random element for each games performance to simulate normal between-game variance and divided the games history into three epochs (1871-1900, 1901-1939, 1950-2009) because of decreases in performance variation among players over time (to be discussed in the Historical Changes section of this chapter), with the 1940s excluded just because the streak occurred during that period. The resulting probability for a DiMaggio streak occurring during that entire history was 15.2 percent. Thomas also attempted to model the less-publicized 84-game on-base streak Ted Williams had on 1949, but was unable to find a solution that recreated the number of 50+ game streaks of this kind. Horowitz and Lackritz (2013) piggy-backed on Chances method in calculating the odds that players with given lifetime batting averages and career lengths would randomly achieve a hitting streak of a given length, with particular attention paid to streaks in the mid-20s due to their relatively rarity when compared to streaks of, say, 10 games. Rockoff and Yates (2009) have done what I think is the best of all these attempts. Their idea was to simulate 1000 seasons of play using actual seasonal game-to-game performance for each of 58 years of Retrosheet data. Out of the 58,000 simulated seasons, a total of 30 (about .005%) included a hitting streak of 56 or more games. Interestingly, Ichiros 2004 season was included in five of them. Using these data, the authors concluded that the odds of a streak of more than 56 games in any of the 58 seasons in the data set was about 2 percent. In a follow-up (Rockoff & Yates, 2011), they performed 1000 simulated baseball histories under a number of different assumptions: the odds of a hit directly determined by player batting average, including the odds of a hit determined by a varying amount centered around the player batting average, and the odds of a hit partly determined by overall batting average but also by performance in 15 and 30 game hitting streaks around each game under question. The latter two methods stack the deck, because part of the point of this work in the first place is to determine whether or not long hitting streaks are likely. As a consequence, the author(s) uncovered 85 56-game or greater streaks using the batting average approach, 88 using the variation around batting average approach, 561 using the 15 game approach, and 432 using the 30 game approach. I do not take the latter two of these seriously, but the former two seem to be valid. To make this point more clearly, the simulated Joe DiMaggio equaled or bettered his real streak once using each of the two methods and twice using an equal at bats approach, but four and nine times respectively for the latter two methods. Anyway, Rockoff and Yates estimated that allowing streaks to carry over across two seasons would increase the overall number by about ten percent. Winning the Triple Crown Between 1901 and 1967, someone won the proverbial Triple Crown (leading the league in HR, RBI, and BA) once every five or so years. Since 1967, only Miguel Cabrera (2012) has achieved this milestone. Daniels (2008) computed the odds for doing so by simulating the performance of possible winners in the context of their competitors, others who are eligible for leading the league in batting average (at least 3.1 PA for each game their team plays). In so doing, he provided a feasible explanation for the disappearance of Triple Crown winners. According to his data, with 30 competitors, the odds that someone will win it in a given year is about 21 percent, which give or take matches the 1901-1967 stretch, but that percentage nosedives by more than half for each additional 10 competitors, down to 2 percent for 60. The actual number of competitors has about doubled since the beginning of expansion, which coincides very well with the disappearance. Daniels estimated that the odds per year for each league actually averaged 8 percent from 1901 to 1967, but only 1.5 percent since then. Hitting for the Cycle Michael Huber (2016) piggybacked on some earlier work by Jeff Sackmann published at hardballtimes.com in describing how to calculate the odds of a given player hitting for the cycle in a given game. The following steps should suffice: 1 Separately compute the probability of the player hitting a single, double, triple, and home run per plate appearance. 2 Multiply the probability of single times the probability of double times the probability of triple times the probability of home run. This gives you the odds of hitting one of the 24 possible orderings of the four hit types, i.e. single/double/triple/home run is one, single, home run, triple, double is another, and so on, in a four-plate appearance game. 3 Multiply the results of Step 2 by 5, giving you the odds of that specific order of the hits in a five-plate appearance game. 4 Multiply the results of Step 3 by 3, giving you the odds of that specific order of the hits in a six-plate appearance game. Huber did not provide information on greater-than-six-plate appearance games, given their rarity. 5 Multiply each of the products from Steps 2, 3, and 4 by the probability of the specific player getting four, five, and six-or-more plate appearance games, respectively. 6 Sum the results of Step 5, giving you the total probability of hitting for that ordering of the hits. 7 Multiply the sum from Step 6 by 24, the number of possible orderings of the hits. This is the final answer. Reaching Goals Quite a few studies described someplace in this book (Davis and Harvey, 1992; Jackson, Buglione, and Glenwick, 1988) were designed to evaluate implications of drive theory, a psychological theory concerning the impact of basic human needs on behavior. An issue stemming from drive theory is how people respond to different levels of psychological stress. Goldschmied, Harris, Vira, and Kowalczyk (2014) examined the 24 players who had reached 505 home runs before the publication date (Albert Pujols got there a bit too late to be included), comparing how many at bats it took for them to hit the last five home runs before their last milestone (either 500, 600, 700, 715 in the case of Henry Aaron and 756 in the case of Barry Bonds) with the first five homers after it. On average, the five leading up took 117.7 at bats and the five afterward 77.5 at bats, consistent with the authors hypothesis that stress before the milestone restricted performance. Turning away from the academic literature, Bill James (1981 Baseball Abstract, pages 48-53; see page 9 of 1983 Abstract) took a fun take on the issue of player prediction with his Favorite Toy, a quick-and-dirty method for predicting the odds that a given hitter would achieve a given milestone, such as 3000 hits or 500 home runs. The method works as follows: 1 compute the number of hits/home runs the player needs to reach that milestone. 2 estimate the number of years remaining in the players career by 24 (.6 X age) which works out to be 9 seasons for a 25 year old, 6 for a 30 year old, and 3 for a 35 year old. There is a proviso stating that all hitters still playing regularly are credited with a minimum of 1 remaining years. 3 compute the established hit/home run level by the following formula (3 x last seasons hits/HRs) + (2 x preceding seasons hits/HRs) + season before that hits/HRs all divided by 6 a sensible strategy, given highest priority to the most recent performance but some to earlier ones. 4 compute projected remaining hits by years remaining x established hit/HR level 5 finally, calculate the probability of attaining the milestone by (projected remaining hits/HRs divided by hits/HRs needed) minus .5 The .5 is subtracted to ensure that the projected hits is not greater than needed hits, which would give the player a greater than 100 percent chance of attaining the milestone. In fact, Bill used .97 as the upper limit to include the possibility of an abrupt end of career a la Kirby Puckett. If nothing else, Bills estimates over the years gave me my moneys worth of entertainment. Holmes (2003) answered an on-line challenge by Tom Tango to estimate the accuracy of the Favorite Toy in a nice way. He took all players who had gotten halfway to two milestones through 2002 (3000 hits and 500 home runs), computed the odds of achieving the milestone using the Toy, and then compared that estimate to the number that succeeded. Of 506 with 1500 hits, his calculation predicted 30, whereas 25 had achieved it at that time. This is very good accuracy, given that since that time seven others playing at that time (Rafael Palmeiro, Craig Biggio, Derek Jeter, Alex Rodriguez, Albert Pujols, Adrian Beltre, Ichiro Suzuki) had joined the 3000 hit club, increasing the total to 32. Analogously, of 151 who achieved 250 home runs, Holmess method guestimates that 26 would achieve it, 17 had, and the relevant players joining since that time include Ken Griffey Jr., Manny Ramirez, Alex Rodriguez, Frank Thomas, Jim Thome, Gary Sheffield, and Albert Pujols (up to 24). Hitting .300 is not a great feat, but it is a goal for many hitters, and Pope and Simonsohn (2011) believed that the desire to do so can serve as motivation for hitters very close to that mark with a game or two left in the season to perform particularly well in those last couple of games. Examining Retrosheet data from 1975 through 2008 for all hitters with 200 or more at bats in a season (comprising a sample size of 8817), the authors showed that a higher proportion of players hitting .298 or .299 got a hit on their last plate appearance (.352) than players hitting .300 or .301 (.224). They were also, however, less likely to be replaced by a pinchhitter (.041 versus .197). The latter leads to an obvious bias; that hitters just over the .300 benchmark have less of an opportunity to drop under than hitters just under to move over it. Scott and Birnbaum (2010) demonstrate that a statistical correction for this bias removes this last at bat advantage, and in fact there is nothing unusual about the performance of players on the cusp of .300 (page 3). And... Huber and Glen (2007) examined whether, overall and during six eras between 1901 and 2004, the three rare baseball events no hitters (1.98 per season), hitting for the cycle (2.16 per season) and triple plays (4.91 per season) appear to occur at random occasions over time. Overall, triple plays did not appear to be randomly distributed, although they were in five of the six specific eras; no hitters and cycles were random overall but not in, respectively, one and two of the eras. Huber and Sturdivant (2010) continued in this vein by attempting to model the time between games in which a team scored 20 or more runs in a game between 1901 and 2008 (2.06 per season). Again, the data needed to be divided into eras as the number instances were generally associated with overall run scoring. The model fit except for the dead ball era, in which there were an unusually high number of instances in 1901-1902 and again in 1911-1912, with the former probably due to the appearance of the American League, which (as Cliff Blau pointed out to me) did not count foul balls as strikes until 1903, increasing the opportunity to get on base. It is hard to reach a definite conclusion either way from these data. Historical Changes in Offense It is obvious that the balance between offense and defense has changed substantially across the history of baseball. Jim Albert (2017) provided visual displays along with figures for runs scoring per team per game for 1900 through 2015. It has had its ups and downs; the first dead ball era bottoming out in 1908 at about 3.4 (mean batting average of .239), then increasing to the 1930 peak at 5.5 (mean batting average of .296), then down through World War II below 4, up in the early 50s to about 4.5, then down through the big strike zone era (Cliff Blaus term) reaching bottom in 1968 again at 3.4, then up to the steroids period 2000 peak of about 5.2, then down to 4.1 in 2015. Laura Schreck and Eric Sickles (2018) provided some more precise figures for recent years; about 4.75 for 2004 to 2007, then down to approximately 4.1 in 2014, then up to around 4.6 in 2017. Joe Mangano (1988) came up with a helpful list of factors that could contribute to variation in offense over time: 1. Longer fly balls and harder hit liners means some will become home runs, doubles, or triples rather than outs. 2. Some singles falling between infielders will scoot on past outfielders and become extra base hits. 3. Some soft line drives (and, adding to Joes list, popups past the infield) will no longer fall in front outfielders but be caught by them. 4. Some even softer line drives will no longer be caught by infielders and will become singles. 5. Harder hit grounders will turn some ground outs into singles. 6. Some infield-hit singles will travel far enough on the ground for infielders to get the runner at first. Some of these factors imply that one signal of increased offense would be additional extra base hits of all three types relative to the number of singles, and the opposite for a decrease. He noted years in which changes in offense approximated those patterns. Going back one step to the causes of such changes, Joe emphasized the possibility of changes in the balls liveliness. Bill James (1988g), after a detailed discussion of the years in which the ball was likely to be relatively dead versus lively, noted the possibility other possible factors and I will add others; new ballparks of different sizes, changes in the strike zone (those two offered to me by Cliff Blau), increases in the number of balls used during a game following Ray Chapmans death, changes in the height of the pitching mound, climate change (hotter and more humid weather increases batted ball distance), and talent dilution during the first couple of years after expansion if it unequally benefits batters versus pitchers. Joe also claimed that fastballs would become faster and increase strikeouts due to a livelier ball, which I dont buy. The literature on changes in specific indices over the history of baseball is difficult to summarize neatly. The general trends are clear, but different analysts will get slightly different numbers due to, for example, different categorizing of eras (for example, one might do so by decade whereas another by observable breakpoints, or sudden shifts in performance). Here is a summary, index by index: Batting Average: As reported by several researchers (Berry, Reese, and Larkey, 1999a; Gould, 1983/1985, 1986/2003), mean BAs for regulars has drifted around over the decades, in the .250s during the first and second decades of the twentieth century, then in the high .280s in the 1920s before falling to around .260 between 1940 and 1960, below that in next decade before popping back toward .260 in the 1970s and remaining about there since. Power: As with so many other issues, Earnshaw Cook (1966) had his say, showing that power, as measured both by home runs and by total bases as a proportion of hits per team, increased over the 1915 to 1960 time interval. The best analysis so far is probably Groothuis, Rotthoff and Strazicichs (2017) examination of seasonal offensive data from 1871 to 2010, searching for both trends over time and breakpoints in these trends. After trending upward during the 19th century, both slugging and the standard deviation of home runs per at bat broke downward at the turn of the 20th century (beginning of the deadball era), the combination implying that as power hitting waned, the performance spread between sluggers and singles hitters markedly decreased. In other words, everyone became a singles hitter. The beginning of the 1920s (the sprouting of Babe Ruth) reversed both trends, as the slugger again differentiated himself from the singles hitter. SA then trended down until another jump in the early 1990s (steroids era), and then drifted down again. Berry, Reese, and Larkey (1999a) uncovered similar trends across the 20th century, Groothuis et al. missed more recent increases observed by Don Coffin (2012; data from 1912 to 2011) and Jim Albert (2017; 1900 through 2015). Jim also observed a short dip around World War II. Laura Schreck and Eric Sickles (2018) were again on top of recent details, with numbers paralleling what they reported for runs scored. Starting in 2004, total homers decreased steadily through the low 5000s to about 4200 in 2014, but had risen to over 6000 by 2017. HRs both per hit and as a percentage of pitch type reflected the same U-shaped pattern. Diane Firstman (2018) charted a very recent jump in the percentage of runs scored due to homers; fairly steady from 2007 (34.2%) to 2014 (33.4%), it exploded to 42.3% in 2017. In 2018 a large and varied group including statisticians, mathematicians, engineers, and physicists (Jim Albert, Jay Bartroff, Roger Blandford, Dan Brooks, Josh Derenski, Larry Goldstein, Anette (Peko) Hosoi, Gary Lorden, Alan Nathan, and Lloyd Smith) contributed an official report to MLB documenting recent trends. In summary: They basically replicated the Schreck/Sickles findings for homers per batted ball from 2012 to 2017. In addition, there was no change in the rate of batted balls with 15 to 45 degree launch angles and 90 to 115 mph exit velocities, i.e. those most likely to reach the stands, between the second half of 2015 and 2017, but HRs on batted balls in these zones have increased anyway, and by hitters of all types. The point is that batted balls traveled farther each of those years, and analysis by the engineer/physicist subgroup uncovered changes in the aerodynamic properties of the baseball itself that are probably responsible. Incidentally, HR rate increased linearly between 60 and 85 degrees Fahrenheit, going up about 1 percent for each 10 degree rise. This rate of increase did not change between 2015 and 2017, with each successive season bringing more homers at each temperature. Hotter weather alone is, then, not primarily to blame for the HR jump. Strikeouts: These have been reported by Jim Albert (2017; 1900-2015) in terms of strikeouts per team per game and Don Coffin (2014; 1900-2015) and Diane Firstman (2018; 1913-2017) as percentage of plate appearances. Ks per team-game were at about 3 (8% of PAs) in 1900, increased to 4 (11% of PAs) in the early 1910s and then back to 3 (about 8%) around 1918 until 1945, up to 6 (around 15%) during the big strike zone era, down to 5 (12%) about 1980, and have been rising ever since (8 [over 20%] in 2015). Diane listed a year-to-year increase from 17.5 percent in 2008 to 21.6 percent in 2017. Laura Schreck and Eric Sickles (2018; 2004-2017) reiterated the recent increase, but interestingly swinging strike percentage was a steady 8.6 to 8.7 percent of pitches through 2010 and then started rising linearly to over 10 percent in 2016. Walks: Jim Albert (2017; 1900 through 2015) and Don Coffin (2014; also 1900-2015) reported that walks per team per game (Jim) and as a percentage of plate appearances (Don) started around 2 (less than 7% of PAs), were up to 4 (10%) by 1950, and have bounded between 3 and 3.5 (8-9%) pretty much since 1955, except for a short rise back to about 4 (10%) around 2000. Laura Schreck and Eric Sickles (2018) reported some short term variation paralleling that for runs scored that I noted above. Don (2012) also observed that intentional walks over the shorter history of the available data at BaseballReference.com (since 1955) have dipped from .35 per game to .30 as of 2011. Three True Outcomes (sum of homers, strikeouts, and walks): Don Coffin (2014; see Diane Firstman, 2018, for replication) demonstrated that the Three True Outcomes figure as a percentage of plate appearances has gone up pretty much in tandem with strikeout rate, its largest component and responsible for most of TTOs variation over time. At the turn of the 20th century, it was at about 16 percent, went up to maybe 21 percent in 1910, but back to about 15 percent in 1918. It reached 20 percent around 1935, 26% in about 1970, dipped to maybe 23% in 1980, then has risen ever since to 32% in 2015. Walks plus hits per innings pitched: Laura Schreck and Eric Sickles (2018) saw the same recent general trajectory for WHIP as they diagrammed for several indices Ive summarized, except that WHIP has not bounced all the way back (dropped from 1.4 to 1.275 then up to 1.35). Stolen Bases: According to Don Coffin (2012; 1912-2011; see Dave Smith, 2012, and Jim Albert, 2017, for general replications), stolen bases have had a roller coaster ride, from almost 2 per game in 1910 to only .5 around 1950, then up toward 1.2 in the late 1980s and then down again to about .8 in the late 2000s, then back to 1 since then. Dave Smith (2012) presented some data on base stealing attempts and success using Retrosheet data from 1945 to 2011. The success rate for stolen bases was at about 55 percent from 1945 to 1955, but has increased steadily since, reaching about the traditional 2/3 breakeven point from about 1980 to 1995 and increasing to over 70 percent by 2010. The proportion of attempts to steal second, the most prevalent of the possibilities, mirrored the overall number of attempts, implying that the former probably accounts for most of the variation in the latter rather than stabs at third and home. It rose from about 86 percent of total attempts in that first decade to a high of 92 percent by 1980, dropped back to 84 percent during the steroids era, and by 2010 was back at 86 percent. Triples: Jim Albert (2017; 1900 through 2015) observed that three-baggers per team per game remained steady at about .47 until 1925 and nosedived after then to less than .2. Sacrifice Bunts: Don Coffin (2012; 1912-2011) reported SHs increasing from about 1.1 to 1.3 per game through 1923, after which the impact of Babe Ruth and other power hitters apparently became clear and the number dropped dramatically to about .6 per game around 1930, then eased down to the current .35 (despite scoring rule changes that twice counted sac flies in this category; see  HYPERLINK "https://www.baseball-reference.com/bullpen/Sacrifice_hit" https://www.baseball-reference.com/bullpen/Sacrifice_hit on this). Hit by pitches: According to Gary Belleville (2018), HBPs in the early years of MLB occurred in at least 0.75 or more percent of PAs, with a peak of 1.25 percent just before 1900. The rate then decreased to between .30 and .40 percent in the 1930s and 1940s, up to .70 in the late 1960s, down again to .40-.45 percent in the early 1980s, up to 1.0 percent around 2000, then down to .8 percent right after 2010, and up once more to .95 percent in 2017. Peak Seasons: In an interesting take on the issue, Clay Davenport (2002b) examined the overall distributions of offensive performance across careers during the 20th century. Although the peak generally remained at about age 27 as usual, there were noticeable tendencies for the distributions to favor younger players during poor offensive eras (the 1910s and 1960s) when speed was relatively more valuable and older players during good offensive eras (1920s through 1950s excepting the World War II years, then again 1975 until present) when power took precedence. In the SABR Statistical Committee blog contribution mentioned earlier in this chapter, Mark Armour (2018a) also included the proportion of total bWAR made by players (including pitchers) aged 35 or greater between 1968 and 2017. As high as 12 percent in 1982 and 14.2 percent in 2002, it had fallen to 1.6 percent in 2017, which Mark claimed was the lowest since 1877. Also, following up on the fact that the average bWAR midpoint of ones career is usually later than the peak in other words, total player productivity is usually greater after age 27 than before that average kicked around between 27.5 and 28 in the 1970s, went up to over 29 in the late 1990s and early 2000s, but decreased back to 28 in the early 2010s. Relative Importance of Different Indices A much different take on changes on offense over time is implied in work by Bill James from the 1983 Baseball Abstract (pages 96-97). James invented an informal measure for the value of a given category of offensive play on team performance. For the years 1969 through 1982, Bill computed the average divisional standings in a given year for the teams that were first and last in a given offensive category that year and then subtracting the former from the latter. For example, the average standings for the team that was last in doubles during that stretch was 5.11 whereas that for the team that came in first in doubles was 2.50. The difference between the two is 2.68, which is the highest differential among any of the categories he examined, with home runs next at 1.96, walks after that at 1.81, triples well behind at 1.18, and stolen bases last at 0.79. The point was to show how relatively unimportant stolen bases was over the period; as triples were not much higher, one could generalize the point to team speed in general. Shiner (1999) performed the same analysis for league standings in earlier (pre-divisional play) decades, revealing how the relative importance of categories has changed over time. The data provide interesting evidence in support of our intuitions about different eras. For example, during the dead-ball 1900s, home runs were by far the least important of the five indices for league standings (1.68, with every other 2.69 or greater). By the offensive-minded 1930s, HRs had become the most important (3.35). Just for the record, Shiner also presented divisional data for 1983 through 1998, and they are indicative of the ascendency of the slugger over the speedster: HRs were 2.22 and BBs 1.91, distantly followed by 2Bs at 1.24, 3Bs at 0.67, and SBs at 0.40. In the same vein, Bruggink (1999) discerned that speed (as measured by stolen bases plus triples) had a greater impact on team winning average in the 1901 through 1919 period than in the 1920 through 1940 era, whereas isolated power had the bigger effect in the latter of the two. Batting average and ERA had equivalent values across eras. Fielding average may have been more important in the later period, but the author did not trust the reliability of his results. Bill James (1988 Baseball Abstract, pages 23-29) also performed a study specifically designed to examine the impact of the strike zone enlargement that occurred between the 1962 and 1963 seasons. Overall, runs created shrank from 4.998 to 4.327 between seasons, with batting average dipping from .274 to .261 and slugging average from .426 to .398. What is particularly interesting was the difference of impact depending on hitter height, with the larger strike zone particularly harmful to the taller players. Bill listed several height increments; I will limit my presentation to the 31 smallest (57 to 510) to the 34 tallest (62 to 65) among the most active position players spanning both years, but the effects summarized here were basically linear across all of Bills categories. In 1962, BAs were identical no matter the height (.273 for both groups), and SAs ranged from .402 to .443. BAs in 1963 shrank 6 points for the shortest group but 15 points for the tallest group; respective SAs dropped 20 and 35 points. In contrast, walks went down for 74 percent of the short group but only 47 percent of the tall group. So far, it looks like increasing the strike zone size hurt hitters in general, which is why it was imposed; but the impact appears to have been disproportionately felt by the taller and more powerful hitters. In a second type of breakdown, those hitting .300 or more in 1962 lost 29 BA points in 1963, with 79 percent of the players in the group striking out more and 54 percent walking less; those at less than .250 saw their average go up 4 points, with 51 percent striking out more and 74 percent walking less. Bill made the point that the change in the strike zone in most of these trends was confounded by regression toward the mean, although neither Bill (nor I) have an explanation for the walk data. Third, of those with 29 or more percent of plate appearances resulting in walks or strikeouts, 33 percent whiffed more often and 78 percent drew fewer free passes; for those with 14 or lower percent BBs plus SOs, 82 percent struck out more often and 36 percent walked less. The strikeout effect could also be regression toward the mean; again, the walk pattern makes no obvious sense. Finally, for those whose strikeouts were at least 70 percent of the strikeout/walk sum, 45 percent struck out more often and 40 percent walked less; of those with 40 or lower percent strikeouts of that sum, 67 percent struck out more and 75 percent walked less. This could also be regression toward the mean. Taken as a whole, most of these numbers are hard to fathom. In a historical analysis specifically relevant to the steroid era, Rader and Winkle (2002), noted that the ratio of strikeouts to homers, which averaged 0.47 HRs for each K in the long 1901 to 1999, had jumped to 0.87 HRs for each K in the last 13 years (1987 to 1999) of that interval, implying that swinging for the fences was far more successful in the 1990s than previously. Dave Smith (2018) observed a correlation of .83 between strikeouts and home runs over more than a century of play (1908-2017 excepting 1917 and 1918). And last, Don Coffin (2012) examined historic changes in power as measured by average isolated power and home runs per game and correlated these with revisions in strategy use over time, from 1912 through 2011 (1955 in the case of intentional walks). Overall, the use of both sacrifice bunts and intentional walks were negatively correlated in the range of -.7 with the two power indices; stolen bases attempts, however, only related at about -.2. And David Kaplan (2008) examined the extent to which trends in performance indices (home runs, doubles, stolen bases, walks, runs scored, runs batted in) were associated, either positive or negatively, from year to year both for each measure alone and across them. For a given season, his analysis was based on averaging these indices across players with 500 or more plate appearances in that season, in so doing ignoring within-season differences across players, which from the Gould-inspired research are known to have decreased over time and perhaps biased the results. In any case, not surprisingly, each index except, for some reason, RBIs was related from year to year. Across indices, RBIs in one season was related with, of all things, doubles in the previous season, and doubles in a given season positively with home runs and RBIs but negatively with runs scored in the previous season. Frankly, these findings make absolutely no sense to me. Has Hitting Ability Improved Over Time? Almost certainly yes. I will begin with evidence in the affirmative. The famed paleontologist and baseball fan Stephen Jay Gould wrote a couple of essays (1983/1985, 1986/2003) attempting to demonstrate through the use of batting averages the steady increase in position player ability over the course of major league baseball history. While average BAs have wandered up and down over the decades, the within-season variation has decreased steadily over that time. In the first essay, Gould showed that the difference between the five highest batting averages and the league average was in the range of about 90 points during the 19th century, 80 points the first three decades of the 20th, and 70 points since; that between the five lowest and thr average began at about 60 or 70 points and has decreased to about 35 since. In the second essay, Gould made the same point in a more statistically rigorous way, presenting the fact that the standard deviation in within-season batting averages has decreased from about .05 at the beginning to about .03 at the time of the essay (Berry, Reese and Larkey, 1999a, also reported this trend). In both analyses, the improvement was asymptotic, occurring quickly at the beginning but slowing down toward some yet-unknown limiting figure. Rob Wood (2001) calculated the improvement function to be Log Standard Deviation = 0.234536 X (year-1870)-.011316 Rob calculated that an evaluation method that ignores this change over time would overvalue players in 1871 by 73%, players in 1900 by 18%, players in 1950 by 5%, and players in 1990 by 1% as compared to the last year in his study, 2000. In their paper mentioned above, Groothuis, Rotthoff and Strazicich (2017) agreed that the standard deviation of batting has eased downward from 1871 until a breakpoint in 1981, followed by a levelling off (did we reach the point in which overall talent is now as great as it can realistically get? I doubt it). More subtlely, there were sharp dips in 1906 (deadball era again), and 1933 (maybe because everyone was hitting so well??), and a quick rise in 1966 (maybe because some people were hitting really badly during the big strike zone era???). Goulds original explanation for this effect was the standardization of play; for example, teams have gotten progressively savvier in positioning their fielders, making it harder to get hits. Also, as Pete Palmer pointed out to me, much of the 19th century decrease is due to the lengthening of the season, which provides players with the a greater opportunity to perform closer to their real talent level. But by the second essay he began to realize the real explanation; the overall increase in talent. We must assume that the greatest stars have been about equally good over the interim, which would follow from Goulds belief that there is a limit to ability based on the very facts of human physiology (an outer limit of human capacity) that the best players can approach but never cross. If so, then the average player has gotten closer to the best, and the greater availability of competent players has increased the replacement level such that .180 hitters are just too poor to play regularly anymore no matter how well they field. There is a critical, and strongly supported, implication of this conjecture. Although the number of major league teams has almost doubled since 1960, the population from which major league players are procured has far more than doubled, keeping up with expansion and refuting the often-made claim that expansion has diluted talent. Quinn and Bursik (2007) have noted that the ratio between the number of teams and U.S. urban population had not increased over the 20th century, indicating that at least relatively speaking it has not become easier to become a major leaguer. In 1900, there were more than five teams per ten million urbanites; this figure decreased to a bit more than one in ten million by 1960. Thanks to expansion, this ratio has stayed about constant since that time. One can of course argue that the entire U.S. population of relevantly-aged men would be a better basis for comparison. In any case, as major leaguers have more and more come from outside of the U.S. in the past few decades, the actual ratio of teams per eligible people has probably continued to decline. In support of the general proposition, Schmidt and Berri (2005) uncovered a correlation between the percentage of players born outside of the United States and the shrinking of within-season variance in home runs and, for pitchers, strikeouts. In other relevant and supportive work, and as I reported above, Groothuis, Rotthoff and Strazicich (2017) documented the standard deviation of home runs per at bat braking downward beginning at the turn of the 20th century, with the exception of the beginning of the 1920s (the sprouting of Babe Ruth) and the early 1990s (starting the steroids era). Also in agreement is Abbott Katzs (2009) finding that the batting advantage of lefthanded hitters over righties, which was in the range of 25 points between 1910 and 1934, has steadily decreased to less than 10 points now, which Abbott speculated could be explained a la Gould as due, in his words, to a general honing of player competence, having the effect of trimming performance extremes at both ends (page 5). Finally, Corinne Landrey (2017) uncovered another recent trend; the decrease in variation among positions in offensive production. To take extreme cases, in the 1970s, shortstop, the least productive position at bat, averaged an OPS+ of about 75, whereas left field, one of the most productive, was at about 115. By 2016, shortstop was up to about 95 and left field down to about 100. This was not just the result of the then-recent influx of offensively-gifted shortstops; in general, the infield positions (excluding first base, which has always been offense first) had risen at the expense of the outfield positions. This would also make sense given the Gould hypothesis; it is too bad that she did not go further back in time for more evidence. Finally, two nay-sayers: Hessenius (1999) attempted to demonstrate that, at least during the 20th century, standard deviations did not go down, such that there is no evidence of overall improvement in batting. The author was purposely trying to disconfirm Gould, but his results are not as different from Gould as he thinks, as Gould claimed that most of the reduction in standard deviation occurred during the 19th century. Hesseniuss work differed in that he distinguished between both leagues and handedness, such that National League righthanded batters were only compared to one another, thus providing four analyses per season. He noted that both the best and worst batting averages among each of these groupings per season are usually about two standard deviations away from the mean. As noted above, Terrys .401 was not even that high; the highest at the time of Hesseniuss work time was Rogers Hornsbys .424 in 1924 (s.d. of 3.98 for N. L righties that year). Hessenius argued that, given a high enough league batting average, a .400 batting average is not that unusual. Horowitz (2000) presented evidence that the difference between the five league leaders and the average player in doubles, triples, home runs, runs batted in, and stolen bases (plus strikeouts and shutouts for pitchers) as a whole did not decrease across the 20th century. Overall, though, research is supportive of Gould. Contra Hessenius, Leonard (1995) made the point that the standard deviation itself is not the best measure of variation because its size is dependent on the mean; rather the coefficient of variation (standard deviation divided by mean) is better as it controls this dependency. Leonard examined a set of offensive measures for one year in each decade from the 1900s through the 1990s and noted a tendency (sometimes significant, sometimes not) for the coefficient of variation to decrease for batting and slugging averages, runs scored and RBIs, hits and home runs, but not for doubles, triples, walks, and stolen bases. The latter four indices are probably too dependent on (for the first two) changes in fashion in ballpark design and (for the second two) strategic considerations to be consistent with Goulds conjecture. Evaluating Hitters across Eras If one wants to compare players from different eras, one absolutely cannot do so using their raw performance measures, because the game has changed too much over the interim for such comparisons to be in any way meaningful. To use Shoebothams (1976, 1983) example, Bill Terry led the National League with a .401 batting average in 1930 whereas Carl Yastzremski paced with American League with a .301 average in 1968, but any direct comparison between these figures means nothing given that the National League as a whole hit .303 in 1930 while the American League hit .230 in 1968. It has turned out to be a challenge for researchers to validly compare players across eras of baseball, and some that have tried have run into difficulties. Shoebotham (see also DAniello, 1999, for analogous work) computed a relative average index representing the ratio of the players average to the league average, but as both Thorn and Palmer (1984) and Mike Sluss implied (described simply in 1999a; there was also a now-unavailable technical version of his work), this only works if the standard deviations of the league distributions are about the same, and the whole point of Goulds work is that it used to be larger. The larger the s.d., the more that the best hitters stand out taller from the field, which is demonstrated in Shoebothams findings; the top ten and 22 of the top 29 seasons in his relative batting average measure for the 20th century were between 1900 and 1919. A few methods for compensating for this problem have surfaced. Merritt Cliftons approach, described well in Thorn and Palmer (1984), compared players to both the average and best batting averages, as follows: 1 Compute player batting average divided by league batting average as in Shoebotham 2 Compute league leader batting average divided by player batting average 3 Divide the results of step 1 by the results of step 2 The z score (the number of standard deviation units above or below mean) would also be a good measure because it compensates for this problem, and Yazs in 1968 (+2.21) was actually more extreme than Terrys in 1930 (+1.78) and so was a relatively better year. Ward Larkin (1982), after demonstrating a la Gould that standard deviations in American League batting average had declined over time, computed z-scores for ten great A. L. hitters; Rod Carew at that time (before end-career decreases) was tied for first with Ty Cobb at 2.45. Bang (2004) used z-scores to rank the greatest single season home run hitters (Babe Ruths 1920 was first at 7.97, and the Babe had the top six plus eleven out of the top twenty-five), but admitted that that measure short-changes one players great performances when there are others almost as great (e.g., Maris in 1961). Even sophisticated statistical analyses are probably unable to do any better than this. Dick Cramer (1980) also tried to compare players over time, but his method was fundamentally flawed. Using Batter Win Average as his index but translating it into batting and slugging average, and based on every player with at least 20 plate appearances between 1876 and 1976, Dick concluded that there was a .120 improvement in BWA over that span, equivalent to a .123 difference in BA and .073 difference in SA. In other words, a player hitting .300 and slugging .450 in 1876 would have hit .180 and slugged .347 in 1979. As many have mentioned (for example, Michael Stagno in a now-unavailable critique), Cramers attempt fundamentally confounds individual player career trajectory with talent changes over time. Given the normal career trajectory, with its relatively fast rise to a peak and slow decent afterward, what looked to Cramer like later players being more skilled than earlier is almost certainly later players in their prime being compared to earlier players a bit over the hill. In addition, as Dallas Adams pointed out to William Rubinstein (1981), Goulds point is relevant here; the fact that variation between the best and worst hitters has narrowed considerably over the years implies that there has been improvement over time in the quality of the weakest hitters but probably not the strongest, and it was the strongest that received Cramers greatest attention. In conclusion, this is one area in which Dicks work (which at its best has been ground-breaking; see the Offensive Evaluation chapter) should not be taken seriously. Scott Berry, with associates Shane Reese and Patrick Larkey (1999a), also tried to solve the career-trajectory problem. As mentioned in the Age Section above, Berry et al. first modeled career trajectories. They then used those trajectories to compute an estimation of players real talents and contrasted them across time, allowing for a ranking of players. However, as both Albert (1999) and Schell (1999) pointed out in responses to Berry et al.s work, this contrast was only possible through a statistical slight of hand. In order to establish a benchmark for comparisons across eras, Berry et al. treated all of the players as if their batting averages peaked at 27 and home run performance at 29. If this treatment was accurate, they would have been able to disambiguate career trajectory from change over time. But it is not accurate (despite their claim to the contrary in a rejoinder [1999b]), and so their method was unsuccessful. One final attempted method is calculating the probability of performance, in other words the odds of achieving a particular feat in comparison with the average performance for the year. Mike Sluss (1999a) discussed the issue in length, including information on computation via the binomial distribution and the assumptions implied (that the average player is a concept that is comparable over time, and that all plate appearances are independent from one another). He used a logarithmic transformation of the odds that the average player can achieve a given index (BA, OBA, HR)., which takes into consideration the performance of the player being evaluated (better performance is less likely achieved by the average player), the mean performance across players that year (a higher mean implies that a given performance is more likely), and how much the player played, because it is harder to achieve a really good (or bad) BA or OBA the more one plays. After the transformation, the probability of performance (POP) for the average player in a year would be .3. A POP of 1 signifies a level that the average player would achieve by chance 10 percent of the time, a POP of 2 would equate to 1 percent of the time, 3 would mean .1 percent, and so on. Extraordinary performances will be rated in the teens or more (e.g., George Sislers .407 in 1920 equates to a logarithmic POP of 11.08. Sluss (1999b) later performed a POP analysis for walks, and Peter Ridges (2001) extended it to slugging average. Petersen, Penner, and Stanley (2011) did something analogous, resulting in ratings for relative annual performance in home runs, hits, and runs batted in claimed to be uncontaminated by baseball era. Rob Wood (2000a) was uncomfortable with Slusss method, because its use implicitly presumes that all players are league average and as such ignores variation around that mean. He proposed a different method that is based on the odds that a player chosen at random would achieve a given index. He admitted that it was far more complex computationally but defended it as more realistic than Slusss. In a rejoinder, Sluss (2000) defended the use of league average, argued that Woods (and Shoebothams) method does not sufficiently account for playing time, and added a variation of his version for players below average in production (negative probability of performance, or POP). Injury I begin this section with an editorial comment. Starting about 2015, medical researchers discovered that one route to vitae building via quick publication is to perform analyses of the impact in playing time and performance of major league players hit by various injuries. As a consequence, there are now far more works of this nature being added to the academic literature than any other topic in baseball (excepting physiological studies). The more serious injuries attract multiple studies that too often repeat one anothers findings based on largely overlapping periods of time. As the point of replication is to see if an effect generalizes over different data sets, not the same one, the duplication is useless. And as you can see below, the wealth is shared, as it is hard to find many articles with fewer than five authors; one wonders how much work each individually contributed. Make no mistake, some attention to major injuries is quite welcome; see in particular what we have learned about ulnar collateral ligament tears and subsequent reconstruction (aka Tommy John) surgery reviewed both below but primarily in the Pitching Issues chapter, and on concussions described below. However, as I write this (2019), the impact of the most critical injuries have been well documented, leaving academics to examine ever rarer and less serious ones. On a positive note, much of the work has been performed consistently with an ideal analytic method exemplified in an analysis by Erickson, Gupta, Harris, Bush-Joseph, Bach, Abrams, San Juan, Cole, and Romeo (2014). Performance comparisons are made for multiple seasons both before and after the injury-relevant season between the sample of injured players and a meticulously-chosen control group matched pitcher-to-pitcher along a wide range of variables; in the Erickson et al. study, these were age, body mass index, years in MLB, performance, pitching position (I believe they meant starter versus closer versus other reliever), and handedness. For each control, an index year in their career is chosen that corresponds to the year in which the matched injured players medical procedure occurred. With a large enough sample size, which is a problem with the more arcane injuries that one finds in recent work, results are pretty much trustworthy. Getting to the issue at hand, a historical change in addition to those just described has been the increase in disabled list time for position players (and pitchers also, see Chapter 8 on that). Pete Palmer (Heeren & Palmer, 2011) presented the following data on the number of disabled position players at some point in the season following one in which they had appeared in at least 100 games: 1946019657.5%198522.7%200527.0%19551.6%197512.5%199633.7%200736.3% The following table presents the top 10 locations of injuries to major and minor league players resulting in missed games between 2011 and 2014 (Dahm, Curriero, Camp, Brophy, Leo, Meister, Paletta, Steubs, Mandelbaum, and Pollock , 2016). Body partNumber%Body partNumber%Shoulder/clavicle495214.7Lower back/ sacrum/pelvis18955.6Upper leg394211.7Ankle17135.1Hand/finger/thumb340910.1Head/face16945.0Elbow31859.5Lower leg/ Achilles tendon15504.6Knee21716.5Foot/toes14294.3All others768322.9 On to some specific injury types. Concussions As mentioned above, research concerning the impact of concussions is particularly valuable given the potential long-term repercussions. Green, Pollack, DAngelo, Schickendantz, Caplinger, Weber, Valadka, McAllister, Dick, Mandelbaum, and Curriero (2015) uncovered evidence of 277 mild traumatic brain injuries in major and minor league baseball players during 2011 and 2012, resulting in a median of nine days out of action. The vast majority occurred either while in the field (53.4%), batting (27.4%), or baserunning (9.0%). Interestingly, the most prevalent causes differed between the minors (collision between players, 30.8%; hit by pitch, 26.7%; hit by batted ball, 18.4%) and majors (hit by batted ball, 29.3%; collision between players, 22%; hit by pitch and diving for balls, both 19.5%, with latter only 4.5% in minors). Not surprisingly, catchers were the most susceptible (41.5% of the total); among these, 41.2 percent were due to collisions, 35.3 percent to being hit by a batted ball, and 13.2 percent to being hit by a bat. Catchers also dominated Wasserman, Abar, Shah, Wasserman, and Bazarians (2015) list of 66 incidents between 2007 and 2013 that met a number of restricting conditions, with 26 instances, followed by 20 outfielders, 13 corner infielders, and only 7 middle infielders. Mean estimates of time missed cited in articles tended to be about a month (Pettit, Navarro, Miles, Haeberle and Ramkumar, 2018; Sabesan, Prey, Smith, Lombardo, Borroto and Whaley, 2018). The fact that mean time missed is far greater than median implies that a substantial number of concussions have resulted in very long absences, as much as a full year. Turning to performance effects of concussions, Wasserman et al. compared 38 players who missed fewer than 10 days with a set of 68 players on the bereavement or paternity lists and thus missing fewer than 10 days for non-health reasons. With no differences in the two weeks prior to injury, the two weeks after returning to action resulted in significantly worse performance for the concussed (BA, .232 vs. .266; SA, .366 vs. .420; OBA, .301 vs. .320; OPS, .667 vs. .746). These figures were basically the same for the entire sample of 66, including those missing more than 10 days. Fortunately, the performance decrement for the concussion victims had become far smaller in the 4 to 6 week period after return. A return to pre-concussion performance has also been noted by Sabesan et al. (2018), Ramkumar, Navarro, Haeberle, Pettit, Miles, Frangiamore, Mont, Farrow, and Schickendantz (2018), and Chow, Stevenson, Burke, and Adelman (2019). In contrast, Pettit et al. (2018) reported that the likelihood of the concussed playing a full season one (61.5%), three (33.8%) and five (16.9%) seasons after the year of injury was noticeably below players missing time due to bereavement or paternity leave (88.3%, 62.1%, and 36.5% respectively). Other Injury Types Although not as prevalent as with pitchers, ulnar collateral ligament tears and resulting reconstruction surgery also occurs to position players. Particularly striking in published reports are its effects on catchers. Camp, Conte, DAngelo and Fealy (2018a) examined 167 UCL reconstructions on major and minor league position players between 1984 and 2015, with 8 of these second-timers. Surgery was far more prevalent among catchers (46 cases) than infielders (62 across the four positions) or outfielders (58 across the three), and rate of return to action (58.6%) was markedly lower than other positions (infielders, 75.6%; outfielders 88.9%), and pitchers (83.7%). In addition, unlike infielders and outfielders, catchers were less likely to return to play and, if they did, failed to return to pre-injury performance levels (Begly, Guss, Wolfson, Mahure, Rokito, and Jazrawi, 2018), were more likely to require a second procedure (Camp et al.), and had much shorter careers after the procedure than a matched set without it (2.8 versus 6.1 seasons; Jack, Burn, Sochacki, McCulloch, Lintner, and Harris, 2018). Those at other positions did suffer in that half of those in Jack et al.s data set changed positions upon return, likely down the defensive spectrum. Finally, median time to return to play according to Camp et al. was highest for pitchers (392 days), equal for catchers (342) and outfielders (345), and lowest for infielders (276). Jack, Sochacki, Gagliano, Lintner, Harris, and McCulloch (2018) uncovered 21 thumb ligament repairs for position players between 1987 and 2015. All returned to play, an average of four months after the procedure. There were no significant differences before and after surgery or with a matched group in games played, WAR, or UZR. Biceps tenodesis is a procedure for fixing biceps tendon ruptures. The track record for pitchers receiving this treatment is poor, but appears to be much better for position players. Chalmers, Erickson, Verma, DAngelo and Romeo (2018) uncovered five such examples between 2010 and 2013; four of whom were able to return to play. Schallmo, Singh, Barth, Freshman, Mai, and Hsu (2018) explored 18 major league players with serious cartilage damage to the knee between 1991 and 2015, including both pitchers and position players. All returned to action after an average of 254 days, but with significant decrements in OPS for position players and WHIP for pitchers the following season. However, with an average age of 31 years, performance drops could be a result of natural career trajectory, and there is no matched group to make any comparison. I found three studies concerning hip arthoscopy covering almost the same time periods; Schallmo, Fitzpatrick, Yancey, Marquez-Lara, Luo, and Stubbs (2018; 1999 to 2016), Frangiamore, Mannava, Briggs, McNamara, and Philippon (2018, 2000-2015), and Jack, Sochacki, Hirase, Vickery, McCulloch, Lintner, and Harris (2019, 2000 to 2017). The number of affected players differed among them (33, 23, and 26 respectively), as did the average age of injury (30, 26, and 30 respectively). About 80 percent returned to action. Both Schallmo et al. and Jack et al. claimed lower OPS in the season after compared to the season before, but given the age of their sample, we again may be seeing the impact of natural career trajectories. Frangiamore et al.s group did not register performance drips (.264 BA the year before, .267 BA the year after), but their data set was a lot younger. I have no idea why the average ages differed so markedly across studies. The hook of hamate is a hand bone that can be injured while swinging the bat. Guss, Begly, Ramme, Taormina, Rettig, and Capo (2018) limited their sample to 18 cases with at least 100 major league PAs in two seasons before and after the injury. There was similar performance in WAR, ISO, OPS and other measures before and after injury and with a control group matched on age and Similarity Score. Finally, Camp, Wang, Sinatro, DAngelo, Coleman, Dines, Fealy, and Conte (2018) examined 2920 major and minor league players who missed at least one game between 2011 and 2015 due to being hit by a pitch. Fortunately, only 1.4 percent needed surgery. The hand was the target for 21.8 percent of these, followed by the head (17%), elbow (15.7%), forearm (9.1%), and wrist (7%). Actual injuries were reported for 31 percent of those hit in the head, compared to 9 percent for the forearm and hand and 2 percent or less for other locations. Pitch velocity correlated about .30 with injuries per HBP and almost that much for days missed per injury. Protective equipment helped return to play. If hit on the elbow, an average of 1.8 days were missed if the player was wearing a pad versus 3.5 days without one, and if hit on the head, 7.3 days were missed if the ball hit the helmet versus 12.7 days if it did not. Performance Enhancing Drugs Research examining the impact of performance enhancing drugs on hitting began appearing after Major League Baseball was finally embarrassed into attempting to route out the problem. The best rationale for this work of which I am aware is a study by a physicist named Tobin (2008). After reviewing past physiological work on the impact of steroids on weight lifters, he decided to assume an increase in muscle mass of ten percent from its use, leading to an analogous increase in kinetic energy of the bat swing and a five percent jump in bat speed (I dont understand why that is also not ten percent). Using an equation for bat/ball collision from past work resulted in a four percent increase in the speed of the ball as it leaves the bat. Next on the agenda was a model for the trajectory of the ball. There is apparently disagreement among past workers on the impact of air resistance, with quite different models following from different assumptions about it. Tobin examined the implications of several, with the stipulation that a batted ball would be considered a home run if it had a height of at least nine feet at a distance of 380 feet from its starting point. Computations based on these models resulted in an increase from about 10 percent of batted balls qualifying as homers, which is the figure one would expect from a prolific power hitter, to about 15 percent with the most conservative of the models and 20 percent for the most liberal. These estimates imply an increase in homer production of 50 to 100 percent. Turning to on-field evidence, some fairly weak studies have been performed that fail to reveal as much about the effect of performance enhancers as their authors claimed. Muki and Hanks (2006) examined career trends in home runs per at bat for the top hundred career home run hitters, which at that time went down to George Bretts 317. They claimed that these trends were significantly different (usually decreasing with age) for the 64 members of this group whose careers ended in 2000 or earlier than those (usually increasing with age) for the 36 whose careers either ended later or who were still active at the time of the study. However, Caillault (2007) pointed out that the 19 of the latter 36 who were still active had not yet had the opportunity so to speak for their trend to turn downward. In addition, Muki and Hanks gave every season equal impact on their trend analysis (including, for example, one totaling 13 at bats for Harmon Killebrew), leading to untrustworthy trends. Baseball Prospectuss Nate Silver (2006a) studied the 40 batters and 36 pitchers Organized Baseball suspended for PED use during 2005, comparing indices adjusted for age and level of play before (i.e., 2004 and 2005 before suspension) and after (rest of 2005) suspension. For position players, Silver noted decreases upon return from suspension that were tiny although, due to small sample sizes, on the verge of being statistically significant (page 335; batting average, -.010; on-base average, -.014; slugging average, -.006; Cliff Blau pointed out to me that this implies isolated power actually went up); for pitchers, decrements were insignificant (+0.3 walks, -0.1 strikeouts, and +.02 home runs per 9 innings, and ERA jump of +0.13). Keep in mind that these results could be due to performance layoff rustiness rather than stopping the use of PEDs; this is an easy comparison to do via using matched players missing time from injury. For an analyst as skilled as Silver, this is disappointedly weak work. A far more rigorous examination was conducted by Schmotzer, Switchenko, and Kilgo (2008, 2009). The authors began by accepting the specific conclusions of the Mitchell Report, i.e., the fingered players for the exact years specified, which included 33 players (over 79 seasons) for steroid use and 26 players (over 70 seasons) for human growth hormone use. Through a series of models with different assumptions and either including or not including the extreme outlier Barry Bonds, they compared these to 1277 non-accused players (over 6508 seasons) who totaled at least 50 plate appearances in a season between 1995 and 2007. Their measure of choice was a version of Runs Created 27 adjusted for the age of the player (i.e., across all players of a given age, the difference between mean RC27 for the age cohort and mean RC27 across all age groups). For steroids, estimated increase in RC27 ranged from 3.9 percent to 18 percent, with their favorite model estimating 7.7 percent without Bonds and 12.6 percent with. The researchers obtained analogous enhancements for home runs and isolated power, but a decrease of up to twenty percent in stolen bases. In contrast, most of the models for HGH predicted no impact, consistent with physiological research they cite. Moskowitz and Wertheim (2011), studying the 249 minor league players who tested positive and were suspended for PED use between 2005 and fall 2010, concluded that this group was 60 percent more likely to be promoted to higher levels in the minors the next year than non-suspended players. Incidentally, shorter and lighter players were more likely to be caught than taller and heavier, a likely function of the fact that the former would have greater motivation to add power to their game. De Vany (2011) presented data that he saw as contradicting the claim that steroid use was widespread in baseball. For him, the feats of Mark McGwire, Sammy Sosa, and Barry Bonds are indeed unusual, reminiscent of Babe Ruths domination of home run hitting during the 1920s. However, he found the ratio of home runs per at bat, per hit, and per strikeout as remaining the same between 1959 and 2004. I think it is safe to say that his methods were not subtle enough to find the obvious. Gould and Kaplan (2011) performed a study specifically about Jose Canseco, inspired by Cansecos claims concerning his personal impact on the steroid use of teammates. Based on data from 1970 through 2009, the authors concluded that after playing with Canseco, power-hitters (poorly operationally defined by position; first base, outfield, designated hitter, and for some reason catcher) increased their home runs by an average of 20 percent and their RBIs by 12 percent in years subsequent to playing Canseco, with no analogous impact for non-power-related performance measures such as walks and batting average. Yet, there was no analogous impact for other players, including some known or highly suspected to be steroid users (Giambi, McGwire, Palmeiro). Finally, even Cansecos impact disappeared starting in 2003, when steroid testing became prevalent. Although the authors do not specifically state that they trust everything Canseco said about others steroid use, it is implied between the lines. Finally, Ruggiero (2010) attempted to predict who was and was not a user based on whether performance differed substantially from career trajectory in specific seasons, particularly toward the end of careers. Mark McGwire, Ken Caminiti, Jason Giambi, and Len Dykstra had seasons that made them look guilty, Sammy Sosa and Gary Sheffield did not, and the data were unclear for Jose Canseco and Barry Bonds. In summary, there is good evidence that steroid use substantially increases power, but attempts to finger specific players are not trustworthy. There is no good evidence that HGH has analogous effects. A related issue of great concern is the motivation for PED use. Moskowitz and Wertheim (2011) noted that, of the 274 professional baseball players who tested positive between 2005 and fall 2010, natives of Hispanic countries were proportionally overrepresented by about 100 percent and natives of other countries underrepresented analogously. Although it is possible that usage is actually comparable but Hispanics are less successful in hiding it, the likely explanation for this is players from Hispanic nations are generally quite poor, and are willing to take a risk in this regard given that baseball is probably their own realistic route out of poverty. In fact, there was a clear linear relationship between per capita gross domestic product for specific countries and likelihood of a positive test. In particular, Puerto Rico, with somewhat higher GDP than the other Hispanic cultures, had a somewhat lower likelihood. Incidentally, the same relationship existed among U.S. players alone, with wealth measured by the average in the players location of birth. In addition, Venezuela and the Dominican Republic had higher rates than Mexico and Colombia. I wonder if, in order to connect with MLB organizations, players from the former two countries are more often at the mercy of middle men (the buscon) who may pressure the youngster (most are mid-teens) to use PEDs. Incidentally, almost all suspensions for recreational drug use were for U.S.-born players. Another finding was the linear relationship between national income levels and the age in which the positive test occurred. The average ages were as follows: for Dominicans and Venezuelans 20-21, for Mexicans and Puerto Ricans 25, U.S. states 27, Japan, Taiwan, Canada, and Australia 30. Some of this variation is due to the average age in which players start playing professional baseball in the U.S., but according to Moskowitz and Wertheim not all of it. The differing incentives are likely the desire to make the majors in the first place among the Hispanics and the desire for marginal players to maintain a career for the others. Protection It is time to puncture another baseball myth. From on-the-air chatter it is clear that traditional baseball people presume that fielding a lineup with two good hitters in a row protects the first of them, meaning that the pitcher is more willing to chance getting him out (and so perhaps give him hittable pitches) than pitching around him (making it likely it he walk and thus be a baserunner for the second to drive in. A weaker second hitter would provide less incentive for the pitcher to pitch around the first. Bill James was the first to question this presumption, reporting a little study by Jim Baker in the 1985 Abstract (page 258) showing that in the previous six seasons, Dale Murphy had actually hit for a better BA with frequently-injured Bob Horner out of the lineup (.283) than in (.269). David Grabiner (1991) examined the performance of 25 American League batters during 1991 who were generally followed in the order by a batter with a .450 slugging average and determined that on average they performed a bit better when unprotected by that .450 slugger. James Click (2006a) noted that batting performance of a given batter was unaffected by the quality of the next batter in 2004. In fact, although almost certainly a random finding, performance was worst of all for the best of five categories of next batters, dropping by 13 OPS points. Mark Pankin (1993) provided a much more rigorous examination of the protection hypothesis. Based on 1984-1992 data, Mark determined that BA and SA for batters of various strengths were unaffected by the strength of the batter after them. In addition, stronger hitters get more walks when batting in front of weaker hitters, which seems to be the opposite of a protection effect. There is an exception, in that everyone hits better and gets more walks before the pitcher bats. John Charles Bradbury and Douglas Drinen (2008) continued in this vein, contrasting the protection hypothesis with an effort hypothesis in which pitchers put more effort into retiring the first hitter to try ensuring that he wont be on base for the second. The protection hypothesis implies that a good on-deck hitter will decrease the walks but increase the hits, particularly for extra bases, for the first hitter; the effort hypothesis predicts decreases in all of these indices. Retrosheet data from 1989 to 1992 supported the effort hypothesis; on-deck batter skill as measured by OPS was associated with decreased walks, hits, extra-base hits, and home runs, with the association increased by a standard platoon advantage for the on-deck hitter. This support, however was weak, as a very substantial OPS rise of .100 for the on-deck hitter amounted on average to a drop of .002 for the first hitter. The authors mention an additional and important implication; contiguous plate appearances appear not to be independent, contrary to so many of the most influential models for evaluating offense. However, if their data are representative, the degree of dependence may be too small to have a practical impact on these models applicability. David C. Phillips (2011) performed the most thoughtful study of protection to date, with analogous implications. He realized that a study of protection based on player movement within a batting order (e.g., moving a cold hitter to a different spot in the lineup) leads to ambiguous findings, because any change in the performance of that hitter could be due to the change in subsequent batter or to random changes in that players performance irrelevant to who is batting behind. In response, Phillips went back to Jim Bakers original thinking by looking at differences in performance for a given player remaining in the same lineup position based on changes in the next batter caused by injury. Based on Retrosheet data from 2002 through 2009 and limited to protectors with an OPS of at least .700 for a minimum of 200 plate appearances (in other words, hitters good enough to count as potential protectors), Phillips noted that injuries to protectors resulted in an overall OPS decrease of 28 points at that lineup position due to a weaker replacement. With the weaker replacement, the hitter being protected tended to receive a lot more intentional walks but fewer extra base hits (but no more hits, as additional singles compensated), indicative of the expectation that a non-protected hitter will be pitched around more often. These two tendencies pretty much cancelled one another out, resulting in little overall protection effect. Streakiness What Counts as Evidence for the Existence of Streakiness? Another question that has led to controversy is the best explanation for batting streaks and slumps. There is no question that there are stretches in which batters perform better (streaks) and worse (slumps) than their average performance. The question is whether streakiness in batting is due to real differences in player skill level over time, such that it can be explained by some physical or psychological factor relevant to performance (e.g., Player A is seeing the ball really well right now; Player Bs mechanics are messed up right now) or is just the result of random processes. To bring back the first chapter example, take coin flipping. Over a long stretch of flips, we would expect about half to be heads and half to be tails. However, over short periods, we are likely to get something much different. The following is the result of a series of flips of a quarter done as I write this: HHHTTHTTHHHTTTTHTHTTTTHHTHTHTTHTTHHHHHHTTHTTTTHTTHHHHHTHTT 1 6 11 16 21 26 31 36 41 46 51 56 Of the 58 tosses, 28 were heads and 30 were tails, almost the expected fifty/fifty split. Yet, it might not look like the result of a random process, because of the streakiness in the data. Note the streak of heads from flip 34 to flip 39, and the slump of heads from flip 12 to 22 (only 2 of 12). Stretches like these always occur naturally in random processes. Similarly, a hitter who tends to get hits in thirty percent of his at bats in the long run (the equivalent of 6 hits every 20 at bats) is bound to have stretches during which he gets hits in fifty percent of his at bats (say, 10 for 20) and others in which he gets hits in ten percent of his at bats (say, 2 for 20). And even if it appears that streaks and slumps occur by chance statistically, two options still remain: they are statistical artifacts only, or they have real causes that occur randomly. Players claims that they are real cannot be trusted, as they are examples of the after-the-fact rationalizations that people always make for random occurrences studied in detail by social and cognitive psychologists. Now, if players were able to predict when they were going to go into a streak or slump before the fact and those predictions turned out to be accurate, then we would be able to trust them. Obviously I would love to see such evidence. Jim Albert has been the leading figure in trying to make sense of this issue. Jim and Patricia Williamson (2001) described two different ways to try to find out. One way is to do what we did above; see if the pattern of, say, hits over a sequence of at bats does or does not resemble a random process. For example, one sees if the number of runs in the sequence is more or less than one would expect by chance. A run is a stretch of events with the same outcome. The first run above is HHH, the second TT, the third H, and so on. There are a total of 28 runs in that sequence of coin flips. More runs than expected by chance would indicate more streakiness than chance would allow, evidence that streaks are real. Fewer runs than expected by chance would indicate more consistency (stability) than chance would allow, which is also evidence that the data are non-random and that something real is going on. There is a statistical procedure called the runs test that allows us to compare a data set such as this with a real random process to see they have the same number of runs. It provides a z-score representing the comparison; if the z if statistically significant, then this sequence can be considered non-random (see Chapter 1 if a refresher on z-scores is needed). The sequence above resulted in a z of -.521, which is nowhere near the statistical significance level of +/-1.96. The second way Albert and Williamson describe is to examine the conditional probabilities of the sequence (if you forgot what those are, see Chapter 2). If the process is random, then the long-term odds of one type of event following another should reflect the events overall probability. For a .300 hitter, the probability of a hit should be about .3 and of an out should be about .7 whether or not each follows a hit or an out. If, say, the probability of a hit following a hit for a .300 hitter is .5, that means that hits tend to come in bunches, evidence of streakiness. In our example, heads are followed by heads exactly half the time; tails by tails 55.2 percent of the time. The difference in proportions is, as with the number of runs, nowhere near statistical significance. Later, Jim (2008a) described a third way to examine streakiness. One can group consecutive at bats into groups of, for example, 20. Using Jims example, during 2005 Carlos Guillens sequence of 20 at bat groupings produced 5, 5, 7, 10, 10, 10, 6, 9, 4, 4, 6, 7, 4, 2, and 6 hits (before finishing with 12 for 34). This looks streaky starting relatively cold, Guillen seems to have heated up during the fourth through eighth of these sequences and then cooled off again. One can then test whether these ups and downs are or are not randomly organized. Jim developed a relevant statistical test (see Albert, 2013) and used it (Albert, 2014) to determine that there was more streaky behavior than one would expect from chance in home run hitting for batters with at least 200 ABs for seasons in the 1960 through 2012 interval. Some of the work on streakiness has been demonstrations of statistical methods concentrating on the performance of individual players. For example, even before his work with Williamson, Jim (1998) compared the streakiness in Mike Schmidts home run hitting during his competent years (1974 through 1987), in which he averaged about seven homers every hundred at bats, with 2000 simulated Schmidts homering randomly at the same rate. Based on the pattern of number of games played between home runs, Schmidts record appeared to match the average of the simulated Schmidts, but based on expected home runs over two-week periods, the real Schmidt appeared streakier than expected by chance. In the 2001 Albert and Williamson (2001) paper, they reported an analogous study of Mark McGwires home runs from 1995 through 1999 and Javy Lopezs hits in 1998 (a year for which he had been labeled in an Internet article as particularly streaky) using five-game periods and found both to resemble the average of the simulations. Scott Berry (1999a) examined whether the game-by-game home run performance of 11 prolific power hitters showed signs of streakiness during the historic 1998 season. In one analysis, he noted whether the number of at bats between home runs varied consistently with a random process. It did not for Sammy Sosa, who was streakier than chance would allow, and Andres Galarraga, who was more consistent than chance. In another, he adopted a Markov model with three states (normal, hot, and cold), in which the odds of a switch from normal to either hot or cold at any one at bat was 5 percent and a switch back to normal at any at bat was 10 percent. Again, Sammy Sosa appeared to have specific hot and cold streaks; this time, Ken Griffey Jr. seemed to have cold stretches. As Berry admitted, it is very difficult to distinguish whether Sosas streakiness was due to an actual hot hand or just a random outlier. Sommers (1999-2000) looked at the distribution of games in which Ruth (1927), Maris (1961), Sosa (1998), and McGwire (1998) had their historic home run years, and noted a random pattern. In his 1986 Baseball Abstract (pages 230-231), Bill James reported on a study he commissioned of seven Astros hitters by Steven Copley, who compared performance after good games (2 for 6 or better) or bad games (an oh-fer), and uncovered a slight improvement in the game after good games (.280) versus bad (.268). Bill felt that any evidence for across-game consistency was very questionable. James was right; the chi-square on that data are a minuscule .37. As mentioned, these studies were intended more as demonstrations of statistical techniques rather than substantive pieces, as one players performance says almost nothing about whether streakiness is a real or random tendency. The analyst needs to examine the performance of a large number of players, which will undoubtedly result in some appearing to be particularly streaky and others seeming particular stable. The researcher then compares the distribution of performances among the players with the distribution that would be expected if streakiness was random (i.e., a normal distribution). If the distributions differ, than performance is not random, and the direction of the differences will indicate whether more players than chance allows are streaky or stable. If the distributions do not differ, then one has evidence that streakiness is random. However, again it is not definitive evidence, because results that look random could also be the result of a situation in which there are a very few players who really are by their nature streaky or stable, too few to appear as anything other than the sort of outliers that will always occur in a normal distribution. If instead the evidence points to non-randomness, then, for example, it makes sense to look for individual differences in streakiness (Player C is more consistent than Player D); if the streaks are random position is correct, then it does not. The issue of randomness versus meaningfulness crops up across seasons. For example, the standard deviation for batting average in a 500 at bat season is about .020, which looks very low, but it means that there is a 66 percent chance that a .275 hitter will end up anywhere between .255 and .295 and a 95 percent chance somewhere between .235 and .315 due to chance. In a poorly conceived study, Gosnell, Keepler and Turner (1996) thought they were examining streaks and slumps lasting entire seasons and accounting for chance, but they failed. They looked at year-to-year batting averages for 100 randomly chosen players active between 1876 and 1987 with at least 300 career at bats and calculated that variation by chance alone can account for differences across seasons for 66 percent of the players and 90 percent of the seasons themselves. The problem with this sort of study is that it ignores the normal career trajectory, and so seasons not accountable by chance could well be those at the beginning and end of careers. Evidence that Streakiness is Not Real Most of the relevant work performed has found little evidence for the existence of non-random patterns, implying either that streakiness is a random occurrence or, if there is some actual non-random cause for it, that cause operates randomly (i.e., the occurrence of batters really seeing the ball well is unpredictable). I will organize this review in terms of studies using each of the three methods Jim Albert proposed. Method 1 Examining sequences of at bats The Hirdts (Siwoff et al., 1989, page 164), noted a slight tendency (translating to a .015 BA improvement) for players to get a hit following an at bat with a hit versus an at bat producing an out within the same game. There was no corresponding tendency across games. As such, it is very likely that these results are due to facing the same pitcher in consecutive at bats. Based on 2013 Retrosheet data, Wolferberger and Yaspan (2015) basically replicated these findings; only the immediately preceding PA had any predictive value for a given PAs outcome, and not more distant-in-the-past PAs. Method 2 conditional probabilities among at bats The most important of the early studies of the issue was by S. Christian Albright (1993a), which was followed by responses by Jim Albert (1993) and Hal S. Stern and Carl R. Morris (1993), and a rejoinder by Albright (1993b). Albright studied the probability of getting on base using Project Scoresheet data from 1987 to 1990 and included the 501 500-at-bat seasons that occurred during that interim. He used three different types of analysis; a runs test, a test of conditional probabilities, and a logistic regression analysis in which he studied whether performance over the previous 1, 2, 3, 6, 10, and 20 at bats predicted whether or not the batter got on base during the next at bat. In each case, he noted the overall distribution of players to see if their performance appeared to reflect a random process or not. Both the runs and conditional probability analyses revealed very slight tendencies toward streakiness. The logistic regression analysis resulted in random findings for predictions based on one previous at bat and a slight tendency for stability (absence of streakiness) for predictions based on twenty previous at bats. Albright concluded that there was no conclusive evidence for either streakiness or stability. Both Albert, and Stern and Morris, proposed alternative statistical models and performed their own analyses, none of which resulted in evidence contradicting Albrights conclusion. Stern and Morris also found that the logistic regression analysis is biased toward results favoring stability, which might account for Albrights 20-game findings. Stern (1995) noted analogous findings in a subsequent reanalysis of part of Albrights data set. Method 3 sequences of groups of at bats In one of their better studies, the Elias folks (Siwoff et al., 1987, pages 97-99) were among the first to take on the issue of streakiness. The Hirdts first defined a streak as a five-game stretch hitting .400 or better and a slump at five games of .125 or worse, and then observed performance in the five games following a streak/slump as just defined. Between 1984 and 1986, 161 players who experienced both a streak and a slump in a given season hit better in the five games following a streak than in the five games following a slump, but 184 hit better for the next five games after a slump rather than a streak. This evidence against long hitting streaks was even more extreme when original streaks and slumps were defined as three games (107 versus 165) and ten games (154 versus 220). My own work was another example of the third, grouping method, using week-to-week performance data over 11 seasons (1991-2001) and including players who had at least 10 at bats per week for ten consecutive weeks over four seasons (93 players qualified, with 549 seasons). In one study (Pavitt, 2002), performed a season at a time to protect against the impact of career performance trajectories, there was slight evidence for more inconsistency across weeks in BA and SA than chance would allow, probably due to the alternation of home and away games and any ballpark effects those would produce. In a second (Pavitt, 2003), including entire player careers corrected for career trajectories with a quadratic regression term, I noted non-random consistency across weeks for slugging average but not batting average, likely due to overall linear increases in power across careers that were not controlled by the curvilinear trajectory projection. Jim Albert (2007) looked at 284 batters during the 2005 season also using the groupings method (with 20 at bats) for hits, strikeouts, and home runs, and found that hit data led to about 20 batters displaying more streakiness than expected when 14 would be expected by chance; no analogous findings occurred for strikeouts and homers. Again, any home versus away effects could conceivably account for this discrepancy. In 2013, Jim used a method based on the distribution of the number of outs between hits for all 438 batters with at least 100 at bats in 2011. Although 70 players had distributions differing from pure chance, further analysis implied a random pattern. Jim replicated analogous findings for each season between 2000 and 2010. In contrast, an examination of the number of times batters made contact between strikeouts revealed significant streakiness for each of those seasons. Jim noted that striking out is more a function of skill and less of luck than getting a hit, and that difference may be crucial in explaining the contrast. Finally, using 2005 data, Lawrence Brown (2008) noted that batting average performance comparing the first and second halves of a season, and again month-by-month, resulted once again in an approximately normal distribution. Tom Tango, Mitchel Lichtman and Andrew Dolphin (2006) in a sense replicated the first Elias study. Working with Retrosheet data from 2000 through 2003, they looked at sequences of five games with at least 20 at bats in which hitters performed in the upper and lower five percent (generally wOBAs of greater than .525 and less than .195) to see if those trends continued in subsequent games. Compared to the players average wOBA in the seasons immediately before, during, and immediately after the sequence, they uncovered wOBAs about 4 points higher for the streakers and 5 points lower for the slumpers, implying a continuation of the hot or cold stretches. However, the differences are slight, and TMA did not correct for strength of opponent or ballpark. As such, the findings are not definitive. The Chinese Professional Baseball League on Taiwan has been the victim of a rash of game-fixing scandals, with one in 1996 and four between 2005 and 2009. As a consequence, eighty-two players were legally indicted, with at least 26 sentenced. Lin and Chan (2015) used data envelopment analysis to determine whether this method could indicate guilty players based on their week-to-week performance (SA for batters, total bases per inning for pitchers). The authors claimed accuracy rates ranging from 61% to 100% for indicating guilt through demonstrating stretches of anomalously poor performance. However, they never compared their findings to week-to-week patterns for non-indicted players to see if it differed, and in the end all they showed was that baseball players have performance slumps. Claimed Evidence that Streaks are Real Some work by Trent McCotter (2008) led to quite a bit of debate. Trents method was as follows: Using Retrosheet data from 1957 through 2006, he recorded the number and length of all batting streaks starting with one game along with the total number of games with and without hits in them. He then compared the number of streaks of different lengths to what occurred in ten thousand random simulated permutations of the games with/without hits in them. There was a consistent and highly statistically significant pattern across all lengths starting at five for more real-life streaks than in the simulations. Trent concluded that hitting streaks are not random occurrences. Although nobody challenged Trents method as such, there has been some criticism of other aspects of his work. He first proposed three alternative explanations for these patterns; batters facing long stretches of subpar pitching, batters playing in a good hitting ballpark, and streaks due to better weather conditions for hitting, i.e. warmer weather. He uncovered no evidence for the first, and claimed the second and third to be unlikely without empirically evaluating them. He instead opted for untestable speculations concerning a change in batter strategy toward single hitting and just the existence of a hot hand. I called him on these (2009); he responded (2009) with helpful analyses inconsistent with the second and third of the testable explanations and basically punted on the untestable ones. Jim Albert (2008b, summarized in 2010a) lauded the method and replicated it, but this time restricting the sample to five seasons of Retrosheet data studied separately (2004 through 2008). Again, real streaks occurred more often than in the random permutations, but only three out of twenty comparisons were significant at .05 and a fourth at .10. This initiated a debate in the Baseball Research Journal Volume 39 Number 2, in which Jim questioned the practical significance of Trents findings giving the huge sample size Trent used, Trent defended the huge sample size as necessary to tease out streaks buried in noisy data, and Jim challenged and Trent (McCotter, 2010a) upheld Trents use of the normal distribution as the basis for comparison. A subsequent publication (McCotter, 2010) added nothing substantive to the debate. Another reported demonstration that received a good bit of publicity was an unpublished study by Green and Zwiebel, based on Retrosheet data from 2000 through 2011. In essence using the second, conditional probability method, Green and Zwiebel wanted to see if the outcome of a particular plate appearance for both batters and pitchers could be predicted more accurately using the outcomes of the previous 25 at bats than overall performance for the given season, minus a 50 at bat window around the plate appearance under question. They provided various operational definitions for hot and cold streaks. Some of these definitions seem to bias the study in favor of finding streakiness; these established criteria based on the assumption that the average player is hot five percent and cold five percent of the time, which strikes me as out of bounds given that it presumes streakiness exists. A more defensible definition required the batter to be hot or cold if in the upper or lower five percent of a distribution based on his own performance. Their equations also controlled for handedness and strength of opposing pitchers and ballpark effects, but not, as Mitchel Lichtman (2016) pointed out, for umpire and weather. Unfortunately, the ballpark effect was poorly conceived, as it was based solely on raw performance figures and did not control for relative strength of the home team (i.e., a really good/bad hitting home team as indicated by performance in away games would lead to the measure indicating a better/worse hitting environment than the ballpark is in truth). The authors results indicated the existence of hot/cold streaks for all examined measures: hits, walks, home runs, strikeouts, and times on base for both batters and pitchers. Interestingly, after noting improved performance after the plate appearance under question than before, the authors attributed half of the reported increase in that PA to a learning effect, in essence true improvement in hitting. As Mitchel Lichtman (2016) pointed out, if so, then it should not be considered evidence for the existence of streakiness. I would guess that this result is due to facing the same pitcher multiple times. Green and Zwiebels work elicited a lot of critical comment. Along with the ballpark problem, which Zwiebel acknowledged in email correspondence with Mitchel Lichtman, one criticism was that subtracting the 50 at bat window biased the study in favor of finding streaks. Heres an example showing why: let us assume that a player is a .270 hitter. If a player happens to be hitting .300 or .240 during that window, then the rest of the season he must be hitting say .260 or .280 to end up at that .270. In this case, the .300 and .240 are being compared to averages unusually low and high rather than the players norm. But it strikes me that this would only be a problem if hot and cold streaks actually existed if not, it would be .270 all the way. It is the case that subtracting the 50 at bat window lowers the sample size of comparison at bats, increasing random fluctuation and again adding a bias in favor of finding streakiness. Whether this loss of 50 at bats is catastrophic during a 500 at bat season for a regular player is a matter for debate. In any case, Mitchel Lichtman (2016) performed his own study using 2000-2014 Retrosheet data, but in this case used the sixth PA after the 25 window in order to insure that it usually occurred in a different game. He also used a standard projection method (i.e. three years of past performance with the more recent weighted over the less) rather than a within-season window. The results were a small hot and slightly larger cold hand effects for BB/PA, OBA, wOBA, and HR/PA, and almost none for BA. Mitchel speculated that changes in both batting (such as swinging for homers after hitting a few) and pitching (such as pitching more carefully to the hot batter and less so to the cold) strategies might be at least partly responsible, along with cold batters playing with an injury. Neither of the first two proposals are realistically testable. Green and Zwiebel were eventually able to publish their paper in 2018, basically unchanged but with an additional analysis claiming that the opposition notices hot streaks and responds by walking the batter in question more often than the batters norm. They also criticizes the TMA method on two counts, both basically implying that TMAs use of a three-year average wOBA as the baseline for comparison to hot streaks is wrong. The implications for them is that the data representing the possible hot streak data are also included in the comparison data, which is statistically invalid, and that the TMA method does not take regression to the mean for players performing over and under their heads during the three year period. The problem in Green and Zwiebels claim for me is the same as it was before; it presumes that hot and cold streaks exist rather than demonstrating that they do. In conclusion, it is possible that streaks and particularly slumps have some reality, but the bulk of performance variation across time is undoubtedly random fluctuation. And A good example of regression to the mean is the supposed Home Run Derby curse. Players are chosen for this All-Star-Game-related event due to their HRs during the previous three months of the season, and it is not unusual that quite a few contestants have been productive well beyond their performance in past seasons. Consistently with the curse, Joseph McCollum and Marcus Jaiclin (2010) noted that contestants had significantly lower HR/AB and OPS after participating than before, and that no analogous differences occurred for the same players when not taking part. However, they also discovered that both the first and second half performance of contestants were on average about equally superior to their normal performance. The point is that contestants tend to be chosen after unusually good starts while enjoying their best seasons. OLeary (2013) examined relevant 1999-2013 data and basically replicated this result, but also noted that this decrease only occurred for losers and not winners. Team Interdependence Along with protection, another platitude one often hears from baseball insiders is that hitting is contagious, such that good performance from one hitter begets good performance from the others such that everyone reinforces one anothers success, and analogously for bad performance. Although not directly studying that specific issue, which would require looking at performance at the level of the game, there is evidence of interdependence in the sense that playing on a good team tends to improve counting indices. Bill James not surprisingly got the ball rolling in the 1985 Abstract (pages 175-177), with an informal study in which he matched 68 players into pairs with similar performance in a given season, with one member of the pair on a very good offensive team (a mean of 834 runs scored over the season) and the other one a very poor one (mean of 568). The players on good hitting teams averaged 13% more runs scored and 15% more RBI than those on bad hitting teams. Bill proposed the idea that this could be partly due to those of good hitting teams having more at bats, which makes sense in general but not here as ABs were part of Bills matching, and also the fact that on good hitting teams one gets up with more runners on base (thus more RBI) and gets on base with better hitters following (thus more runs scored). David Kaplan (2006) performed a far more rigorous analysis, including two seasons (2000 and 2003) with players totaling at least 200 plate appearances. David found evidence in support of interdependence for a wide array of cumulative indices hits, walks, total bases, runs scored, runs batted in, runs created but not for a set of related average measures on-base and slugging averages, isolated power, on base plus slugging. As cumulative measures are affected by individual opportunity to play whereas average measures are not, what seems to be implied is that, with a season, there are teams who tend to allot most of the playing time to a few regulars, increasing their cumulative indices as a set, and others, either by design or happenstance (injury, personnel changes) that divide opportunity among more players, limiting their individual ability to accumulate counting statistics. A different type of interdependence among players is the performance of hitters with and without baserunners. Here, there is evidence consistent with the platitude that baserunners disrupt the defense and improve the fortunes of hitters. Based on 1999-2002 Retrosheet data, the TMA group (2006) determined that mean wOBA, .358 overall, was .372 with runners on first and fewer than two outs. Again not surprisingly, that broke down to .378 for lefthanded hitters and .368 for righties. Transactions Contract Status Is it true that players shirk after signing long term contracts and perform better the year before entering free agency? There have been quite a few studies of this issue, but unfortunately many of them have been seriously flawed. In the majority of cases, free agency arrives after players have past their peak, and distinguishing shirking from general skill erosion requires more than just comparing what happens before signing a long term contract with what happens afterward. Comparisons between free agents and non-free agents can help determine whether performance differences are indeed a function of free agency or a function of overall changes in player skill level across years. The issue itself is of more than passing interest to both organizational psychologists and economists, as it is relevant to significant theoretical issues within their domains. For example, organizational psychologists have approached the issue of contract status through pitting two theories with opposing implications, equity theory and expectancy theory, against one another. Advocates of expectancy theory, which basically implies that the opportunity to improve ones salary would lead to harder work, would argue that players would be motivated to perform better the last year of their contract in order to try to gain attractive offers the following year, and that they would be less motivated during other years and underperform (Krautmann & Solow, 2009). On the other hand, equity theory predicts that people who feel that they have not gotten the rewards they deserve for their effort become angry whereas those who feel they do more than they deserve become guilty. This could imply that players in their last year may suffer feelings of inequity in pay given salary increases others have received for comparable performance during the length of their contract, and respond by underperforming during the final year of their contract as payback for their lower pay. They would then bounce back to normal performance with what they consider a more equitable new contract the next season (Lord & Hohenfeld, 1979). A majority of the early evidence favored the equity theory prediction. Specific hypotheses and relevant evidence both for and against included the following: Hypothesis 1: As stated above, the most basic hypothesis is that the year before free agency, players will perform worse than before and then bounce back after signing. During the early years of free agency, players who wanted to test the waters had to wait one additional season with their team (the option year). Lord and Hohenfeld (1979) supported the implied hypothesis of poorer performance for 13 relevant players in 1976 in the cases of home runs, runs scored, and runs batted in, but not batting average, as compared to the three previous years and the one following. However, Duchon and Jago (1981) extended the analysis to 30 position players with option years from 1976 to 1978 and found no difference across years, suggesting that the Lord/Hohenfeld findings had been a small-sample fluke. Harder (1991), considering 106 position players in the 1977 through 1980 stretch, noted a decrease in batting average but no discernible change in home runs per at bat. In another analysis, based on the contracts achieved by the very first (1976) cohort of free agents, he observed that home runs per at bat but not batting average were related to salary; note the inconsistency across the two findings. Hypothesis 2: Winners will feel guilt and improve their performance to assuage it, whereas losers will feel robbed and perform worse. In the case of salary arbitration, the extent to which the players arbitration offer differs from the teams will predict the extent of that over- or underperformance. Hauenstein and Lord (1989) noted some support for this hypothesis for 81 players who went through arbitration between 1978 and 1984. Looking at all position players with new contracts in the years 1976, 1977, 1987, and 1988 with new contracts in those years, Harder (1992) claimed some evidence that players paid better than predicted by the before-contract performance/salary relationship had a higher runs created figure, and in some years a higher total average, than would be expected in the year after the contract signing. However, for players who were underrewarded, there was only a slight decrease in total average the next year and no impact on runs created. Bretz and Thomas (1992) examined performance before and after arbitration cases for 116 position players between 1974 and 1987. Their performance index was a very strange amalgam of several disparate measures purposely biased toward power, which they felt has a disproportionate impact on salary. Arbitration winners performance afterward was markedly better than it had been both two years before arbitration and across their entire career, whereas arbitration losers performance was not better a year and worse two years afterward. Complicating the picture is work by Weiner and Mero (1999), in a study based on 205 position players with at least two years of experience who were in the major leagues in 1991 and 1992. After controlling for a few performance indices (with experience, at bats, and career runs created per at bat significant covariates), they uncovered evidence that players who are paid more than average in 1992 relative to others at the same position tended to increase their runs created per at bat and their total player rating between 1991 and 1992 whereas those below average tended to decrease theirs. These findings are consistent with equity notions, but there was no control for whether or not the players changed teams, and one would suspect that only a minority did. The implication is that changing teams may not be as significant as relative pay alone. Hypothesis 3: Guilt from leaving their old team would result in performance decrements against it. Kopelman and Schneller (1987) examined 54 players who switched teams during the first nine years of free agency (between 1976 and 1985), but noted only negligible differences in batting average. As for expectancy theory, the basic hypothesis is the opposite of equity theorys; better performance the year before returning to normal afterward. Examining 110 position players who between 1976 and 1983 signed a contract lasting at least 5 years, Krautmann (1990) first noted that the number of players performing above (68 versus 71) and below (42 versus 39) their career means in slugging average did not differ between the year before and year after the signing. Second, he observed that only 5 of the 110 in the sample performed above their expected range of random variation in slugging average, as calculated across seasons for each player, the year before, and only 2 produced below that range. The implication is that performance differences between the two seasons were random. Scoggins (1993) responded by claiming that any shirking after the signing would be reflected in time spent on the disabled list, arguing that total bases is a better measure than slugging average because it includes both hitting prowess and endurance as causal factors. Using the same sample, he demonstrated that, combined across players, total bases decreased from the year before to the year after the signing. In reply, Krautmann (1993) contended that it is wrong to combine player data, and uncovered only 6 out of the 110 in the sample who had fewer total bases the year after than the range of their expected random variation. Krautmann and Donley returned to this issue again (2009), based on position player signings for the 2005 and 2006 seasons, noting no performance decrement the year after signing as measured by OPS but some as measured by a players estimated monetary value. Turning to other analysts, Ahlstrom, Si, and Kennelly (1999) specifically pitted the two theories against one another. They uncovered no change between the free agency year (mean BA = .261, mean SA = .391) and the previous season (mean BA = .258, mean SA = .392) for 172 free agent hitters changing teams from 1976 through 1992, but a significant decrease going into the season after (mean BA = .247, mean SA = .367), leading them to support expectancy theory. Sturman and Thibodeau (2001) examined a paltry 33 players who gained a 30 or more percent increase in salary in a multi-year contract averaging at least one million dollars per year that was signed between the 1991-1992 and 1997-1998 off seasons. Batting average, home runs, and stolen bases decreased in the first year after signing from the previous two years but bounced back the second year after signing. Martin, Eggleston, Seymour, and Lecrom (2011) examined 293 free agents between 1996 and 2008 and noted that batting average, on-base average, slugging average, runs created per game, and adjusted batting wins were all greater in the walk year than the seasons before and after. Finally, White and Sheldon (2014) looked at players on their first multiyear contract for 66 players between 2006 and 2011. There was only a slight increase in performance as measured by BA, SA, OBA, HRs, and RBIs for the last year of the old contract as compared to the previous season, but a significant decrease between year of and year after. In summary, given evidence for and against both theories, we cannot reach a conclusion one way or another based on this research. All of it, however, is fundamentally flawed. As mentioned earlier, you cannot just examine performance relative to free agency; you have to place it in the context of the general trajectory of performance over a career, and you should compare your sample with a matched set of non-free agents. Even non-academic baseball researchers who should know better have made this error. Dayn Perry (2006a) of the Prospectus group demonstrated that 212 free agents between 1976 and 2000 had a Wins above Replacement Player 0.48 greater than both the year previous and the year after, while noting their average age (31) but without correcting for it, or at least compensating for differences in ages among them. In addition, WARP, as a counting index rather than an average, is strongly affected by number of games played, and the sample did indeed play more games the free agency year than those around it. Perry claimed that this only makes up part of the free agent performance increment but does not provide relevant data. A couple of researchers did slightly better. Woolway (1997) calculated a production function relating team winning average to batter OBA, SA, and SBs, pitcher ERA, and unearned runs (representing team defense), and then used that function to estimate the number of wins the team would have both with and without 40 players who signed multi-year contracts between 1992 and 1993. He concluded that these 40 players were worth an average of 1.191 fewer wins to their teams in 1993 than in 1992, and to his credit found no difference in this decrement between players in their primes (26-30 years of age) and past it (older than 30). But he failed to compare these 40 to other players, and to other seasons in these players careers, so for all we know many of these players had career years in 1992 that made them particularly attractive for multi-year deals. Finally, Paulsen (2019) examined 535 position players with 3 or more years of service between 2010 and 2017, with a total of 1068 contracts, and concluded that the more seasons left on a contract, the worse the performance, to the tune of a tiny .07 rWAR per year. This tendency was not impacted by whether players changed teams. Happily, he controlled for past rWAR (because higher rWAR means longer contracts) and experience (not surprisingly negatively related with rWAR). The results were substantially the same when limited to players with 6 or more years of service, i.e. free agent years. Some research by economists has attempted to compensate for the problems stated above through computing expected career performance trajectories for players and then seeing if the expected performance differed reliably from actual performance for hitters before and after the signing of a new contract. Maxcy (1997) examined slugging average for more than 2200 player-seasons for an unknown number of players between 1986 and 1993 (and some seasons back to 1983 for long-term contracted players) and uncovered no decrease in performance independently of the effects of aging. Analogously, Maxcy, Fort, and Krautmann (2002) used slugging average for 1160 seasons for 213 position players from unnamed seasons and observed no impact for contract status; they did note that players spend less time on the disabled list and play more than expected during the last year of a contract. Marburger (2003) compared before-and-after performance, as measured through bases gained through hitting and base stealing, by 279 free agent position players in the 1990 through 1993 interim with that for a matched set of players from before the free agency period, and discerned no differences between the groups. In a sample of 527 free-agent position players between 1997 and 2007, Krautmann and Solow (2009) noted OPS adjusted for home-field and league effects was a scant 3 points greater in the year before free agency than career trajectories would imply, and a still-small 10 points less in the first of five-year contracts (which was a rare length; 80 percent were for either one or two years and only 5 percent for five years or greater). ONeill was involved in two studies that were substantially the same as Krautmann and Solows, including predictors for age and age squared along with a dummy variable indicating whether the players career ended at the conclusion of his contract. The first (Hummel & ONeill, 2011) included 227 position players reaching free agency between 2004 and 2008, the second (ONeill, 2013, 2014) consisted of 546 instances for 256 position players between 2006 and 2011. The first indicated a substantial 4.2 to 5.5 percent increase in OPS during the contract year; the second a smaller 1.1 to 1.8 percent improvement. However, Phil Birnbaum (2015) described a problem with ONeills work; let me describe it in my own terms. As noted above, older players are often unable to get another contract after a particularly poor year. For the best of these players, that year may be randomly bad and not representative of their true remaining talent level. A randomly good year would of course lead to another contract. So, careers are likely to end with randomly bad seasons. This results in projections that are unrealistically pessimistic for the last years of good but aging players (Phil used Moises Alou as an example). For this reason, Phil was for good reason convinced that ONeills projections could not be trusted. Earlier, Phil (2007), using runs created per 27 outs as his offensive index for 399 free agents between 1977 and 2001 with at least 300 batting outs that year, corrected for random variation in performance by weighing RC/27 for the free agent year for regression toward the seasons around it, and then compared them to 3692 non-free agents. The free-agents were 1.2 runs per year better, which Phil likened to turning one out into a triple; in short, very little. ONeills latest work (ONeill & Deacle, in press as I write this) took care of that problem, but is further plagued with another; the failure to control for player age. The sample consisted of players eligible for free agency (6+ years) who played for at least two seasons between 2007 and 2011; a total of 225 players and 822 player-seasons. Without going into too much detail, here are her estimates for differences between predicted and obtained OPS+ for position players overall, for those in the top quartile, and for those in the bottom quartile, for their contract year and the subsequent season depending on the length of their new contract: ContractContract yearOne year contractTwo year contractThree year contractFour year contractFive year contractSix year contractOverall+6+12+30-3-6-8Top quartile+7+15+6+5+3+1-1Bottom quartile+2+1-7-14-20-27-34 The overall effect remains as tiny as in previous work, and additionally looks like players, during the first year of a new contract, shirk a bit more the longer that contract lasts. But check out the distinction between quartiles. It looks as if the strongest hitters are inspired by their new contract whereas the weaker ones shirk significantly that first year after signing contracts of any considerable length. ONeills interprets these data along the extra effort/shirking lines. That could well be correct, but I am not at all convinced, and the quartile distinction reveals why. As I noted, ONeill failed to sufficiently control for player career trajectories. As described in the Age section above, stronger players on average enter the majors a couple of years earlier than weaker ones. Putting the two together, I hypothesize the following alternative: 1 - The top quartile players hit their contract year a couple of years earlier than the bottom quartile players. Therefore, the former are coming into their peak whereas the latter are at theirs. This is why the data are higher for the top quartile than the bottom. 2 The more that a player randomly happens to overproduce above expectation given their career trajectory during the contract season, aka has well-timed career year, the longer a contract they are able to sign. After that signing, they return to normal. The greater the overproduction, the greater the fall to normal. That is why productivity is lower the longer the contract. The relevant data are available for both of these hypotheses to be tested on their own. In any case, ONeill has one more study to do, this time including age and age squared as controls. Arbitration has remained a fertile area to study. Dumble (1997) performed an analogous but simpler study including player eligible for arbitration from 1986 through 1992. In this case, winners did better the year before arbitration than the year after, losers a bit better the year after than the year before, and eligible players who did not experience the arbitration process had no change in performance, as measured by Palmers Batting Runs. The author noted the obvious explanation: those playing over the heads the year before won their cases and then reverted to normal, those playing below their heads the year before lost and then reverted to normal. Dumble ought to have included previous and subsequent seasons to the analysis to be sure. Marr and Thau (2014) hypothesized that what they called status loss has a more negative impact on performance for those with previous high status when compared to those with previous low status, and used MLB final-offer arbitration as one of their tests. Their sample was 186 players who experienced arbitration once only between 1974 and 2011. They concocted a status scale by summing the number of All-Star Game appearances, MVP, Rookie of the Year, Silver Slugger, and Gold Glove awards. Status itself was totally unrelated with odds of the player winning or losing arbitration. Anyway, those toward the higher end of the status scale tended toward lower OPS the season after an arbitration loss as compared with the season before, whereas those toward the lower end of the status scale had no such tendency. Importantly, player age was included as a control in their models, which is critical because older players would tend to have high status and have decreasing performance, biasing the results. Phil Birnbaum (2007) used the same sort of method for arbitration as that just described in his work on free agency, and learned that those who lost actually slightly outperformed the winners, and both of those groups were outdone by those not undergoing arbitration. Finally, and in contrast with both equity and expectancy theories, the very experience of arbitration may hurt performance. Based on 1424 filings between 1988 and 2011, Budd, Sojourner, and Jung (2017) proposed that the 1182 that ended with agreement before the case resulted in better subsequent performance by those players than the 98 who won and 144 who lost their case due to greater trust in and more positive feelings about the team. The evidence did not support that hypothesis, as there were no consistent differences in batter BA, OBA, and runs created, pitcher ERA (both regular and fielding-independent), and overall WARP. They did, however, note a tendency for players who went before the arbitrator to be released or traded before the end of the next season, and likewise before the next season started, particularly if the player won the case. Interestingly, the greater the difference between the teams and the players offers, the worse the performance the next year (do relatively lousy players unrealistically inflate salary expectations?). In conclusion, the best research suggests results consistent with expectancy theory and the general idea of shirking, but if there is an effect, it is tiny. Switching Teams The impact of switching teams on subsequent performance has received a bit of attention from academics, who noted immediate improvement right after the transaction. Bateman, Karwan, and Kazee (1983), with a sample of 97 batters between 1976 and 1980, noted increased BA, HR, and RBI if the transaction occurred midseason but not between seasons. Jackson, Buglione, and Glenwick (1988) looked at 59 batters switching teams within season and observed BA and SA to be lower in the months during that season before the transaction than in previous years and higher in the subsequent months that season than in the next year. Explanations for the effect include one similar to expectancy theory (Bateman et al., job transfers lead to increased motivation to do well, with the absence of improvement across seasons explained away as motivation dissipating over the winter) and psychological drive theory (Jackson et al., fear of a trade beforehand leads to over-arousal and poor performance whereas pleasure at being somewhere they are wanted leads to extremely good performance afterward). A much better explanation, analogous to Bill Jamess for team performance under new managers, is that these findings are an artifact of the probability that players are more likely to be discarded when they are randomly performing below form and thus disappointing their team, and then bouncing back to normalcy after the move. Muddying the waters, the Hirdts (Siwoff et al., 1991, pages 4-5) examined the records of 49 batters who were traded to a different team in the other major league a year after hitting at least 20 home runs, and uncovered a strong tendency for their home run totals to decrease with the new team; 40 of the 49 for raw homer numbers and 37 of the 49 for HR per AB. However, Pete Palmer implied in a personal communication, much of this decrease could be regression to the mean, and the Hirdts did not examine batters who were not traded as a comparison. Reasoning that as a consequence of anger and/or the desire to protect self-esteem, Kopelman and Pantaleno (1977) hypothesized that performance the year after being traded or sold would be better against the players former team than other teams. Note that this thinking is the opposite of the equity-based prediction for free agency movement above. The authors, based on 47 players traded or sold in 1968 or 1969, obtained marginally significant findings supporting this conjecture for batting average, particularly for players who had never been traded before, players with the former team at least three seasons, younger players, and players in the upper half of the sample in batting average, all of whom would hypothetically be more upset by the transaction. In addition, these effects faded away during the second and third year after the move, supposedly because the anger would have faded away. As with the shirking issue, this work means little due to the absence of comparisons with players remaining with their teams. As a consequence the best work is by Rogers, Vardaman. Allen, Muslin, and Baskin (2017), although it is still compromised due to insufficient thinking about the relevant issue. They compared position players who, after two seasons with at least 100 PAs for one team during the 2004 through 2015 stretch, switched to another team after declining (422 cases) versus stable or improving (290 cases) performance across those two seasons. Performance indices included BA, OPS, weighted runs created plus, and fielding average, with a set of controls including among others career batting average and age to account for the effects of a players declining ability (page 552); I think it would have been better if the authors had computed a career trajectory and then used the relevant seasons estimate from that trajectory as a stand-in. There were also comparisons to players with declines (922 cases) versus stable or improving performance who did not change teams. The findings: On average, those who declined across years 1 and 2 improved in year 3 on all three batting indices, a simple case of regression to the mean. Those who switched teams declined more than those that stayed, which I hypothesize indicates that those who declined more were more likely to switch teams than those who stayed; the authors do not seem to have performed this easy test. On average, those who were stable or improved across years 1 and 2 declined in year 3 on all three batting indices, again more so if they changed teams than if they did not, with the same implications as beforehand. Reason for movement, i.e. trade versus free agency did not matter in either case. These regressions to the mean remained stable for year 4 for those remaining with the year 3 team. Fielding average changed in unison with the batting indices, but was never significant. Rogers and associates realized that the issue here was the absence of variance in fielding average; obviously, they should have used one of the range factor-type measures instead. In summary, there is no reason to believe that performance is affected by switching teams. True Ability A players batting performance in a given year is partly due to skill and partly due to luck (some years all the grounders find holes, other years all the line drives are caught). Assuming random processes at work, in a 600 at bat season, the expected standard deviation for batting average is 17 points (Heeren & Palmer, 2011, page 31). This means that, through luck alone, the odds are about two-thirds that someone who is really a .275 hitter will end up between .258 and .292 and 95 percent that he will finish between .241 and .309 in a given year. If 200 regular players get 600 at bats, then about five would be expected to end up 34 or more points higher and another five 34 points lower than their true ability. Pete Palmer also noted that the 600-AB s.d. for slugging average is 35 points. Given the non-independence of slugging average and one-base average (hits are included in both), figuring out the s.d. for OPS is complicated; through simulation, Pete came up with an s.d. of 51 points. In the chapter on offensive evaluation, I described Jim Alberts (2005, 2006b, 2007) work on consistency in performance, which he wisely interpreted as relevant to the question of distinguishing how much of performance reflects true ability rather than luck. Although not originally interpreted as such, the first relevant work in this area was by Carl Morris (1983, 1998), who proposed a method for estimating what batting averages should have been without normal random variation. He determined that Ty Cobb was 88 percent likely to be, in his words, a true .400 hitter at some point in his career; the only other player with a reasonable chance was Rogers Hornsby. Earlier, Jim Albert (1992) demonstrated a method for estimating a batters true peak and career home run performance that discounts unusually good and poor seasons and accentuates the general trajectory of a players performance during his career. In truth, the true career totals of 12 hitters Albert used in his demonstration did not differ much from their actual totals. Casella and Berger (1994) demonstrated a method with the same goal for situations in which the data used for estimation are known to be biased; using their example, for when an announcer has mentioned that Dave Winfield has 8 hits in his last 17 at bats and you know that exact number of at bats was chosen rather than others because it best accentuates his recent hitting success. Frey (2007) provided a model for estimating a batters true ability based only on his batting average. The point of the model is that a batting average of, to use his example, .833 is achievable with only 6 at bats (5 hits), and as such provides far less information about the players true ability than an average of .338, which requires at least 65 at bats (with 22 hits) to achieve. To be honest, the method may be of interest to pure statisticians but does not offer as much to the baseball analyst as those attempts that explicitly include number of at bats in the model. As part of his work, Frey also noted that mean batting average in a season increases with number of at bats, particularly for those with fewer than 200 at bats. This finding is the result of the fact that players with lower batting averages are more likely to be benched or dropped from a team before amassing a large number of at bats than players with higher batting averages. Null (2009) attempted to project future performance through distinguishing among fourteen different abilities (for example, tripling through fly ball versus doubling through fly ball versus tripling ground ball) plus considering age effects. Finally, based on implications of the binomial distribution (his method is described in more detail in the Team Evaluation chapter), Pete Palmer (2017) estimated that variation in batting average among major leaguers is about half skill and half luck; as we know, whether a batted ball leads to a hit or an out has a fair element of randomness. Variation in normalized OPS among major leaguers is more in the order of 60 percent skill and 40 percent luck, given that getting walks and hitting for power have a smaller randomness element than getting hits. Walks No, a walk is not as good as a hit; it is about two-thirds as good according to the bottom-up regression methods cataloged last chapter, making it a significant offensive weapon. Further, whereas the variation among batters in BA is about 1 (from a high of say .350 to a low of .200), the variation among batters in the proportion of walks per plate appearance is about 5, from the high teens (Bryce Harper was at 18.7% in 2018) to less than 5 percent (the same season, Salvatore Perez was at 3.3%). Obviously, disparities such as this would not be allowed to exist for hits per plate appearance, as someone with a batting average one-fifth of the league leader would not last in the major leagues very long. One way to think about walks is in relation to strikeouts. Long-time SABR stalwart Cappy Gagnon (1988) proposed what he called the Batting Eye Index, a simple measure of walks minus strikeouts divided by games played (at that time, Ted Williams was the career leader at .572). Jonathan Katz (1990) argued for the advantages of dividing by plate appearances or, even better, walks plus strikeouts (as the latter is by definition bounded by +1 and -1 and so easier to interpret). Given that the three indices intercorrelated around .9 in his data, none has a real statistical advantage over the other. What is surprising is that neither Gagnons or Katzs versions had any correlation whatsoever with lifetime batting or slugging average or career length. This implies that batting eye seems to be a skill independent of, and thus not predictive of, the skills involved in hitting the ball. This work was actually predated by a monumental examination of the walk by then Texas Ranger employee Craig Wright (1984a, 1984b, 1985). In the two 1984 pieces, Wright analogously with Katz found few discernible differences in overall offensive performance between high (mean of 11.4 walks per 100 plate appearances in 1983; I assume similar for 1982), average (mean of 8.1 walks), and low (mean of 5.2) walks among American League players with at least 400 PA in 1982 and 1983; the best tended to have a couple of more home runs and slightly fewer triples, but nothing really glaring. It did turn out that the best walkers were slightly older (average 29.9 years as opposed to 29.3 for average and 28.4 for low), which was consistent with the evidence described in the Age section of this chapter. There was also evidence here that rising walk rates might be correlated with better offensive performance during the latter stages of a career, prolonging both the players peak and slowing his inevitable decline. There was, however, evidence that high walkers were more successful than average or low in situations meeting Eliass clutch definition (late inning pressure), with a only slightly higher batting average but a good 50 percent more home runs, which Wright hypothesized might be an indication of the high walkers being more adaptable in batting style than the others. The 1985 article is a historical study on this last theme; implying that walking is a learnable skill that at least some players are able to take advantage of (whereas others are too stubborn), and that some managers have recognized that, in so doing improving their teams. Due to the availability of PITCHf/x plus data from organizations such as Baseball Info Solutions, we are now able to make extremely detailed studies of plate discipline. New indices measuring the percentage of swings at pitches outside (O-Swing%) and inside (Z-Swing%) the strike zone and contact rates on these swings (O-Contact% and Z-Contact%; see Plate Discipline, n.d. for a list of indices and the following data) allow ratings of batter discipline along with pitcher effectiveness. For the record, reported mean figures are 30% for O-Swing%, 65% for Z-Swing%, 66% for O-Contact, and 87% for Z-Contact% for 2002 through (I think) 2017. Using these data, Ryan Pollack (2017) defined an Aggression Factor based on a box representing the median deviation along horizontal and vertical axes of pitches swung at, normalized for different numbers of plate appearances, with lower numbers representing only swinging at pitches in a relatively tighter area. Aggression Factor ranged from about 4 (extreme discipline) through about 20 (the opposite). The correlation between AF and wRC+ was negative .35, implying an average gain of almost three wRC+ points for each square inch in which the box narrowed, based on (I believe) 2015-2017 data. Jim Albert (2017, Chapter 7) presented data relevant to overall batter swing and contact rates for the 2016 season. That year, batter swing rates varied from 35 percent to 60 percent; contact rates went from 70 percent to 90 percent. Higher swing rates were associated with lower contact rates for pitches both in and out of the real strike zone, more strikeouts, and fewer walks. Vock and Vock (2018) constructed a method for computing what a players BA, SA, and OBA would be with differing amounts of plate discipline, based on his actual performance as influenced by his odds of swinging against different types of pitches of different speeds/movements/locations. Using PITCHf/x data for 2012 and 2014, they estimated what Starlin Castro would have done if he had Andrew McCutcheons discipline. As for the opposite of plate discipline; in a study receiving plenty of publicity in the popular press, Kutscher, Song, Wang, Upender, and Malow (2013) noted a linear relationship in batters swinging at pitches out of the strike zone across the months of the season, from 29.2 percent in April 2012 to 31.4 percent in September 2012, demonstrating what they interpreted as a fatigue effect. Importantly, they added that the same tendency existed for the years 2005 through 2011 in previously reported data. And Winter, Potenziano, Zhang, and Hammond (2011) were able to convince 16 major league players to fill out the Morningness Eveningness Questionnaire, which based on self-report answers can type people according to the time of day during which they perform best (I did an on-line version and it absolutely nailed what I thought is my best bedtime). Based on their 2009-2010 BA performance with game start time adjusted for travel between time zones, the nine counting as morning type players averaged .267 in games starting before 2 p.m. versus .259 for games starting at 8 p.m. or later; the corresponding numbers for the seven evening type players were .252 versus .306. A small sample size, but suggestive nonetheless. Does playing every day result in performance decrements? Harold Brooks (1992) used Cal Ripken as an example, showing that his September/October performance between 1984 and 1991 was somewhat poorer than his April-to-August production (incidentally, I did a little study [1994] showing that when the Orioles had a long stretch of games without an off-day between 1984 and 1989, Ripkens hitting suffered during the last few days [starting day 17]). Harold went on to show evidence that this seems to be true across the board. While there is normally a decline in September, he showed that the more games played, the greater the late-season drop-off for throwing infielders (2B, SS, 3B) relative to league average for those starting 153-157 and particularly 158-162 games at those positions. Outfielders did not suffer the same decline. References Adams, Dallas (1981a). The probability of the league leader batting .400. Baseball Research Journal, No. 10, pages 82-84. Adams, Dallas (1982c). Some patterns of age and skill. Baseball Analyst, No. 2, pages 3-9. Adams, Dallas (1983). On the probability of hitting .400. Baseball Analyst, No. 7, pages 5-7. Ahlstrom, David, Steven Si, and James Kennelly (1999). Free-agent performance in Major League Baseball: Do teams get what they expect? Journal of Sport Management, 13, 181-196. Albert, Jim (1992). A Bayesian analysis of a Poisson random effects model for home run hitters. American Statistician, Vol. 46 No. 4, pages 246-253. Albert, Jim (1993). 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