The authors have declared that no competing interests exist. The authors are affiliated with a small startup company, “Scalaton”. This company is developing ideas and designs for cloudbased computing systems. The authors declare that they intend to adhere to all PLOS ONE policies on sharing data and materials.
Conceived and designed the experiments: JB. Performed the experiments: JB. Analyzed the data: JB AM. Wrote the paper: JB DG.
Data analysis is used to test the hypothesis that “hitting is contagious”. A statistical model is described to study the effect of a hot hitter upon his teammates’ batting during a consecutive game hitting streak. Box score data for entire seasons comprising
Baseball folklore has long included the belief that
Hot hitting is a transient phenomenon, inherently related to an observation interval. An archetype of unusually hot hitting in baseball is the individual consecutive game batting streak. In particular, streaks of 30 or more games’ duration are rare–only
The question examined here is whether or not hitting is contagious. The infrequent, long hitting streak provides a model situation to study this question. However, the methods and results of this research have broader implications. If empirical evidence for a contagion effect in baseball were shown to exist, it could provide insights into the psychology of motivation in other team sports, and perhaps more generally into the dynamics of propagation of positive behaviors in sociological, organizational management or economic studies.
Are sports streaks real phenomena, or merely views of random sequences of events misinterpreted by a desire to detect temporal patterns? The “hot hand” has been a bountiful topic for sportsrelated statistical research.
Streakiness has been studied in connection to many different sports. In a 2006 review, BarEli
Subsequent to the BarEli review
Baseball hitting streaks were studied by Albright
These investigations applied statistical methods to analyze an individual streaky player, or that of aggregate behavior. The identification of contagion effects requires consideration of a streak’s effect as it spreads to teammates. The metaphor of contagion suggests utilizing analytical methods developed in epidemiology as a framework for scientific investigation.
The first Surgeon General’s report in 1964
Contagious feelings in social groups have been widely studied. Hatfield
Barsade
There may be a neurobiological mechanism explaining such observations, suggesting a connection between observation of sports behavior and its propagation to observers of that action.
Rizzolati
Gray and Beilock
It is important to recognize the potential impact of unobserved variables before declaring that a causal relationship is fundamental in any contagious hitting effect. The work of Shalizi and Thomas
The purpose of this study was to examine the hypothesis that “hitting is contagious” in baseball. A retrospective analysis was undertaken on box scores from
There appear to exist no previous empirical studies within the framework of confirmatory data analysis to quantify the spread of hot hitting in baseball. This approach appears to be novel, and could be applied to studies of performance enhancement in other team sports, or extended to sociological, organizational and economic investigations.
The implied population for the present study is the set of all Major League Baseball (MLB) players active since 1945. The experimental sample culled from this population comprises players who were teammates of one of the players achieving a consecutive game batting streak of length
We partition the sample into two groups: a treatment group
A sufficiently randomized sample is achieved by collecting data from both Major Leagues, over many seasons, thereby mitigating undue influence of potential sample biasing factors including: (a) the raised mound and expanded strike zone (ca. 1963–1968) which tended to favor pitchers; (b) the “steroid era” (approx. 1988–2010) which favored batters using performanceenhancing substances; and (c) subjective differences in strike zones between the two leagues favoring pitchers (National League: low zone) and batters (American League: high zone), respectively
Finally, our sample streaks postdate the Lively Ball Era (1920–1945) where batters were at a distinct advantage.
The box score data analyzed in the present study were obtained from the online resource BaseballReference.com (
ID  Player  Team  Year  Finish 


1  T. Holmes  BOS Braves  1945  6/8  37  4 
2  D. DiMaggio  BOS Red Sox  1949  2/8  34  7 
3  S. Musial  STL Cardinals  1950  5/8  30  6 
4  V. Pinson  CIN Reds  1965(6) 
4/10  31  6 
5  W. Davis  LA Dodgers  1969  4/6  31  6 
6  R. Carty  ATL Braves  1970  5/6  31  5 
7  R. LeFlore  DET Tigers  1975(6) 
6/6  31  5 
8  P. Rose  CIN Reds  1978  2/6  44  5 
9  G. Brett  KC Royals  1980  1/7  30  9 
10  K. Landreaux  MIN Twins  1980  3/7  31  7 
11  B. Santiago  SD Padres  1987  6/6  34  6 
12  P. Molitor  MIL Brewers  1987  3/7  39  6 
13  J. Walton  CHI Cubs  1989  1/6  30  5 
14  H. Morris  CIN Reds  1996(7) 
3/5  32  4 
15  N. Garciaparra  BOS Red Sox  1997  4/5  30  7 
16  S. Alomar, Jr.  CLE Indians  1997  1/5  30  7 
17  E. Davis  CIN Reds  1998  4/6  30  6 
18  L. Gonzalez  ARI Diamondbacks  1999  1/5  30  5 
19  V. Guerrero  MON Expos  1999  4/5  31  4 
20  L. Castillo  FLA Marlins  2002  4/5  35  5 
21  A. Pujols  STL Cardinals  2003  3/6  30  4 
22  J. Rollins  PHI Phillies  2005(6) 
2/5  38  6 
23  C. Utley  PHI Phillies  2006  2/5  35  4 
24  W. Taveras  HOU Astros  2006  2/6  30  7 
25  M. Alou  NY Mets  2007  2/5  30  5 
26  R. Zimmerman  WSH Nationals  2009  5/5  30  3 
27  A. Ethier  LA Dodgers  2011  3/5  30  4 
28  D. Uggla  ATL Braves  2011  2/5  33  4 
In streaks spanning two seasons, only the season with the majority of games comprising the streak is considered.
Raw box scores were downloaded manually in commaseparated value (CSV) format. These files were annotated according to the dates of activity of the associated hitting streak; this annotation formed the basis for partitioning the batters into the two sample groups. The aggregate sample sizes for each group were identical (
This database of box score data was subsequently analyzed using the statistical methods that are described in the section on
How shall we define “hotness”? Our model situation for studying the contagion of hot hitting is the consecutive game hitting streak. By definition, it is the
By extension, in the putative measurement of hot hitting contagion throughout the dugout, it is possible that the batting average statistic alone may not be a sufficiently sensitive indicator.
Long runs of consecutive games with at least one hit are not realized by the streaking batters’ teammates; otherwise they would constitute noteworthy streaks in and of themselves. However, short bursts of “microstreaks” coincident with the hot batter’s streak are observed and can be quantified.
To assess offensive production by the core lineup players constituting each sample group, we propose a statistic that expresses both microstreak length (run length of consecutive games with
As an illustration, consider sequences of hits per game as produced by two different hitters. Suppose that for a notional 13 game interval, each player records 4 atbats per game. The hit totals for each player are, respectively,
The first hitter’s 13game hitting streak yields statistics
The statistics
Comparative distributions of raw numerical values for these statistics for the two groups are presented in
Original sample data representing the population from which resampled statistics are drawn, and ultimately used to construct bootstrap distributions and confidence intervals.
Original sample data representing the population from which resampled statistics are drawn, and ultimately used to construct bootstrap distributions and confidence intervals.
In the next section, we describe hypothesis tests applied to the distributional differences between groups based on this sampling of the population.
Let us define hitting statistics for the differences in group wise means of the distributions of batting average and batting heat index:
The null hypothesis
The null hypothesis was tested using bootstrap resampling to calculate nonparametric confidence intervals (CIs) around the statistics for the differences in group wise means,
Efron introduced bootstrap methods
In principle the distribution of nearly any realvalued statistic may be examined using the bootstrap procedure. The statistics
Locations of the values of the statistics observed from the original sample were compared to these CIs in order to infer the presence or absence of a significant (
Our twosample bootstrap procedure is described by the following steps
Draw distinct resamples of size
Compute statistics (
Repeat
For each statistic, construct sampling distributions and estimate
The main results of the experiments are summarized in
Blue lines denote
Blue lines denote
The bootstrap distribution for batting average difference between groups
For the batting “heat index” metric, bootstrap distribution results for differences between the treatment groups (
Our results show that for the batting average, the null hypothesis of “no difference between groups” is rejected at the
The null hypothesis is also rejected for the heat index statistic
We reject the null hypothesis. However, this does not prove the truth of the alternative hypothesis; that is, we cannot claim to have demonstrated a direct causal relationship between a hot hitter’s streak and improved hitting performance of his team.
The observed results may be generated by any number of latent factors. Some of these are discussed below.
A player’s position in the batting order relative to that of the streak hitter might correlate with the quality of pitching he experiences, thereby contributing to the observed effect. To study this variable, mean values for offensive statistics posted by players in each sample group were associated with their average relative lineup position for the games under consideration.
The results are summarized in
Offensive statistics for players in each sample group as a function of batting order. Negative values along the abscissa correspond to batting before the streak hitter.
Offensive statistics for players in each sample group as a function of batting order. Negative values along the abscissa correspond to batting before the streak hitter.
Consider
Similar trends are seen in terms of the heat index
We conclude that batting order position is not an explanatory factor in the hitting contagion phenomenon.
One of the criteria for causal infererence that was identified in the Surgeon General’s report
In terms of the present study, a reasonable question is whether the streaking batter’s teammates notice a change in behavior in the early games of a nascent hot streak. We wondered if the contagion effect would manifest as statistical improvement after some period of latency following the official onset of the hitting streak.
The procedure followed to partition the sample groups in this investigation assumed that the treatment group immediately recognizes that their teammate is “hot”. In practice, a hot batting streak in baseball usually eludes diffuse media attention before having progressed for at least
To simulate this situation, we carried out the bootstrap resampling procedure and computed
The results of this analysis are compiled in


Reject 

Reject 





1  0.009  N  0.025  Y 
3  0.005  N  0.018  Y 
5  0.005  N  0.018  Y 
7  0.000  N  0.011  N 
Mean values of bootstrapped group wise mean differences for various streak recognition delays
ID  Player  Year  Δ 
Δ 
1  T. Holmes  1945  0.0898  0.1491 
2  D. DiMaggio  1949  0.0582  0.0516 
3  S. Musial  1950  0.0543  0.1432 
4  V. Pinson  1965  0.0803  0.0972 
5  W. Davis  1969  0.1654  0.3488 
6  R. Carty  1970  0.1116  0.2635 
7  R. LeFlore  1976  0.0975  0.1860 
8  P. Rose  1978  0.1168  0.1930 
9  G. Brett  1980  0.1063  0.3332 
10  K. Landreaux  1980  0.1497  0.2982 
11  B. Santiago  1987  0.0560  0.1456 
12  P. Molitor  1987  0.0957  0.3035 
13  J. Walton  1989  0.0639  0.2546 
14  H. Morris  1996  0.0729  0.1558 
15  N. Garciaparra  1997  0.0975  0.1470 
16  S. Alomar, Jr.  1997  0.1329  0.2392 
17  E. Davis  1998  0.1019  0.2651 
18  L. Gonzalez  1999  0.0848  0.0684 
19  V. Guerrero  1999  0.0877  0.1541 
20  L. Castillo  2002  0.1303  0.1642 
21  A. Pujols  2003  0.0389  0.0221 
22  J. Rollins  2005  0.1172  0.2016 
23  C. Utley  2006  0.1260  0.2107 
24  W. Taveras  2006  0.0977  0.2302 
25  M. Alou  2007  0.1015  0.3384 
26  R. Zimmerman  2009  0.1145  0.1831 
27  A. Ethier  2011  0.1223  0.2675 
28  D. Uggla  2011  0.1838  0.2547 
Values are expressed as differences in/out of their streaks (see
For the
A different result is seen for the heat index
The quality of opposing pitching facing the sample groups is another variable potentially influencing the results of this study. Although unlikely over the course of a
Analysis of the isolated importance of “pitching quality” would be complicated. One difficulty lies in the task of separating pitching from hitting–the two factors are clearly not independent of one another. As baseball philosopher Casey Stengel once remarked, “
Including pitching quality as an essential variable was deemed beyond the scope of the present investigation. Future research into the role of pitching to suppress hot hitting may be informed by the following notes.
A detailed perspective could be obtained by expanding the present analysis to consider the individual batter versus pitcher matchups for each atbat. This type of information is available from at least two readily accessible resources (
Owing to the fact that a large number of different pitchers are seen by a team over course of a season, an important design parameter would be to establish rational criteria for a requisite number of pitchers’ innings. Managers can always bring in fresh arms from the bullpen when necessary.
A more general view might be accomplished by the formulation of a composite Earned Run Average (ERA) realized against both hitting groups. However, the ERA statistic includes many means to reach base and ultimately score (walks, hit batters, sacrifices) not accounted for using the present hot hitting statistics; this would still be problematic for drawing inference from pitching as a factor. One of many other quality statistics that might be considered is the socalled “Pitcher Dominance Factor” proposed by former MLB pitcher Curt Schilling
The conditional dependence of hitting and pitching could be partially mitigated through the implementation of a Nave Bayes computational approach, for example, as discussed in Duda and Hart
It is conceivable that the observed hot hitting results might be due to a greater concentration of skilled players on certain teams relative to the competition. This relates to the dilemma of discerning homophily from influence
As an indicator of overall team skill, we considered the final standings for each team in the present study. These standings are listed in the
The mean finishing position for the streak teams was
We observed evidence of a statistical contagion effect. The preceding discussion considered a number of possible latent external covariates that might account for our observed results. If hitting contagion does have a concrete basis, it is likely be motivated internally; some neurobiological or psychological mechanisms then would translate the identification and observation of hot hitting by the streak hitter into an improvement in hitting performance by the observer.
We briefly point to four distinct studies from the scientific literature that attempt to explain mechanisms of the transduction of observation into performance by the observer.
In a study particulary germane to our investigation, Gray and Beilock
A fascination with statistics is one of the hallmarks of fans of American baseball. Several interesting extensions to the present work can be envisioned.
Other statistics indicative of hot hitting might be used to augment those used here (
Supplemental studies might investigate the time course of a hot hitting “epidemic” as a streak extends in duration, perhaps carrying the metaphor forward by employing analytical methods from epidemiology to the extent that additional insight may be achieved into the mechanisms of transduction.
Finally, it is of interest to note how the streaking batters themselves performed using our statistical indicators during their long hitting streaks. In
The hitting streak box scores data are provided in
Box score data used in this research. The archive file contains one folder for each of the
(ZIP)
The authors would like to thank the editor and anonymous reviewers for their insightful comments on the original manuscript.