Streakiness in sports.

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 sports-related statistical research.

Streakiness has been studied in connection to many different sports. In a 2006 review, Bar-Eli et al. [1] surveyed a large number of studies providing both support and non-support for the belief that “success breeds success and failure breeds failure” in diverse sports. They concluded that most of the empirical research on hot hand effects supported an earlier conclusion by Gilovich et al. [2], namely that the probability of a successful shot in basketball was independent of the outcomes on previous shots. Hoewever, simulation studies in different sports [1] suggested that rates of success are non-stationary over time, providing evidence in favor of the hot hand.

Subsequent to the Bar-Eli review [1], reports have appeared in the literature that quantify possible hot hand effects in several sports. Raab and co-workers [3] found evidence for streakiness in some volleyball players’ hit-and-miss patterns; scoring a point made a player more likely to score another in future chances. Moreover, when players were “on a roll” this was detected by their teammates, who were found more likely to pass the ball to the streaking player. In cricket, Ribeiro et al. [4] found long-term memory effects by analysis of event-wise scoring in over matches. They concluded that a hot hand phenomena exists in cricket, and that this diffusion-like process may unfold over a very long temporal scale. Recently, Yaari and Eisenmann [5] analyzed a large dataset of sequential success/failure rates on free throw attempts documented in NBA basketball. They reported evidence for a hot hand effect, as the probability of success on a free throw improved by on the second attempt, conditioned on the fact that the first attempt was successful. This was interpreted as strong evidence for a hot hand effect in free throws by NBA players [5].

Baseball hitting streaks were studied by Albright [6], where runs tests and logistic regression models were used to evaluate the existence of batting streakiness. Four years’ Major League batting results were analyzed; the author concluded that batting performance is better explained by a model of randomness, as opposed to objective evidence in support of prevalence in streaky hitting. Albert [7] proposed a consistent-p model assuming that for each at-bat during the Major League season, the probability of that player successfully getting a hit was constant; it was further assumed that outcomes for different at-bats were independent. Using various metrics, players were evaluated and ranked by streakiness with respect to hits/outs, strikeouts, and home runs. The model in [7] was shown to explain most of the intra-seasonal variation in streaky hitting. Quintana et al. [8] developed Bayesian models to investigate sequential hitting success (incorporating hits, walks and sacrifices) spanning four complete seasons. They did not find evidence that streakiness of individual players persisted from season-to-season. The most important covariates with situational hitting success were found to include: (1) the number of outs at time of plate appearance; (2) the number of runners on base; and (3) game location at batter’s home field; and (4) the earned run average (ERA) of the opposing pitcher [8].

Previous studies of statistical contagion.

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 [9] established the U.S. government’s position that scientific evidence suggested a causal relationship between cigarette smoking and lung cancer. This report advanced a number of criteria to identify causal relationships between variables, including: consistency (reproducibility over time and location), strength of association, specificity of association, temporality (cause precedes effect), and coherence (concurrence of collective evidence). These criteria for evaluation of either an association or true causative effect between environmental feature A and a consequent event B were reviewed extensively by Hill [10]. In a recent study of epidemiological literature, Parascandola et al. [11] found that strength and coherence were most often used in practice to establish causal inference; consistency was moderately used, and temporality and specificity were not applied at all in some cases. This suggests that over time, scientific approaches to causal inference in an epidemiological setting have a reduced emphasis on temporality as an essential prerequisite to statistical demonstration of causality.

Contagious feelings in social groups have been widely studied. Hatfield et al. [12] described the process of emotional contagion as one in which “…people nonconsciously and automatically mimic their companions’ fleeting expressions of emotion…people can and do `feel themselves into’ the emotional landscapes inhabited by their partners.” They concluded that from moment-to-moment, people tend to “catch” others’ emotions, and cite literature from a wide spectrum of fields in support of this conclusion.

Barsade [13] conducted experiments on different aspects of mood propagation amongst groups, and concluded that emotional contagion does exist within groups. Positive emotional contagion was correlated with better cooperation, reduced conflict, and enhancements in perceived task performance. In the sporting world, Moll and investigators [14] uncovered association between team celebrations after successful soccer penalty kicks and the ultimate outcome of a penalty shootout. This was attributed to the spread of a positive attitude throughout the team during the sequence of shots. The opposite effect was seen on the opposing team–after a successful kick, the opponents’ next try was more likely to result in a miss if certain behaviors were exhibited by the previous, successful kicker. On the cricket pitch, Totterdell [15] found evidence for what he called “mood linkage” on sports teams, which contributed to correlation between a positive overall team mood and a players’ mood as well as self-appraisal of his performance. Experiments reported by Lee et al. [16] showed that golfers who believed they were using a club previously used by a professional golfer realized improved putting performance. Specifically, subjects perceived that the golf cup itself had increased in physical dimension. The authors [16] assigned this to a positive contagion effect from using the pro golfer’s equipment.

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 et al. [17] reviewed a large number of “mirror neuron”-related studies, which show that simple conceptualization of limb movements produces activity in the same brain areas that are involved in producing the actual movements themselves. The ventral premotor cortex has both cognitive (space perception, action understanding and imitation) and motor functions, the latter of which transform object properties into hand actions, and spatial locations into head and arm actions. Cross and co-workers [18] found experimental evidence for a common neural substrate for both observational and physical learning. The authors concluded that it is possible to achieve new action learning from passive observation.

Gray and Beilock [19] reported on a simulation experiment on the psychological mechanism of “action induction”, whereby observation of the actions of a hot hitter in turn improve the batting performance of the observer. While real game data was not part of this study, action induction was proposed as a sensorimotor explanation for the belief that hitting is contagious.

Homophily or social influence.

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 [20] suggests that in general, it is virtually impossible to distinguish between influence and “latent homophily” in social networks. Under social influence (or contagion), The diffusion of behaviors corresponds to the idea of contagion, as behaviors change in order to be more similar to others in the group. Homophily, on the other hand, suggests the formation of social connections due to pre-existing similar attributes among individuals. As applied to the present study, important analogic incongruities are present. Unlike online social networks, in the baseball setting, the population on each team is small and fixed; associations are controlled by the manager who constructs the lineup. Baseball teams don’t self-organize; baseball ownership draft or trade for players based on economics and skillset requirements at different positions. The effect of location in the batting order is the closest conceptual analog to a linkage change in a dynamic social network; however, this change is controlled by the manager, not the players, precluding the application of statistical tests to discriminate homophily versus influence (for example, see [21] and [22]). Despite these discrepancies in analogy, we recognize that unobserved covariates may be important in shaping the observed results. Our discussion considers a number of possible latent factors, including batting order position, opposing pitching quality, latency in streak recognition, and overall team skill level.

Contribution of this research.

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 entire seasons during which long hitting streaks were accomplished. An hypothesis test was formulated to investigate the treatment effect of the hot hitter on his teammates. Results obtained from the aggregated sample () suggest the demonstration of a “statistical contagion effect”. This work offers a contribution to the literature where investigators found evidence for positive contagion in other sports, such as soccer [14], cricket [15] and golf [16].

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.

Hot hitting statistics.

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 length of the streak itself that is the primary distinguishing factor. The batting average, the ratio of number of hits to at-bats ,(1)is the most widely understood and fundamental measure of hit production by a player. According to MLB rules, just a single base hit per game (with at least one qualifying at-bat) is enough to perpetuate a consecutive game hitting streak. While virtually unobserved historically, it is possible under the rules to post a low batting average during a long () game streak.

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 ) and batting average to express the quality of batting performance. Let us define a batting heat index over the microstreak by the core lineup member as(2)where is the run length in games, and is the player’s batting average for games occurring over this interval. For runs where no hits are produced, the value of , precluding heat accumulation over hitless microstreaks. The core lineup batter realizes many such clusters (total ) of short-term streaks within the course of the hot hitter’s streak, which lasts for games. The overall heat index for this player is compiled and normalized as

(3)The heat index of Eqn. 3 represents both the persistence and density of hit production by the core lineup player over the interval .

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 at-bats per game. The hit totals for each player are, respectively,(4)

The first hitter’s 13-game hitting streak yields statistics , ; the second hitter’s statistics over this interval are , . The batting averages are identical. As measured by , the second player’s ephemeral hotness as compared to the streak hitter can be quantified, although his 2 microstreaks are not extraordinary events.

The statistics and were computed from box score data for the core lineup players comprising each sample group (treatment streak active; control streak inactive).

Comparative distributions of raw numerical values for these statistics for the two groups are presented in Figures 1 and 2. These figures display, side-by-side, distributions of the hot hitting statistics between groups. These are the original sample data representing the population from which resampled statistics are drawn, and ultimately used to construct bootstrap distributions and confidence intervals. They provide a visual description the relative distributions of values of the statistics between groups.

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Figure 1. Distribution of BA statistic for sample groups.

Original sample data representing the population from which resampled statistics are drawn, and ultimately used to construct bootstrap distributions and confidence intervals.

https://doi.org/10.1371/journal.pone.0051367.g001

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Figure 2. Distribution of Q statistic for sample groups.

Original sample data representing the population from which resampled statistics are drawn, and ultimately used to construct bootstrap distributions and confidence intervals.

https://doi.org/10.1371/journal.pone.0051367.g002

In the next section, we describe hypothesis tests applied to the distributional differences between groups based on this sampling of the population.

Hypothesis tests.

Let us define hitting statistics for the differences in group wise means of the distributions of batting average and batting heat index:(5)(6)

The null hypothesis assumed no difference between groups. In this investigation, the interpretation of the null hypothesis was that hitting is not contagious. The alternative hypothesis was that hitting is contagious. Symbolically, these hypotheses are written(7)(8)for the batting average and heat index tests, respectively.

Bootstrap confidence intervals.

The null hypothesis was tested using bootstrap resampling to calculate nonparametric confidence intervals (CIs) around the statistics for the differences in group wise means, (Eqn. 5) and (Eqn. 6).

Efron introduced bootstrap methods [25], which have been shown to be useful for a large variety of statistical estimation problems. Here, bootstrapping is used to esimate the value of a statistic describing a population by repeated resampling of the original sample representing the population, computing the statistic for each replicate, and finally constructing a “bootstrap distribution”–an approximation of the shape, variance and bias of the sampling distribution of the sample statistic.

In principle the distribution of nearly any real-valued statistic may be examined using the bootstrap procedure. The statistics and express the difference in means of distributions for the treatment and control groups. We used the percentile method [26] with bootstrap replicates to estimate empirical distributions of the resampled statistics; the and percentiles of these distributions were taken to constitute the limits of the CIs. Analysis of bootstrap differences in means between the sample groups was carried out using the simpleboot package [27] within the R statistical computing environment [28].

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 ()“treatment effect” of streaky hitters upon their teammates.

Our two-sample bootstrap procedure is described by the following steps [29]:

1. Draw distinct resamples of size , with replacement from treatment and control samples, respectively;
2. Compute statistics (, ) from these resamples;
3. Repeat times;
4. For each statistic, construct sampling distributions and estimate confidence intervals.