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Research Quarterly for Exercise and Sport Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/urqe20

The Hot Hand Belief and Framing Effects a

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Clare MacMahon , Jörn Köppen & Markus Raab a

Swinburne University of Technology

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German Sport University Cologne

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London South Bank University Published online: 20 Aug 2014.

To cite this article: Clare MacMahon, Jörn Köppen & Markus Raab (2014) The Hot Hand Belief and Framing Effects, Research Quarterly for Exercise and Sport, 85:3, 341-350, DOI: 10.1080/02701367.2014.930089 To link to this article: http://dx.doi.org/10.1080/02701367.2014.930089

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Research Quarterly for Exercise and Sport, 85, 341–350, 2014 Copyright q SHAPE America ISSN 0270-1367 print/ISSN 2168-3824 online DOI: 10.1080/02701367.2014.930089

The Hot Hand Belief and Framing Effects Clare MacMahon Swinburne University of Technology

Jo¨rn Ko¨ppen Downloaded by [UOV University of Oviedo] at 01:51 17 October 2014

German Sport University Cologne

Markus Raab German Sport University Cologne London South Bank University

Purpose: Recent evidence of the hot hand in sport—where success breeds success in a positive recency of successful shots, for instance—indicates that this pattern does not actually exist. Yet the belief persists. We used 2 studies to explore the effects of framing on the hot hand belief in sport. We looked at the effect of sport experience and task on the perception of baseball pitch behavior as well as the hot hand belief and free-throw behavior in basketball. Method: Study 1 asked participants to designate outcomes with different alternation rates as the result of baseball pitches or coin tosses. Study 2 examined basketball free-throw behavior and measured predicted success before each shot as well as general belief in the hot hand pattern. Results: The results of Study 1 illustrate that experience and stimulus alternation rates influence the perception of chance in human performance tasks. Study 2 shows that physically performing an act and making judgments are related. Specifically, beliefs were related to overall performance, with more successful shooters showing greater belief in the hot hand and greater predicted success for upcoming shots. Conclusions: Both of these studies highlight that the hot hand belief is influenced by framing, which leads to instability and situational contingencies. We show the specific effects of framing using accumulated experience of the individual with the sport and knowledge of its structure and specific experience with sport actions (basketball shots) prior to judgments. Keywords: contextual effects, decision making, sequential judgment, sport belief

The research objective of this article is to theoretically integrate the hot hand belief and hot hand behavior into one perspective. Two studies share the perspective that decisions in sports can be influenced by experience with the sport and knowledge of how it functions. This experience can be previously accumulated strategic knowledge, or more immediate knowledge tied to experienced actions and provides a frame for the decision, which influences beliefs and behavior. The studies add evidence that this new Submitted March 6, 2013; accepted December 31, 2013. Correspondence should be addressed to Clare MacMahon, School of Biomedical and Health Sciences, Faculty of Health, Arts and Design, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia. E-mail: [email protected] Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/urqe.

perspective, derived from the exposure view of the hot hand belief, allows a test of whether the hot hand belief is a generalized belief or differs based on experience.

LITERATURE REVIEW In the hot hand belief, successful performance is used to predict continued successful performance. For example, in the classic hot hand work by Gilovich, Vallone, and Tversky (1985), 91% of basketball fans agreed that a basketball player has a better chance of making a shot after having just made his (or her) last two or three shots. Research on the hot hand belief in sport confirms its prevalence in outcome prediction and pattern interpretation. A review by Bar-Eli, Avugos, and Raab (2006), a recent meta-analysis (Avugos,

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Ko¨ppen, Czienskowski, Raab, & Bar-Eli, 2012), and a book by Reifman (2012) illustrate the growing volume of work across a variety of sports such as baseball (e.g., Frohlich, 1994), basketball (e.g., Adams, 1992), and hockey (e.g., Morrison & Schmittlein, 1998). What has not yet been addressed within the volume of work on the hot hand pattern and belief is an understanding of when and why the belief in the hot hand persists. Indeed, we argue that understanding the belief and behavior associated with the hot hand belief are more crucial than understanding the pattern itself. We argue that the hot hand belief lies beneath some influential decisions such as to which player to allocate the ball, which player to select for a team, and the choice of defensive strategies. Predicting when this belief will come about and the behaviors to which it is linked provides useful strategic information as well as ideas on correcting it when it proves deleterious. There are two main views of the origin of the hot hand belief: the evolutionary view and the exposure view. Wilke and Barrett (2009) proposed an evolutionary approach, suggesting that cognitive adaptations to our environment have led us to detect streaks. They proposed that the human propensity to detect “clumps” comes from its benefit in foraging. This argument is supported by experiments that showed a greater sensitivity for detecting clumps (or streaks) of “hits” rather than “misses,” with the analogy that detection of areas where there is little fruit (“cold areas”), for example, is less important than detection of areas in which fruit is abundant (“hot areas”). In the sport domain, however, Ko¨ppen and Raab (2012) found an equal ability to detect “hot” and “cold” hands. Specifically, volleyball players allotted fewer balls to a player who exhibited a streak of misses (cold hand). It can be argued that for sport, detecting a cold hand is adaptive and provides a strategic advantage in allocating fewer passes to a less successful player. The cold hand finding in sport thus illustrates that the evolutionary approach to explaining the hot hand may be too generic. The exposure approach presents an alternative to the evolutionary explanation of the hot hand belief and can also account for cold hand findings. For example, Gilovich et al. (1985) labeled the hot hand belief an error attributed to the misperception of short sequences, or small samples (the law of small numbers), which observers believe should be “representative of their generating process” (p. 295). Tversky and Kahneman (1971) suggested that the law of small numbers is a manifestation of the representativeness heuristic, in which people access their own experience to make judgments. This misperception is believed to show base-rate neglect, in which the background frequency of events is disregarded and more local, exposure-based information such as one’s own experience is used. Although Gilovich et al. did not define a “small sample” or elaborate on exposure-based experience, the perception of a hot hand or streak has been shown in response to short sequences of

success, with two or three hits (e.g., Larkey, Smith, & Kadane, 1989). Although the exposure view of the hot hand belief accounts for findings not explained by the evolutionary perspective, the exposure view may still be rather generic and in need of greater depth. For example, it is unclear whether task, experience, and expertise differences are considered. Moreover, it is unclear exactly how “one’s own experience” is used to make judgments, as Tversky and Kahneman (1971) discuss, what these experiences are, and how the nature of this experience influences beliefs. In particular, both individual and task factors can influence whether someone shows a hot hand belief and whether their behavior is affected by this belief. For example, Tyszka, Zielonka, Dacey, and Sawicki (2008) found that strategy choice (i.e., hot hand or gambler’s run) for prediction of uncertain events was related to beliefs about randomness in general (i.e., belief in trend continuation such as hot hand vs. belief in trend reversal such as gambler’s run). Similarly, the authors also found that with the particular task of basketball shooting, individual differences did not have an influence, with all participants using a hot hand pattern for prediction. Task and individual factors were thus both significant. This previous research shows a reasonable expectation that other task and individual effects are influential on the hot hand belief beyond these rather generic effects (basketball, beliefs about randomness). Moreover, other, more specific exposure-based effects have yet to be explored. For example, exposure to a domain through either observation or action (e.g., watching or playing a sport) strengthens the representativeness heuristic, which lies beneath base-rate neglect. Experience may also be accessed more readily and easily with less direct information, such as when using a symbolic representation of outcomes (e.g., the # and & symbols to represent alternating outcomes in Ayton & Fischer, 2004). The potential influence of task and individual difference factors on beliefs is captured in the idea of framing. As described by Soman (2004), framing refers to the mental model that is used to perceive and structure a judgment. This mental model may differ depending on who is making the judgment (e.g., observer or actor), when the judgment will be made, and in what context. In addition, a problem or decision scenario itself may be framed in different ways, which may alter the mental model and final solution. Manipulating individual and task variables and examining outcome behaviors can thus be used to identify how particular frames influence the model and its outcomes. Such manipulations will provide insights into the hot hand belief and its associated behaviors and, in particular, the exposure view of this belief. Despite an abundance of evidence of the effects of framing on judgments, this variable has not been examined for the hot hand belief, much less acknowledged within either the evolutionary or exposure views. Because the hot

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hand belief involves physical performance, it is relevant to consider who is making the judgment and the context of the judgment. Ayton and Fischer (2004) found a pattern in which the gambler’s run was associated with chance tasks, such as a coin toss, and the hot hand was associated with human performance tasks. However, Langer and Roth (1975) showed an illusion of control in which tossing a coin oneself, rather than watching someone else toss a coin, created an expectation of success, as well as a more positive evaluation of past performance. The hot hand belief can thus be explored for both others’ and one’s own performance, as well as for generic tasks and those with which one has knowledge and experience. More specifically, previous work on the hot hand has either used controlled laboratory experiments to test whether this pattern exists or has tested observers’ beliefs and predictions (e.g., Gilovich et al., 1985; Raab, Gula, & Gigerenzer, 2012). Performance and belief in streaks of one’s own successful behavior (i.e., hotness) have very rarely been tested together (cf. Avugos, Bar-Eli, Ritov, & Sher, 2013). Research in other domains provides evidence that perceptions of others’ actions differ from perceptions of one’s own actions (e.g., Flach, Knoblich, & Prinz, 2003; Knoblich & Flach, 2001), but to our knowledge, this has not been explored in the hot hand belief. Examining the hot hand belief after one’s own performance is tied to the idea that this belief may be an exposure-based functional adaptation, given that a perception of greater control and success (self-efficacy) is related to superior performance (Bandura, 1997; Moritz, Feltz, Fahrbach, & Mack, 2000). Moreover, in terms of individual framing factors of the hot hand belief, only two studies have assessed experience or expertise in a sport (Gula & Ko¨ppen, 2009; Ko¨ppen & Raab, 2012). For example, Gula and Ko¨ppen (2009) provided participants with information on past performance of volleyball players and then assessed allocation decisions (to whom to pass the ball). They showed that both expert and novice players were influenced by streaks, but the experts were influenced less so than the novices. Experts were less inclined than novices to allocate the ball to the player who exhibited a long streak of successful performance. Ko¨ppen and Raab (2012) followed up on this work and showed that although both expert and novice players allocated more balls to a player exhibiting a “hot hand,” the influence was stronger on novices. This work also delved deeper into experience and compared general sport experience to sport-specific experience by again using a volleyball task and compared allocation decisions of athletes in team versus individual sports for both hot and cold hand scenarios. In both of these tests, participants allocated more balls to players showing a hot hand and fewer balls to those displaying a cold hand, with no difference according to their experience based on general sport structure (team vs. individual). Based on the reviewed literature, it is apparent that a structured, systematic examination of framing, as related to

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an exposure-based view of the hot hand belief in sport, would provide a contribution to this area and advance our understanding. Knowing what factors are related to the behaviors and outcomes of the hot hand belief will help us understand its function and whether it facilitates or hinders performance and decision making. This understanding will also facilitate efforts to change this belief where it is detrimental.

RATIONALE FOR THE STUDIES We conducted two studies of the hot hand belief in sport to systematically investigate broader and more domainspecific factors related to exposure and framing of judgments. Study 1 considered individual factors in the hot hand belief in general and compared participants with and without sport experience. Study 2 examined the impact of actual basketball shooting performance on the hot hand belief as well as how the hot hand belief influences prediction of one’s own performance. We used Raab et al.’s (2012) relative rankings of sports most likely to exhibit a hot hand. Although not addressed by Raab et al., we suspect that the perceived likelihood of hot hand patterns in sports is related to the timing and sequential nature of scoring events. For example, basketball games are high-scoring relative to soccer, with sequences of scoringrelated events (made and missed baskets) occurring frequently. Baseball was chosen for Study 1 as a sport with a moderate ranking in likelihood to exhibit the hot hand (ranked 5 out of 10). The “score-related” event of an at-bat, in which a batter faces a pitcher in a series of pitches with outcome sequences (e.g., two ball pitches, three strike pitches), presents a component of the game in which the hot hand belief and/or pattern may be elicited. Baseball is also a sport for which there is relatively low general population knowledge but also a pool of individuals with greater knowledge. This thus allowed us to manipulate knowledge and exposure/experience with the sport. Moreover, the use of baseball allows examination of a sport in which strategy may influence performance outcome patterns—a point that will be elaborated. For Study 2, we examined basketball as the sport ranked most likely to exhibit the hot hand. We also examined basketball because it allows field-based manipulation of individual success rates by altering shooting distance.

STUDY 1: ALTERNATION RATES AND TASK ASSOCIATION This study replicated Ayton and Fischer’s (2004) Experiment 2, in which sequences of outcomes of human performance-based (e.g., basketball shots) and chancebased (e.g., roulette wheel) tasks were presented to participants. Both of the tasks chosen have dichotomous

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TABLE 1 Examples of Sequences at Different Alternation Rates for Study 1 Series &&#####&&&& &&&&####&&# #&&##&&&&## &#&&&&#&### &##&##&&#&& &##&#&&#&&# &##&#&&#&#&

Alternation Rate .2 .3 .4 .5 .6 .7 .8

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Note. Alternation rates were calculated as the number of switches divided by n – 1, where n is the number of symbols in the sequence. Thus, for the first sequence in this table, the sequence switches once from & to #, then once again from # to &. The alternation rate is thus calculated as 2/(11 – 1) ¼ .2.

outcomes (e.g., hit – miss, red – black), and thus, the outcomes can alternate between these possibilities. For instance, in a sequence of 20 trials, a basketball player may show a 50% shooting accuracy, with 10 hits and 10 misses. These can alternate frequently (e.g., hit –miss –hit – miss – hit –miss) or can show more streak-like outcomes (e.g., hit – hit –hit – miss – miss – miss). The rate at which the outcomes switch between the two possibilities is the alternation rate. A high alternation rate (e.g., .9) indicates frequent switching and few streaks of one outcome, and a low alternation rate (e.g., .1) indicates infrequent switching and streak performance. The sequences of outcomes were presented in symbolic, cumulative form. That is, they were simply represented as a string of two neutral (#, &) alternating symbols on a paper. The procedures of the original study were used here, so that each sequence was composed of 6 “&” symbols and 5 “#” symbols, for a total length of 11 symbols. Sequences were created using a spreadsheet in which a column containing the 11 symbols was sorted by a column with a random number. The alternation rate of the sequences was calculated. This process was used until there were four different sequences for each alternation rate between .2 and .8 (see Table 1 for examples and information on the calculation of the alternation rate). In this replication of the original study, however, we explicitly controlled for sport experience. We considered that experience with a specific sport contributes to the perception of patterns in that sport (e.g., Williams, 2000) and that an alternation rate as high as .5 for basketball shots is labeled a streak performance (Gilovich et al., 1985). The symbolic representation of outcomes (#, &) and cumulative summary information of data (sequences presented as a whole, rather than in a time lag, as a contrasting example) meant that participants would be more likely to rely on past exposure and experience. Although it is possible that participants in the original study had either spectating or playing experience, or both, this was not controlled or measured. Moreover, we were also interested in any differences in these specific types of experience by role (player, umpire), which can loosely be

considered actors (players) and observers (umpires). We hypothesized that Ayton and Fischer’s (2004) patterns of associating chance tasks with the gambler’s fallacy and human performance tasks with the hot hand pattern would be replicated for a general population sample but would not hold for participants with experience in the domain, given their knowledge of and experience with the complexity of sport tasks. This hypothesis was based on Gula and Ko¨ppen’s (2009) finding that experts were less inclined to show a belief in the hot hand compared with novices. Moreover, one can argue that the intention in two of the sports that Ayton and Fischer examined—soccer goals and basketball shots—is generally always for success. The third sport and task they examined, however—the tennis serve—is open to strategic outcome choices. That is, a tennis player may be willing to miss a first serve in an attempt to place it close to the line in order to gain a strategic advantage through manipulating her opponent’s expectations. The player will not purposefully miss the serve, but may make more risky choices, thereby increasing the chance of a missed serve. Knowledge of the sport can thus be argued to influence expectations of outcome patterns. This is also the case in baseball, where, similar to tennis, there is the opportunity to “sacrifice a pitch” by intentionally throwing a ball or a pitch very close to the edge of the strike zone (rather than a definite strike), with a higher probability of missing the strike zone (pitching a ball) to manipulate an opponent’s expectations. Comparing experienced and inexperienced participants tests this supposition. Method Participants Baseball players and umpires were recruited for participation in this experiment as a group with experience in the sport. The difference between these two groups also allowed us to test any domain-specific, role-based differences by loosely comparing roles associated more so with observing (umpires) or with acting (players). Eighteen players were recruited from a baseball program at a state sport institute, and 32 umpires were recruited at a state baseball preseason training camp, for a total of 50 participants with sport experience. The players had an average of 8.9 years playing baseball and no umpiring experience. Umpires had an average of 12.5 years of umpiring experience. In addition, 59 control participants with no baseball playing or umpiring experience were recruited from an Australian university (total N ¼ 109). All participants signed an informed consent form detailing the study protocol and potential risks arising from participation in this study. The informed consent form and the study protocol were approved by the institutional ethics committee. Materials and Procedure All participants were provided with information sheets and gave informed consent prior to participation. Participants

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were then presented with a sheet of paper on which there were 28 sequences of 2 alternating symbols (#, &). Each sequence can be considered a trial, which was at one of seven alternation rates between .2 and .8, inclusive (see Table 1 for an illustration with an example of a trial at each alternation rate). There were four trials per alternation rate. For each 11-symbol sequence trial, participants were asked to indicate whether they felt the sequence represented the outcome of a series of baseball pitches (balls, strikes) or a series of coin tosses (heads, tails). The probability of indicating baseball pitch was calculated for each alternation rate using the four trials per alternation rate. To control for the effect of trial order, two formats or versions of the test were created, each with a random ordering of trial sequences. Participants were assigned randomly to one of the two formats. Results and Discussion There were no statistically significant differences in how umpires and players performed in this study. Their data were thus collapsed into one group with experience in the sport, and all subsequent analyses compared them to the inexperienced control group. To address the goal of understanding the influence of experience on perceptions of sequences of dichotomous outcomes in human performance and chance

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tasks, a 2 (group: experience, control) £ 7 (alternation rate: .2, .3, .4, .5, .6, .7, .8) analysis of variance (ANOVA) with repeated measures was used. This tested the degree of likelihood for labeling an alternation rate as the representation of a series of baseball pitches. The model revealed a statistically significant main effect of alternation rate, F(6, 102) ¼ 7.44, p , .01, with a large effect size of h2 ¼ .30. This main effect was qualified by an interaction between alternation rate and group, F(6, 102) ¼ 3.05, p , .01, h2 ¼ .15, indicating a large effect. This specifies that the association of sequences of outcomes with the likely generating task (i.e., coin toss vs. baseball pitches) differs depending on the experience of the participant making the judgment. Specifically, post-hoc testing using Tukey’s Honestly Significant Difference tests showed a statistically significant difference at the .5 alternation rate ( p , .01). For this alternation rate, baseball-experienced participants were more likely than the baseball-inexperienced participants to identify sequences as the results of baseball pitches. This interaction and significant difference is illustrated in Figure 1. To compare our results more directly with those of Ayton and Fischer (2004), who used only one group and did not control for experience with different sports, we also considered the two groups in two separate ANOVA models. This secondary analysis increased the number of comparisons between performance, and thus, we used a Bonferroni ∗

FIGURE 1 Interaction of group and alternation rate in Study 1, with standard error of values indicated. *Denotes a statistically significant difference between groups ( p , .01).

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correction to guard against Type I error. In these two ANOVAs, the criterion value of p was adjusted to .025 to test for significance (.05/2). If our hypothesis that experience influences attributions in this task is supported, the control group will show the same pattern of performance as in Ayton and Fischer’s study—a pattern of declining probability to attribute sequences to baseball as the alternation rate increases (a linear trend). Our hypothesis predicts a nonlinear trend for the baseball-experienced group but does not specify the pattern. When the control group was examined in isolation, there was a statistically significant linear trend across alternation rate, F(1, 58) ¼ 9.91, p , .01, and a large effect, h 2 ¼ .14. MSE ¼ .36. This finding replicates that of Ayton and Fischer, with controls more likely to attribute low alternation rates or “streaks” to baseball pitches and high alternation rates to coin tosses. This is again illustrated in Figure 1. When the domain-experienced group was examined in isolation, there was a statistically significant quadratic trend across alternation rate, F(1, 49) ¼ 20.29, p , .01, h2 ¼ .29, MSE ¼ .07. This finding shows that experienced participants did not fall into the hot hand belief association in the same manner as inexperienced participants, and it reflects the results of the ANOVA. More specifically, domain-experienced participants were most likely to identify sequences as the outcome of a series of baseball pitches at a moderate alternation rate (.5). This quadratic pattern was also borne out in the likelihood of associating the .7 alternation rate with baseball pitches. Although it was not statistically significant, this likelihood was higher than those for the .6 and .8 alternation rates, again showing a different overall pattern than the linear pattern in the control group. In a broad-based examination of the associations between outcome sequence patterns and type of tasks (chance, human performance), Ayton and Fischer (2004) showed the association of a hot hand for sport and a gambler’s run for chance tasks. What they did not account for, however, is the influence of experience and an associated knowledge base related to the processes generating outcomes. Study 1 shows a very strong effect of domain experience (but not role-specific experience) on the associations of outcome patterns with tasks. Even when using a simplified representation, the outcome of a series of baseball pitches was not strictly associated with the hot hand belief for those with experience in baseball. This result was driven by the specific finding that those participants with experience associated an alternation pattern of .5 with baseball pitches much more often compared with those participants without experience. There was also a quadratic trend in this group, wherein the .7 alternation rate had an increased probability of association with the baseball task than did the .6 or .8 rates. The fact that this pattern is different can be interpreted to show that experienced baseball participants are aware of the subtleties of the sport

such as the possibility that a pitcher may intentionally throw a ball pitch for strategic purposes. Although it is unclear what drives the specifics of this pattern (i.e., the quadratic trend), these results show that it is not strictly the case that streaks are associated with human performance tasks and alternating outcomes are associated with chance tasks, as Ayton and Fischer contended, but that experience influences the associations and expectations. This fits with a more sophisticated exposure view of the hot hand belief; exposure alters expectations and patterns. Study 1 thus shows that using symbolic representations of outcomes for a sport and a chance task elicits different perceptions for those with and without experience in the sport task, although the specific role currently occupied (umpire, player) does not seem to have an influence, at least using this method of investigation, as indicated by the lack of statistically significant differences between these groups. These results indicate that the hot hand belief appears to be moderated by general experience and knowledge of outcome patterns and strategy. For this study, however, outcomes were symbolically represented, and for the experienced participants, predictions were based on the perception of more generic outcomes accumulated through long-term experience of the sport. Study 2 was designed to address the relationship between the hot hand belief and perception of one’s own performance at different levels of expertise.

STUDY 2: THE INFLUENCE OF EXPERIENCING OUTCOMES According to Hales (1999), not only can spectators detect when a player is performing above average, but players themselves can also detect their own performance elevations and streaks. The purpose of Study 2 was to examine: (a) hot hand patterns of performance in a controlled shooting experiment with explicit, methodical manipulation of distance relative to ability; and (b) predictions and beliefs based on one’s own behavior as a third consideration in framing. We were additionally able to examine the influence of expertise based on the level of play of our participants (and the levels of performance during testing). Given previous research on the hot hand pattern, the illusion of control in outcome predictions (Langer & Roth, 1975), and previous findings around expertise and the hot hand belief, we hypothesized that participants would not show a hot hand pattern of performance, but rather they would predict greater success, show a high belief in the hot hand in general, and also display a positive relationship between performance and hot hand ratings. Based on Gula and Ko¨ppen’s (2009) results, we hypothesized that players with greater expertise would show a lesser belief in the hot hand.

FRAMING EFFECTS AND THE HOT HAND BELIEF

Method

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Participants The participants (n ¼ 30) were all male basketball players aged 13 to 17 years old (M ¼ 15.4, SD ¼ 1.1), with a mean of 4.0 years of experience playing basketball (SD ¼ 2.1). This allowed us to explore skill by dividing the group into those from a lower level of play (n ¼ 13, M ¼ 3.3 years in a district league, SD ¼ 1.9), and those from a higher level of play (n ¼ 17, M ¼ 4.4 years in a regional league, SD ¼ 2.2), referred to as experienced and subelite, respectively. All participants signed an informed consent form detailing the study protocol and potential risks arising from participation in this study. The informed consent form and the study protocol were approved by the institutional ethics committee. Procedure Before the actual experiment, participants took part in eight basketball shooting sessions of 30 min each during 4 weeks to determine the optimal individual distance from the basket for a typical 50 – 50 performance. In each session, participants took 30 shots, for a total of 240 shots during the 4-week period. Two basketball coaches were present and recorded performance. The starting point for shooting was the free-throw line. Based on the outcome of the 30 shots, the distance for the next session was decreased or increased. At each new distance, the participant took another 30 shots. Following this method, the coaches determined 30 individual distances with an average shooting performance between 48% and 52%. A onesample t test against 50% showed no statistically significant differences, indicating that these distances produced 50%like performance for all participants. During the experiment, participants were individually tested in sessions of approximately 60 min. After a written introduction to the experiment and the provision of demographic information, the participant was asked to take 100 shots from his individual distance. Before making each shot, each participant rated his certainty that he would make the next basket on a scale from 0 (not certain) to 100 (very certain). After each shot, participants were required to move one step to the side on an arc, so that consecutive shots were never taken from exactly the same spot but always from the same 50% pretested distance. This controlled for learning and block effects, ensuring a test of the general skill and eliminating the influence of specific factors such as context.1 After the experiment, each participant indicated 1

Due to the large number of practice trials from the specific location of the free-throw line, basketball shots from this location develop a unique memory representation different to those from other locations. The free throw has thus been dubbed an especial skill, with implications for factors controlling its success, such as the stronger representation. See Keetch, Schmidt, Lee, and Young (2010) for a discussion of especial skills.

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his general degree of hot hand belief on a scale from 1 (very weak) to 6 (very strong). Data Analysis Our main data analysis on autocorrelation, conditional probabilities, and a runs test was identical to that of the original Gilovich et al. (1985) study, with additional analyses based on level of expertise. These were used to test the performance for the existence of a hot hand (streaks). The hit rate and participant certainty ratings served as dependent variables. These were used to understand whether perception of performance and actual overall success were related to the pattern of performance (hot hand or streaks). To analyze sequential performance dependencies (the hot hand), we calculated the conditional probabilities of hits and autocorrelations, where a hot hand performance is indicated by a positive autocorrelation of sequential success. To analyze performance stability, we conducted a runs test, counting the number of alternations between hits and misses. If fewer and hence longer runs than expected by chance are found, then this is evidence of the presence of a hot hand. Results and Discussion The average overall hit rate, used to examined overall performance, was 48% (SD ¼ 11). A one-sample t test against 50% revealed that this was not statistically significantly different, t(29) ¼ 1.07, p ¼ .293, d ¼ 0.20. The method was thus effective in producing 50 – 50 performance in the group as a whole, which creates a situation in which the hot hand can be examined. We note, however, that the range of individual performance was 27% to 68%, indicating a potential weakness for determining skill on an individual level. A reanalysis of the data using a median split based on test performance (less successful, more successful) showed the same pattern of results as reported here, however, with the exception of one result, which we will present later. Considering overall performance by expertise level, the experienced group shot 44% (SD ¼ 11), and the subelite group shot 51% (SD ¼ 10). This difference was not statistically significant, t(28) ¼ 1.88, p ¼ .070, d ¼ 0.76. We also used a median split to compare groups who shot from a “near distance” (3.5 m to 4.8 m from the basket) and a “far distance” (4.9 m to 6.0 m). The “near-distance” group performed statistically significantly better than the “fardistance” group, t(28) ¼ 7.70, p , .01, d ¼ 0.57, with 57% (SD ¼ 6) and 39% (SD ¼ 6) hit rates, respectively. To examine whether there was behavioral evidence of a hot hand, we examined the autocorrelations and runs test. The autocorrelations of performance were not statistically significant, therefore confirming the independence of each single shot to the basket and in line with our hypothesis that we would not find a hot hand pattern. In addition, a runs test

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TABLE 2 Means and Standard Deviations for Conditional Probabilities of Hits in Study 2

Subelites Experienced p value* Cohen’s d

After One Hit

After Two Hits

After Three Hits

51.9% (12.4) 41.1% (10.6) ,.05 0.19

52.9% (14.6) 39.9% (11.2) ,.05 0.20

54.0% (15.0) 43.0% (13.1%) .055 0.15

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Note. *Group-based analyses of variance using t(28); p values , .05 indicate a statistically significant group difference.

provided a detailed look at alternations between hits and misses. Overall, and with expertise level considered, the results extended those of Gilovich et al. (1985), stating that the sample had as many runs as expected by chance, rather than a hot hand outcome. In contrast to the results of the runs test, a conditional probabilities analysis of performance showed some moderate support for a hot hand pattern in some cases. Specifically, subelites were statistically significantly more likely to hit a shot after one or two hits compared with the experienced group. In addition, there was a pattern, though not statistically significant, for subelites to show a greater likelihood of a hit after three misses compared with the experienced players. These results are shown in Table 2. When examined by distance, the “near –shooting” group was always statistically significantly more likely to make a successful shot compared with the “far” group, regardless of the situation (i.e., after one, two, or three hits or misses). To examine beliefs around performance as related to predicted success, the certainty ratings were examined. The overall certainty rating, averaged across players (M ¼ 66.4, SD ¼ 23) was statistically significantly greater than 50%, t(29) ¼ 3.373, p ¼ .002, d ¼ 1.23, but did not differ by expertise level or distance. Belief in the hot hand was quite high, with an overall mean of 4.7 (SD ¼ 1.1) on the 6-point scale. This did not differ by level of expertise (experienced ¼ 4.5, SD ¼ 1.0; subelite ¼ 4.8, SD ¼ 1.3), but it did differ statistically by the distance from which the ball was shot, t(28) ¼ 2.91, p , .01, d ¼ 0.22, with the players shooting from a “near” distance averaging a rating of 5.2 (SD ¼ 0.6), compared with the 4.1 (SD ¼ 1.3) average rating for those shooting from a “far” distance, although this is a small effect. Similarly, those players who performed more successfully during the test, based on a median split, showed a statistically significantly higher belief rating, t(28) ¼ 3.4, p , .01, d ¼ 0.25, of 5.3 (SD ¼ 0.4), compared with those who performed less successfully during the test (M ¼ 4.1, SD ¼ 1.3). Again, this effect is on the smaller side. The greater hot hand belief in those who were more successful and those who shot from closer to the basket is reflected in the moderate correlation between belief in the hot hand and hit rate, r ¼ .57, p , .01. Although both

expertise groups showed this correlation, it was stronger in the experienced group, r ¼ .73, p , .01, than in the subelites, r ¼ .49, p , .05. These results show that belief in the hot hand accounted for approximately 32% (r 2 ¼ .57) of the variability in hit rate, but more specifically, approximately 53% (r 2 ¼ .53) in the experienced group and 24% (r 2 ¼ .24) in the subelites, indicating a large effect. Hit rate was also moderately correlated with certainty ratings for the group overall, r ¼ .45, p , .05. This relationship was driven by the experienced players, however, with a moderately high correlation, r ¼ .66, p , .05, and with 44% of the variability accounted for (r 2 ¼ .44), rather than the subelites, wherein the correlation was low (r ¼ .33) and only accounted for a little more than 10% of the variability (r 2 ¼ .11). To examine whether certainty is related to the outcome of the last shot, rather than a general performance measure (hit rate), we used a lag-1 autocorrelation. The negative autocorrelation showed that ratings were not related to the outcome of the previous shot. These results partially fit with our hypotheses. Although participants predicted greater success than they had, this did not differ by expertise and there was generally a strong belief in the hot hand, as predicted. There was also an overall relationship between performance (hit rate) and the hot hand belief. However, whereas Gula and Ko¨ppen (2009) found weaker hot hand beliefs in volleyball experts compared with novices, we found a different pattern. Although in these results, overall belief in the hot hand did not differ based on expertise, the higher correlation between belief and hit rate for the experienced group shows this variable had an influence in the opposite direction to that of Gula and Ko¨ppen. These different patterns may be attributable to different designs and methods and provide insights into the hot hand belief. Namely, Gula and Ko¨ppen used the sport of volleyball, participants did not perform the task themselves, and allocation decisions were made based on observation of players in video. Thus, sport, task, and perspective appear to make a difference in the hot hand belief, and particularly in differences based on expertise level. The combination of results shows that both immediate performance and accumulated experience influence the belief at a given time. This adds a deeper understanding to the current state of literature on the hot hand belief and identifies variables that influence the belief both within and between participants. These findings also suggest that the relationship between performance and belief and more specifically, exposure, will be better understood through continued study of different sports, varying tasks, and different manipulations of exposure and framing.

GENERAL DISCUSSION We used two studies to explore the hot hand belief in sport, revealing a complexity and instability that has not been

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FRAMING EFFECTS AND THE HOT HAND BELIEF

previously explored but that indicates the influence of exposure and perspective. More specifically, these studies showed that exposure is a variable that is influential in how framing changes the outcome belief and decisions. In Study 1, participants with experience in baseball perceived baseball sequences differently, with a .5 alternation rate considered the most probable pattern of alternation between the two outcomes of ball and strike. These findings support the idea that exposure influences pattern predictions and perceptions. The pattern for control participants was a simple linear trend with more streaky outcomes associated with baseball and less streaky outcomes with a coin toss. The complex quadratic trend for baseball participants shows a more complex association, perhaps acknowledging varying strategies. For instance, experts may use knowledge of pitchers’ tendencies to aim primarily for strikes but diversify pitches to some degree. The factors behind the exact pattern associations (e.g., a higher association with baseball for the .7 alternation rate than the .6 or .8 alternation rates) are unclear at this time. Study 2 used an extensive pretest to determine individual 50% shooting distances and to extend the previous shooting study in Gilovich et al. (1985). This is one of the first times that the hot hand belief has been examined alongside actual performance. There was no behavioral evidence of hot hand performance, which is in line with both Gilovich et al.’s findings and the current weight of evidence in the literature (see Avugos et al., 2012, for a recent meta-analysis). However, the perception of a streak as compared with statistical designation of a streak is a nontrivial point when it comes to beliefs and may go some ways in explaining some of our findings, as well as the general persistence of the belief and of success within the literature. Moreover, the finding that subelite players showed a “hot hand behavior pattern” with shorter streaks is notable in that Study 2 also showed that performance was related to beliefs, with a statistically significant moderate positive correlation between hit rate and hot hand belief. This is also reflected in the finding that the “near-distance” and more successful shooters were more likely to have strings of two, three, and four hits and stronger hot hand belief ratings. Taken together, then, these data move beyond a simplified view that the hot hand belief is related to exposure. Indeed, it shows that specific experiences of success and of intentions and strategy can frame and influence beliefs for outcome patterns for oneself as well as those who are observed. This helps explain why some situations are more likely and some situations are less likely to elicit the hot hand belief; exposure can be examined as a factor that describes general accumulated experience over time (e.g., familiarity with a specific sport, as tested in Study 1), as well as more immediate experience within one task (e.g., performance in a shooting task, as tested in Study 2). What we now know about the hot hand belief can be further explored by addressing remaining questions due to limitations in the current studies. For example, future work may compare sports in which strategy may play a part, where

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risk-taking behavior increases the probability of a less successful outcome for strategic reasons related to opponent expectations, such as tennis and baseball, to those sports in which the goal for actions is always success, such as in basketball shots. Combined with this consideration of the use of strategy for success is a second factor: altering the perspective of the participant. For example, whether the observer considers himself or herself the pitcher or the batter in an at-bat situation in baseball may influence predicted outcomes and hot hand beliefs. In addition, individual differences in strategic choices (e.g., preference for riskier pitches and sacrificed ball pitches vs. preference for clearer strike pitches) may also play a part. Study 1 did not measure these factors, and future research may use instructions to more directly manipulate the actor or observer perspective, as well as delve into individual strategic preferences. Collection of the rationale or reason behind the choice of baseball or coin toss for each alternation rate or trial may also limit some of the speculative reasoning from Study 1. Additionally, we noted that the range of individual performance of free throws in Study 2 was 27% to 68%, indicating a potential weakness in determining skill level, which may have limited our ability to confidently draw conclusions on this variable. Additional measures, as well as processes such as the post-hoc analyses of performance that were performed here, may be advisable in future work. A greater range of expertise levels with greater contrast between groups will also facilitate examination of expertise as a variable in the hot hand belief. A further clarification to address potential limitations here is the use of questions that clearly examine the hot hand belief in general, as well as the hot hand belief specifically in relation to one’s own performance. Notwithstanding limitations in this work, the emphasis here on the exposure view of the hot hand belief and framing effects has opened up several lines of future research to continue to explore when and why beliefs exist or may change. Our results show that experience with a task influences the perception of outcomes (Study 1). Within one task, however, the perspective can vary both between and within individuals. This is reflected in our use of baseball umpires and players and basketball players. As alluded to, it may be that whether one is an actor or an observer has an influence on predictions for upcoming actions, as previously discussed in the work of Langer and Roth (1975). This work shows that one’s own actions prompt an illusion of control, which results in an expectation of success and inflated positive evaluation of performance. This influence on the prediction of streaks indicates that the hot hand belief can be further explored for both others’ and one’s own performance and for generic tasks, as well as those tasks with which one has knowledge and experience. The actor – observer distinction may be a way to continue exploring both individual experience frames and problem frames. For instance, a participant may be asked to make performance predictions of individual shots, or allocation decisions

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C. MACMAHON ET AL.

(to whom to allocate the ball, based on perceived probabilities of success), from the perspective of an actor, for his or her actions, or an observer. In this manner, both beliefs and behaviors or strategies are examined. A second line of future work builds upon the overestimations we found in Study 2, which may point to a facilitation effect of the hot hand belief. Specifically, certainty ratings were an overestimation of performance for the group as a whole, but particularly for the experienced group, which had less expertise compared with the subelites. These overestimations occurred even though players were aware they were shooting from a 50% distance. The 16% overestimation of success falls in line with Bandura’s (1997) argument that overestimations that only slightly exceed actual capabilities are functional. Although the relationship between efficacy beliefs and performance is complex (e.g., Moritz et al., 2000), there may be a similarly adaptive function to the hot hand belief itself, given the overall high correlation between the rating of the hot hand belief and hit rate. A shooting experiment that includes pretesting and posttesting the belief may begin to explore its potential functions and provide an interesting link. In this manner, our understanding will move from when, where, and with whom the hot hand belief exists to why it exists. WHAT DOES THIS ARTICLE ADD? This work provides a new perspective on the hot hand belief in sport. More specifically, this work reveals that the hot hand belief is more complex than previously thought. This is the first time, to our knowledge, that variables such as exposure and framing have been revealed to influence the belief. This greater understanding of the belief in turn provides greater understanding of related behaviors that can influence the outcome of sports contests, such as selection and strategic decision making on the part of coaches and players. The identification of these key variables has also opened up a path for future research into the hot hand belief in sport. ACKNOWLEDGMENTS We thank the performance psychology group at the Institute of Psychology at the German Sport University in Cologne, Germany, for comments on an earlier version of this manuscript.

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