Mathematical Statistics and Data Analysis (Rice), Chapter 13, Q24: the answer to
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Mathematical Statistics and Data Analysis (Rice), Chapter 13, Q24: the answer to parts a, b, c, and d would be appreciated!
24. Is it advantageous to wear the color red in a sporting contest? According to Hill and Barton (2005): Although other colours are also present in animal displays, it is specifi- cally the presence and intensity of red coloration that correlates with male dominance and testosterone levels. In humans, anger is associated with a reddening of the skin due to increased blood flow, whereas fear is as- sociated with increased pallor in similarly threatening situations. Hence, increased redness during aggressive interactions may reflect relative dom- inance. Because artificial stimuli can exploit innate responses to natural stimuli, we tested whether wearing red might influence the outcome of physical contests in humans. In the 2004 Olympic Games, contestants in four combat sports (box ing, tae kwon do, Greco-Roman wrestling, and freestyle wrestling) were randomly assigned red or blue outfits (or body protectors). If colour has no effect on the outcome of contests, the number of winners wearing red should be statistically indistinguishable from the number of winners wearing blue. They thus tabulated the colors worn by the winners in these contests:Explanation / Answer
I used some of 2008 beijing olympic stats to elaborate
Bright red coloration is a signal of male competitive ability in animal species across a range of taxa, including non-human primates. Does the effect of red on competition extend to humans? A landmark study in evolutionary psychology established such an effect through analysis of data for four combat sports at the 2004 Athens Olympics [1]. Here we show that the observed pattern reflects instead a structural bias towards wins by red in the outcomes of the competition.
Year Test Sport(s) nred n fred p-value 2004
Bouts BOX 147 267 0.551 0.056
Bouts TKD 43 75 0.573 0.124
Bouts GRW 25 48 0.521 0.443
Bouts FSW 27 51 0.529 0.390
Bouts ALL 242 441 0.549 0.023
Rounds ALL 16 21 0.762 0.013
Weight ALL 19 29 0.655 0.068
classes
Tests of a red effect in Olympic contests.
Results for tests of a red effect in the male divisions of boxing (BOX), taekwondo (TKD), Greco-Roman wrestling (GRW), free-style wrestling (FSW), and aggregated over the four sports (ALL), at the 2004 Athens Olympic. Tests denoted “bouts” compare the number of bouts won by red, nred, to the n total wins. Tests denoted “rounds” compare the number of rounds with a majority of red wins, nred, to the n total rounds. Tests denoted “weight classes”compare the number of weight classes with a majority of red wins, nred, to the n total weight classes. In all cases,
fred = nred/n. Reported are the results of one-sided binomial tests (H0 : fred 0.5; HA : fred > 0.5), with = 0.05.
None of the results are significant under a Bonferroni-adjusted threshold c = 0.003 (Supplementary Information).
Hill & Barton [1] reasoned that the effect may extend to artificial stimuli, for example wearing red during a physical contest. In an ingenious first test of this hypothesis, they exploited a structural feature of tournaments in four Olympic combat sports: boxing, taekwondo, Greco-Roman wrestling, and free-style wrestling. In these sports, contestants compete in pairs as red vs.blue, with distinctively colored clothing and/or equipment. In the 2004 Olympics, colors were assigned to
contestants independent of ability. If red does confer a competitive advantage, as predicted, then contestants wearing red would be more likely to defeat their opponents, and more than half the contests would end in a win by red. Data on outcomes in the men’s divisions for the four sports at the 2004 Athens Olympics upheld this prediction [1], and no effect was found in the two sports with women’s divisions (taekwondo and free-style wrestling) [6]. These patterns were taken to support the hypothesis of a red advantage in human competitive interactions: red enhances performance, possibly acting as a cue of relative dominance when factors such as skill or strength are equally matched. At the proximate level, the effect was posited to operate through psychological or physiological (e.g., hormonal) influences on the red- wearing competitor, on his opponent, or both [6]. We present an alternative explanation, which fully accounts for the observed pattern without recourse to an effect of red on competitive outcomes. In the four sports analysed, the competition for a given weight class is arranged as a single-elimination tournament (Fig. 1b). While details vary across sports (Supplementary Information), generally the winner of a contest, or bout, proceeds to the next round in the competition “tree”. Inboxing and wrestling, the contestant placed at the top of the bout wears red, the one placed at the bottom wears blue; the pattern is reversed in taekwondo. A contestant’s relative position, and thus the color he wears, may change between bouts, as he progresses through rounds in the tournament. When the tournament structure is incomplete and contestants vary in skill level, the null distribution for the fraction of red wins can depart from 0.5, due to a bias towards wins by one color in the outcomes of the competition (Supplementary Information). Two sources of incompleteness are byes and walkovers, both of which result in “missing” bouts. Using a Monte Carlo simulation of competition [7] on the actual 2004 tournament structures, we numerically calculated the distribution of red wins under the null hypothesis (no effect of red), for different degrees of variance in competitor skill (Methods).
Testing the effect of red in Olympic contests. a, Fractions of bouts won by contestants wearing red vs. blue in
the male divisions of boxing (BOX), taekwondo (TKD), Greco-Roman wrestling (GRW), and free-style wrestling (FSW) at the
2004 Athens and 2008 Beijing Olympics, along with fractions when outcomes are aggregated over the four sports (ALL) by year
(2004, 2008) or over the two years (both). The number of bouts in each group is reported above the corresponding bar. The
horizontal line shows fred = 0.5. See Table I for details. b, Schematic representation of the structure of a single-elimination
tournament for n = 7 contestants. Because n is not a power of 2, the outermost round is incomplete. In this case, contestant 1
does not compete in the quarter-final round, i.e., he is byed to the semi-final round. In each contest, or bout, the contestant at
the top wears red, the one at the bottom wears blue. For example, contestants 2 and 3 wear red and blue, respectively, in the
quarter-final round. The winner of this bout proceeds to the semi-final round (in blue), where he faces contestant 1 (in red).
The bout between contestants 4 and 5 is won by walkover (dotted red line, indicating that contestant 4 withdrew or failed to
show up). Contestant 5 proceeds to the semi-final round (in red), where he faces the winner of the 6–7 bout (in blue). c,d,
Quantiles for the distribution of the fraction of red wins fred under the null hypothesis on the actual asymmetric tournament
structures in the 2004 Athens and the 2008 Beijing Olympics (red line) and an equivalently sized symmetric tournament (grey
fill). In both cases, the distributions were evaluated by Monte Carlo at the locations of the red dots. Asymmetries in the
tournament structures shift the null distribution away from a mean of fred = 0.5 as skill variance increases. These asymmetries
induced a bias towards red in 2004 and towards blue in 2008. See text for details.equivalent tournaments with no missing bouts, the null
distribution for the incomplete tournaments shifts in favor of red wins as skill variance increases (Fig. 1c). This implies that a standard hypothesis test will overstate the statistical significance of any observed pattern favoring red (Supplementary Information), and a correctly parameterized test of the red hypothesis cannot be constructed without knowing the true variance in skill. A conservative interpretation, however, is that the pattern reported by Hill & Barton [1] reflects this underlying bias in the null distribution (Table I; Supplementary Information). This interpretation is further supported by equivalent data for the 2008 Olympics (Supplementary Information). We find no evidence of a red effect, and Monte Carlo simulations show that in this case the pattern of incompleteness induces a bias towards wins
by blue in the outcomes of the competition (Fig. 1d).Furthermore, data pooled over both years show no evidence of a red effect (Fig. 1a and Table I). Finally, an estimate of the statistical power indicates that if an effect does indeed exist in these data, it must be small, altering the outcome in no more than 1.3% of bouts relative to
peer-reviewed) is the author/funder.
TABLE I. Tests of a red effect in Olympic contests.
Results for tests of a red effect in the male divisions of boxing
(BOX), taekwondo (TKD), Greco-Roman wrestling (GRW),
free-style wrestling (FSW), and aggregated over the four
sports (ALL), at the 2004 Athens and 2008 Beijing Olympics.
Tests denoted “bouts” compare the number of bouts won
by red, nred, to the n total wins. Tests denoted “rounds”
compare the number of rounds with a majority of red wins,
nred, to the n total rounds. Tests denoted “weight classes”
compare the number of weight classes with a majority of
red wins, nred, to the n total weight classes. In all cases,
fred = nred/n. Reported are the results of one-sided binomial
tests (H0 : fred 0.5; HA : fred > 0.5), with = 0.05.
None of the results are significant under a Bonferroni-adjusted
threshold c = 0.003 (Supplementary Information). natural variation. In fact, this value likely overestimates the true impact, as it is calculated without accounting for the structural biases described above (Supplementary Information). These findings suggest that red does not affect the outcomes of Olympic contests, challenging past claims about the role of color in human competitive interactions based on analysis of this system [1, 6]. Moreover, our analysis illustrates that cofounding effects arising from non-independence and biases in the data-generating process, multiple hypothesis testing, and low statistical power can be subtle. Extreme caution is thus required in interpreting related results derived from other systems [reviewed in 8, 9]. This perspective is corroborated by a large-scale test of the effect, based on outcomes of contests in the online game Halo: Reach [10]. In this simulated combat game, which is played primarily by young men [11, 12], competitors belong to teams wearing red or blue uniforms, with colors randomly assigned. In a sample of 8,800,000 such contests, the fractions of wins by red teams (fred = 0.49953) and of points scored by red teams (fred = 0.50004) are indistinguishable from 0.5, and red can provide no more than a 0.01% effect over natural variation. A large body of work has developed over the past decade, building on the hypothesis of a sexually selected response to red in humans [reviewed in 8, 9] — indeed, the effect of red on human behavior has come to be regarded as one of the best established, and most salient, in the field of color psychology, with important practical applications [13]. Our results refute the foundational finding to this body of work [1], calling into question claims that any effect of red on human competition has an evolutionary basis.Methods Details of the data collection and analysis are in the Supplementary Information. Null distributions for fred were obtained by Monte Carlo simulation of single-elimination tournaments, by weight class, sport, and year. Results were then aggregated for analysis. Each simulated weight class used its observed tournament structure, including asymmetries (byes, walkovers). Competitors were as- signed randomly to initial tournament positions, with skill levels drawn i.i.d. from a symmetric Beta distribu- tion: x Beta(, ). Bout outcomes were evaluated progressively over rounds. When a pair of competitors r and b faced off, r advanced to the next round with probability xr/(xr + xb) [7] (Supplementary Information).