BMJ

Confidential: For Review Only

Dissecti ng the curse of the : historical cohort study

Journal: BMJ

Manuscript ID: BMJ.2015.027973

Article Type: Christmas

BMJ Journal: BMJ

Date Submitted by the Author: 07-Jul-2015

Complete List of Authors: Perneger, Thomas; University of Geneva, Division of Clinical Epidemiology

Keywords: regression to the mean, causal inference, myths

https://mc.manuscriptcentral.com/bmj Page 1 of 13 BMJ

1 2 3 Dissecting the curse of the rainbow jersey: historical cohort study 4 5 6 7 Thomas Perneger 8 Confidential: For Review Only 9 Division of clinical epidemiology, Geneva University Hospitals, CH-1211 Geneva, Switzerland 10 11 Thomas Perneger, clinical epidemiologist 12 13 14 Correspondence to Thomas Perneger [email protected] 15 16 Copyright: The Corresponding Author has the right to grant on behalf of all authors and does grant 17 18 on behalf of all authors, a worldwide licence to the Publishers and its licensees in perpetuity, in all 19 20 forms, formats and media (whether known now or created in the future), to i) publish, reproduce, 21 22 distribute, display and store the Contribution, ii) translate the Contribution into other languages, 23 24 create adaptations, reprints, include within collections and create summaries, extracts and/or, 25 26 27 abstracts of the Contribution, iii) create any other derivative work(s) based on the Contribution, iv) to 28 29 exploit all subsidiary rights in the Contribution, v) the inclusion of electronic links from the 30 31 Contribution to third party material where-ever it may be located; and, vi) licence any third party to 32 33 do any or all of the above. 34 35 Competing interests: I have read and understood BMJ policy on declaration of interests and declare 36 37 that I have none 38 39 40 Authorship: TP has conducted the study, written the paper, and approved the version submitted for 41 42 publication 43 44 Transparency: The paper contains an honest, accurate and transparent account of the study 45 46 Ethics approval: none was obtained (not required, publicly available data) 47 48 Funding: none 49 50 Data sharing: the dataset will be available on datadryad.org 51 52 53 54 55 56 57 58 59 60 1

https://mc.manuscriptcentral.com/bmj BMJ Page 2 of 13

1 2 3 Abstract 4 5 Objectives: To dissect the underlying mechanism of the “curse of the rainbow jersey”, i.e., the lack of 6 7 wins that purportedly affects the current cycling World champion 8 Confidential: For Review Only 9 Design: historical cohort study 10 11 Setting: on the road 12 13 14 Participants: professional cyclists who won the World championship road race or the Tour of 15 16 Lombardy, 1965-2013 17 18 Interventions: test of 3 hypotheses – the “spotlight effect” (i.e., people notice when a champion 19 20 loses), the “marked man hypothesis” (i.e., the champion, who must wear a visible jersey, is marked 21 22 closely by competitors), and “regression to the mean” (i.e., a successful season will be generally 23 24 followed by a less sucessful one). 25 26 27 Main outcome measures: Number of professional wins per season in the year when the target race 28 29 was won (year 0), and in the two following years (year 1 and 2); the World champion wears the 30 31 rainbow jersey in year 1. 32 33 Results: On average, World champions registered 5.04 wins in year 0, 3.96 in year 1, and 3.50 in year 34 35 2; meanwhile winners of the Tour of Lombardy registered 5.08, 4.22 and 3.77 wins. Globally, the 36 37 baseline year accrued 31% more wins than the other years (95% CI: 9% to 59%), but the year in the 38 39 40 rainbow jersey did not differ significantly from other cycling seasons. 41 42 Conclusions: The cycling World champion is significantly less successful during the year when he 43 44 wears the rainbow jersey than in the previous year, but this is due to regression to the mean, not to a 45 46 curse. 47 48 49 50 51 52 53 54 55 56 57 58 59 60 2

https://mc.manuscriptcentral.com/bmj Page 3 of 13 BMJ

1 2 3 What this paper adds 4 5 What is already known on the subject 6 7 • People love a good curse 8 Confidential: For Review Only 9 • Cycling World champions have a horrible year wearing the the champion’s stripes (a.k.a. the 10 11 12 curse of the rainbow jersey) 13 14 15 16 What this study adds 17 18 • World champions indeed win significantly less when they wear the rainbow jersey than 19 20 during the previous year 21 22 • … but this is no different from the following year, and similar to the experience of winners of 23 24 25 the Tour of Lombardy 26 27 • Regression toward the mean explains this pattern best 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 3

https://mc.manuscriptcentral.com/bmj BMJ Page 4 of 13

1 2 3 We humans struggle with randomness (1). When we see a pattern, we look for a causal 4 5 agent. When we cannot find one, we make up a curse (2). 6 7 Consider the “curse of the rainbow jersey” (3). In 1965 British cyclist won the 8 Confidential: For Review Only 9 World championship road race, then broke his leg while skiing during the following winter, and lost 10 11 his 1966 season to this and other injuries; famously, Simpson died in 1967 during the . 12 13 14 In the ensuing years several world cycling champions encountered all manner of misery while 15 16 wearing the distinctive white jersey bearing five horizontal color bands: lack of racing success, injury, 17 18 disease, family tragedy, doping investigations, even death (3). It soon became obvious that the 19 20 rainbow jersey was cursed. 21 22 Several explanations can be entertained for the curse. One is that the world champion is as 23 24 likely to encouter sporting and personal difficulties as anyone, but being the champion, people notice 25 26 27 more. This is the “spotlight effect”. Another explanation is that the world champion, very noticeable 28 29 in the hard-to-miss rainbow jersey, is marked more closely by his rivals, which lowers his chances of 30 31 winning. This is the “marked man hypothesis”. Finally, random variations in success rates ensure that 32 33 a very successful season, such as one during which the rider has won a major race, will likely be 34 35 followed by a less successful season. This is the “regression to the mean” phenomenon (4). In this 36 37 study I explored to what extent these hypotheses are supported by racing results of cycling 38 39 40 champions during the past 50 years. 41 42 43 44 Methods 45 46 The study population includes winners of the Union Cycliste Internatinale mens’ World 47 48 championship road race from 1965 to 2013, and for comparison the winners of the Tour of Lombardy 49 50 of the same years. The latter race is of comparable importance – it is one of five “monuments” 51 52 53 among classic one-day races – and takes place at the end of the racing season, just like the World 54 55 championship. 56 57 58 59 60 4

https://mc.manuscriptcentral.com/bmj Page 5 of 13 BMJ

1 2 3 The outcome variable was the number of individual wins in professional races during a given 4 5 year, obtained from a publicly accessible database ( www.procyclingstats.com ). All wins were given 6 7 the same weight. Wins in team time trials were not counted. 8 Confidential: For Review Only 9 Win counts were obtained for 3 calendar years: 10 11 a) Year 0, at the end of which the rider won the championship or the Tour of Lombardy. 12 13 14 b) Year 1, during which the world champion wore the possibly ill-fated jersey. 15 16 c) Year 2, which follows the possibly cursed year, when all riders returned to curse-free status. 17 18 19 20 Study hypotheses 21 22 The hypothesized patterns for the average numbers of wins are as follows (Figure 1): 23 24 a) “Spotlight effect” hypothesis: the problems of the world champion are only apparent, due to 25 26 27 increased meadia attention, so the numbers of wins should remain at the same level for the 28 29 three years. 30 31 b) “Marked man” hypothesis (or, equivalently, the rainbow curse, which is indistinguishable): a 32 33 decrease in wins should affect the current world champion, but this effect should disappear 34 35 in the following year when someone else is the champion, and should not affect the winner 36 37 of the Lombardy race. 38 39 40 c) “Regression to the mean” hypothesis: the baseline year 0 (when the championship was won) 41 42 should be a high outlier, and the number of wins should return to a lower level in years 1 and 43 44 2. The pattern should be identical for the Lombardy winner. 45 46 47 48 Statistical analysis 49 50 The mean numbers of professional victories per rider for years 0, 1 and 2 were tabulated 51 52 53 separately for the winners of the World championship and of the Tour of Lombardy. The Wilcoxon 54 55 paired test was used for year-to-year comparisons, and the T test for between-race comparisons. 56 57 58 59 60 5

https://mc.manuscriptcentral.com/bmj BMJ Page 6 of 13

1 2 3 Mixed Poisson regression was used to evaluate the three hypotheses (generalized linear 4 5 model with logarithm link and Poisson errors)(5). The dependent variable was the number of wins, 6 7 and each rider was afforded an individual tendency to win, represented below by the random 8 Confidential: For Review Only 9 intercept α . Because the data were overdispersed (overdispersion remained even when between- 10 i 11 rider variance was factored out, using variance components analysis), I used a robust estimation of 12 13 14 standard errors, to obtain correct estimates. Four models were built. 15 16 The first model represented the “spotlight effect” hypothesis and added to the random 17 18 intercept only a fixed effect for the race (World chamionship=0 versus Tour of Lombardy=1): 19 20 Model 1: log(wins) = α ⋅ rider + β ⋅ Lombardy 21 i i 1 22 23 The second model represented the “marked man” hypothesis and added a fixed effect for 24 25 the year in the rainbow jersey (rainbow=1 for year 1 of the world champion, and =0 otherwise): 26 27 Model 2: log(wins) = αi ⋅ rideri + β1 ⋅ Lombardy + β2 ⋅ rainbow 28 29 30 The third model represented the “regression to the mean” hypothesis and added a fixed 31 32 effect for the baseline year of both races (baseline=1 for year 0, and =0 for years 1 and 2): 33 34 Model 3: log(wins) = αi ⋅ rideri + β1 ⋅ Lombardy + β3 ⋅baseline 35 36 The fourth model represented both the “marked man” and the “regression to the mean” 37 38 39 hypotheses together: 40

41 Model 4: log(wins) = αi ⋅rideri + β1 ⋅ Lombardy + β2 ⋅rainbow+ β3 ⋅baseline 42 43 In these models the regression coefficients β correspond to expected differences in 44

45 β 46 logarithms of wins, and e expresses the ratio of wins. The a priori hypotheses put no constraint on 47 48 β1, but required a negative β2, and a positive β3. 49 50 The goal of this analysis was to select the model that fit the data best. The measure of fit was 51 52 the likelihood of each model. Because all models were run on the same data the log-likelihood 53 54 statistics were directly comparable. Furthermore, models 1, 2 and 3 are nested within model 4, and 55 56 model 1 is nested within models 2 and 3. Nesting means that the simpler model is a special case of 57 58 59 60 6

https://mc.manuscriptcentral.com/bmj Page 7 of 13 BMJ

1 2 3 the larger model, where the omitted covariates have regression coefficients set to 0. Nesting allows 4 5 direct statistical testing of the improvement of fit: twice the difference in log-likelihoods has a chi- 6 7 square distribution with the number of degrees of freedom (d.f.) equal to the number of added 8 Confidential: For Review Only 9 parameters (6). In addition, to evaluate the amount of variance left unexplained by each model, I 10 11 compared them to a fully saturated model where each year of each race defined a separate level of a 12 13 14 factor (6 race-years, hence 5 d.f.). Models 1-4 are all nested within the saturated model. 15 16 The analyses were run on SPSS version 22 and Stata version 13. 17 18 19 20 Results 21 22 The dataset incuded annual win totals for 289 rider-years: for each race, 49 results in year 0, 23 24 49 in year 1, and 46 (World championship) or 47 (Tour of Lombardy) in year 2. Totals were lower in 25 26 27 year 2 because winners of 2013 only contributed years 0 and 1 (the 2015 season was incomplete at 28 29 the time of analysis), and 3 additional win totals were missing due to rider retirement. Several riders 30 31 won more than one target race (either the World championship or the Tour of Lombardy), and 63 32 33 different riders contributed data: 40 riders had one target win, 14 had 2 wins, 7 had 3 wins, one had 34 35 4 wins (Paolo Bettini, 2 wins in each race) and one had 5 (, 3 World championships and 2 36 37 Tours of Lombardy). Six riders won both races in the same season. 38 39 40 Globally, for the 289 rider-years, the average number of professional wins was 4.27 per 41 42 season (standard deviation 4.81, total variance 23.14, within-rider variance 11.51), and this was 43 44 similar for winners of the World championship (4.18) and of the Tour of Lombardy (4.37, p=0.57). For 45 46 winners of both races, the annual win total was higher in year 0 than in years 1 and 2 (Table 1); the 47 48 difference between year 0 and the following years was statistically significant, but the difference 49 50 between years 1 and 2 was not. 51 52 53 The first regression model confirmed that the average number of annual wins was similar for 54 55 World champions and for Tour of Lombardy winners (Table 2); the log-likelihood statistic of this 56 57 model was -737.56. Model 2 tested whether the year in the rainbow jersey was a special case, and 58 59 60 7

https://mc.manuscriptcentral.com/bmj BMJ Page 8 of 13

1 2 3 while the win ratio was indeed less than one, the reduction was trivial (only a 7% decrease) and 4 5 statistically non-significant. The log-likelihood statistic was almost unchanged. Model 3 tested the 6 7 regression to the mean hypothesis, and showed that the baseline year of both races was indeed 8 Confidential: For Review Only 9 more sucessful than the ensuing years; the number of wins was higher by more than 30%. This 10 11 difference was statistically significant, and the log-likelihood statistic was improved by about 10 12 13 14 points to -726.68. Model 4 confirmed that the rainbow year did not differ significantly from other 15 16 years (this time the win ratio was above 1), but that the baseline year of either race was significantly 17 18 more successful. The fit of this model was not much better than that of model 3, with a log-likelihood 19 20 statistic at -726.27. 21 22 The log-likelihood statistic of the fully flexible model, where each race-year was assigned its 23 24 own regression coefficient, was -725.01. Thus this model did not explain the data significantly better 25 26 27 than either model 3 (chi-square 3.34, 3 d.f., p=0.34) or model 4 (chi-square 2.52, 2 d.f., p=0.28). 28 29 30 31 Comment 32 33 The curse of the rainbow jersey does not exist. True, the current road racing world champion 34 35 wins less on average than he did in the previous season, but this phenomenon is best explained by 36 37 regression to the mean. The relative lack of success was not restricted to the season in the rainbow 38 39 40 jersey, but persisted in the following season, and affected equally the winners of the Tour of 41 42 Lombardy. There was nothig remarkable about the year spent wearing the rainbow jersey. 43 44 Regression toward the mean is unavoidable whenever the variable under study (here, 45 46 sporting success) fluctuates over time, the correlation between observations is less than 1, and the 47 48 baseline observation is defined by an arbitrarily high or low value (here, a season marked by an 49 50 important win). Regression to the mean may explain, for instance, why a patient with a very low 51 52 53 bone density at baseline is likely to have a higher reading at follow-up (6), or why HIV-related risk 54 55 behaviors improve after erollment into a prevention trial (7). This phenomenon occurs regularly in 56 57 clinical medicine, research, or programme evaluation, but in other walks of like as well. For instance, 58 59 60 8

https://mc.manuscriptcentral.com/bmj Page 9 of 13 BMJ

1 2 3 some flight instructors believe that praising a pilot afer a smooth landing is counter-productive but 4 5 reprimanding a pilot after a rough landing leads to improvement (1). Their observation is correct – an 6 7 extreme performance will be followed by a more average one – but the causal inference is not. 8 Confidential: For Review Only 9 Neither is this reaction particularly new. Quite possibly the proverb “Pride goeth before destruction” 10 11 (King James Bible, Proverbs 16:18) should be credited with the first description of regression toward 12 13 14 the mean, and not Francis Galton (8), who merely showed that chance and correlation, not the Lord 15 16 or a large ego, were to blame. 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9

https://mc.manuscriptcentral.com/bmj BMJ Page 10 of 13

1 2 3 References 4 5 1. Tversky A, Kahneman D: Judgment under uncertainty: Heuristics and biases. 6 7 Science 1974, 185:1124-1131 8 Confidential: For Review Only 9 2. Hutson M. The 7 Laws of Magical Thinking: How Irrational Beliefs Keep Us Happy, 10 11 Healthy, and Sane . Hudson Street Press, 2012 12 13 14 3. Healy G. The Curse of the Rainbow Jersey: Cycling’s Most Infamous Superstition. 15 16 Halcottsville NY: Breakaway Books, 2013 17 18 4. Barnett AG, van der Pols JC, Dobson AJ. Regression to the mean: what it is and how to 19 20 deal with it. Int J Epidemiol. 2005;34:215-20 21 22 5. McCullagh P, Nelder J. Generalized Linear Models, 2nd ed. Boca Raton FL: Chapman and 23 24 Hall 1989. 25 26 27 6. Cummings SR, Palermo L, Browner W, Marcus R, Wallace R, Pearson J, Blackwell T, Eckert 28 29 S, Black D. Monitoring osteoporosis therapy with bone densitometry: misleading changes 30 31 and regression to the mean. JAMA 2000, 283:1318-1321 32 33 7. Hughes JP, Haley DF, Frew PM, Golin CE, Adimora AA, Kuo I, Justman J, Soto-Torres L, 34 35 Wang J, Hodder S. Regression to the mean and changes in risk behavior following study 36 37 enrollment in a cohort of U.S. women at risk for HIV. Ann Epidemiol. 2015;25:439-44 38 39 40 8. Galton F. Regression towards mediocrity in hereditary stature. J Anthropol Instit 1886; 41 42 15:246-63 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 10

https://mc.manuscriptcentral.com/bmj Page 11 of 13 BMJ

1 2 3 Table 1. Mean number of professional racing wins for world champions and for Tour of Lombardy 4 5 winners of the preceding year. The index year corresponds to the year during which the “curse of the 6 7 rainbow jersey” applies. 8 Confidential: For Review Only 9 10 N rider-years Mean number of P value P value 11 wins (SD) vs year 0* vs year 1* 12 13 World champions 14 15 Year 0 49 5.04 (4.32) - 0.021 16 17 Year 1 49 3.96 (5.61) 0.021 - 18 19 Year 2 46 3.50 (5.24) 0.011 0.53 20 21 Lombardy winners 22 23 Year 0 49 5.08 (4.04) - 0.030 24 25 26 Year 1 49 4.22 (4.94) 0.030 - 27 28 Year 2 47 3.77 (4.57) 0.004 0.34 29 30 31 * Wicoxon paired test 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 11

https://mc.manuscriptcentral.com/bmj BMJ Page 12 of 13

1 2 3 4 5 Table 2. MixedConfidential: Poisson regression models with robust estimation of standardFor errors Review Only 6 7 8 Hypothesis represented Model 1: Model 2: Model 3: Model 4: 9 Spotlight effect Marked man or rainbow Regression to the mean All 10 curse 11 12 Ratio of wins P value Ratio of wins P value Ratio of wins P value Ratio of wins P value 13 14 Lombardy (vs Worlds) 1.20 (0.93 – 1.54) 0.16 1.17 (0.91 – 1.50) 0.21 1.20 (0.93 – 1.54) 0.15 1.24 (0.99 – 1.54) 0.064 15 16 Rainbow year (vs all others) 0.93 (0.75 – 1.14) 0.46 1.09 (0.83 – 1.43) 0.53 17 18 Baseline year (vs year 1 or 2) 1.31 (1.09 – 1.59) 0.005 1.34 (1.07 – 1.70) 0.013 19 20 Degrees of freedom 1 2 2 3 21 22 Log-likelihood -737.56 -737.16 -726.68 -726.27 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 12 43 44 45 46 https://mc.manuscriptcentral.com/bmj 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 13 of 13 BMJ

1 2 3 Figure 1. Three hypotheses under consideration: expected average number of wins in the year when 4 5 the race took place (year 0), the following year (year 1), and the year after that (year 2), for the 6 7 winner of the World chamionship road race (empty circles) and the winner of the Tour of Lombardy 8 Confidential: For Review Only 9 (full circles). 10 11

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 13 https://mc.manuscriptcentral.com/bmj