Skin tone and red cards — Claude Fable 5.1

Abstract

Question. Do soccer referees give more red cards to dark-skin-toned players than to light-skin-toned players?

Method. Binomial mixed model of red cards per game in 124,621 player–referee pairs (1,585 players with skin-tone ratings). Adjusted for position and league country, with crossed random intercepts for player and referee.

Result. Very dark-skin-toned players have higher odds of a red card per game than very light-skin-toned players: OR = 1.38, 95% CI [1.12, 1.70]. Across ten alternative analysis choices, the OR ranges from 1.29 to 1.40, and every CI excludes 1. The estimate lies near the centre of the 29 teams' estimates (median OR 1.31).

Conclusion. The association is positive and stable across these choices. The data are observational, so they do not show that referee bias causes it.

Files: analysis.R (main model + variants → results_own.csv, fig_robustness.png; about 18 min on 6 cores), teams.R (comparison with the 29 teams → fig_teams.png), robustness_plan.md (variant list and expectations, written before any model ran). Run from a clean start with Rscript analysis.R && Rscript teams.R.

1. Main analysis

Approach. The outcome is the red-card count per player–referee dyad, out of games played together. I fit a binomial GLMM (logit link, glmmTMB). Skin tone is the mean of the two raters’ scores (0 = very light, 1 = very dark). Fixed effects are player position and league country. Crossed random intercepts are included for player and for referee.

Why this model. - Red cards are rare counts, bounded by the number of games. A binomial model with games as trials fits this directly. - Dyads are not independent. The same player appears with many referees, and the same referee with many players. Crossed random intercepts take account of both, so the standard errors are not too small. - Position affects exposure to fouls (defenders get more red cards), and skin tone may differ by position. League country accounts for differences in refereeing norms and in the player pool. - I did not adjust for yellow cards. They are plausibly on the causal path between referee bias and red cards.

Result. For a very dark-skinned player compared with a very light-skinned player (the full 0→1 range of the scale), the odds of a red card per game are higher by OR = 1.38, 95% CI [1.12, 1.70]. This is based on 124,621 dyads and 1,585 players.

2. Decisions the question and data did not settle

Estimand scale. I report the OR for the full 0→1 range of the skin-tone scale (very light vs very dark), not the OR per rating step.
Two raters. I averaged them.
Missing skin tone. I dropped players without a photo or rating (complete cases, 21,407 dyads). I did not impute.
Outcome. Only direct red cards count. Second yellows (yellowReds) are excluded.
Model family. Binomial logit with games as trials, not Poisson, negative binomial or a linear model.
Clustering. Crossed random intercepts for player and referee. There are no random slopes, and there is no referee-country level.
Covariates. Position and league country only. Player size and age, club, and yellow cards were not included.
Missing position was coded as its own category (“Unknown”), so those rows were kept.
No sample filtering by referee or player (no minimum number of games or dyads).
Referee bias measures (IAT/explicit) were not used. They address a different question (moderation).

3. Robustness

The plan was written to robustness_plan.md before any model was run. It lists ten alternatives. My expectation was that none would change the direction of the result. A6 (dichotomising) and A10 (dropping referees with few dyads) could plausibly widen the CI so that it includes 1.

Model	OR	95% CI	Dyads	Players
Main	1.38	1.12–1.70	124,621	1,585
A1 No covariates	1.39	1.13–1.72	124,621	1,585
A2 + height, weight, age	1.35	1.09–1.67	123,868	1,564
A3 Rater 1 only	1.39	1.13–1.70	124,621	1,585
A4 Rater 2 only	1.33	1.08–1.64	124,621	1,585
A5 Raters agree exactly	1.40	1.11–1.75	95,714	1,206
A6 Dark (≥0.75) vs light (≤0.25), middle dropped	1.29	1.08–1.53	107,606	1,359
A7 Any sending-off (red + yellow-red)	1.33	1.12–1.58	124,621	1,585
A8 Poisson GLMM, log(games) offset (rate ratio)	1.37	1.12–1.69	124,621	1,585
A9 Logistic GLM, no random effects, player-clustered SEs	1.30	1.05–1.61	124,621	1,585
A10 Referees with ≥ 22 dyads	1.34	1.08–1.66	112,484	1,584

Range. The ORs range from 1.29 to 1.40, and every CI excludes 1. None of the ten variants changed the conclusion. This includes the two I flagged as risky (A6 and A10).
A6 is on a different scale. It compares the two extreme groups, but the “dark” group includes 0.75 ratings, so its contrast is narrower than 0→1. Its lower OR is expected for that reason.
Limits of this check. All variants share the complete-case sample and the same observational design. Unmeasured confounders would bias them all in the same way. Examples are playing style or aggression that correlates with skin tone within a position, or the player’s nationality.
Figure note. fig_robustness.png was drawn with geom_errorbarh. I then changed that line in analysis.R to the equivalent geom_errorbar(orientation = "y"), because the former is deprecated. I did not re-run the script after that cosmetic edit.

4. Comparison with the 29 teams

Source: teams_estimates.csv, downloaded in this session from https://osf.io/download/fa743/. I used its OR, OR_lo and OR_hi columns. For the teams that reported a different effect-size unit (d, r, IRR), these columns hold converted values. I did not check how the conversions were made.

The teams’ ORs have a median of 1.31 and range from 0.89 to 2.93.
20 of 29 teams’ CIs exclude 1.
My main estimate (1.38) is higher than 17 of the 29 team estimates. It lies within the central cluster of multilevel logistic and Poisson models (Teams 5, 24, 28 and 32 are all at 1.38–1.39).
All my variants lie within the range 1.29–1.40, which is the middle of the team distribution.

I did not open the paper in this session. From memory only: the paper reports that 20 of 29 teams found a significant positive effect and that the estimates ranged from about 0.89 to 2.93. The downloaded file agrees with both figures.

5. Conclusion

In these data, dark-skin-toned players receive more red cards per game than light-skin-toned players, with an odds ratio of about 1.4 (95% CI 1.12–1.70) for the extremes of the scale. The estimate stays between 1.29 and 1.40 across ten defensible alternative choices, and it lies near the centre of the 29 teams’ estimates. The data are observational, so this is a robust association rather than evidence that referee bias causes the difference.