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Week 2 — A club that fouls with impunity, and the five ways I was wrong about it

Published: 2026-08-30 · Snapshot: 2026-W35 · Claim level: L2 (hypothesis, adjusted)

The pre-registered question

A viral post implied a Portuguese club was being favoured by referees. Is there a measurable difference in how often that club is booked, relative to the fouls it commits?

Written before the data was pulled. The analysis below reports what happened, including the parts where the answer changed under its own tests.

I am not linking or naming the post. The claim is the object of study; the person who made it is not. See EDITORIAL.md.

Every number below is produced by scripts/discipline_numbers.py and written to numbers.json. Nothing is quoted here that is not in that file. That constraint exists because of Wrong (5).

The first answer, which looked good and was wrong

Porto's raw booking index — yellow cards received divided by cards expected from their foul count at the league's own rate — comes out at 0.884. Comfortably below 1.0. Benfica sits at 0.880, Braga at 0.931, Sporting at 1.042.

On its face: two clubs booked around 12% less than their fouls imply. Roughly the shape the original claim predicted.

It is also wrong, five times over.

Wrong (1): most of it is being the favourite, and it applies to everyone

Context effect

Booking rate per foul depends heavily on how strong you were expected to be in that specific match. Across the eleven leagues, the median heavy underdog is booked at 1.112 times its foul count; the median heavy favourite at 0.865. The gradient is monotone in 11 of 11 leagues, and no club identity is involved — this is every team, everywhere, purely as a function of the role it is in that afternoon.

Dominant clubs are favourites most weeks. So a large part of Porto's apparent leniency is simply being the better side, a property they share with every strong club in Europe.

The figure shows Portugal, because that is the league the question was about. The eleven-league table is in numbers.json; Portugal is close to the median, at 1.122 and 0.851.

Wrong (2): the expectation model was misspecified

The expectation assumed cards are proportional to fouls: expected = rate × fouls. They are not.

Fitting cards = a + b × fouls per league-season gives an intercept worth 13% to 47% of mean cards, depending on the competition. Observed cards per foul falls monotonically from 0.195 at six fouls to 0.127 at eighteen.

That intercept is real and interpretable: it is the cards that have nothing to do with your foul count — dissent, delaying the restart, entering the field of play, simulation — plus whatever the data feed does or does not record as a foul. In Greece and Italy it is nearly half of all cards.

Forcing the line through the origin therefore inflates every low-foul team and deflates every high-foul team, with no behaviour involved. A test on synthetic teams with identical card processes shows the proportional model inventing a gap of over 0.10 between them.

Dominant clubs foul less than average. The model was flattering exactly the clubs the finding was about.

Affine is enough, though. Adding a quadratic term improves RMSE by 0.44%, so the curve is straight over the range that matters and no further correction is warranted.

Wrong (3): it is an asymmetry, not a shared shift

The cheap test is to compute the same index for the opponents, and it is decisive.

median across 11 leagues consistency
corr(club strength, own booking index) −0.173 negative in 11 of 11
corr(club strength, opponents' booking index) +0.314 positive in 11 of 11

The two have opposite signs. When a strong club plays, it is booked less per foul and the other team is booked more. That is not both teams drifting the same way; it is the two sides of one fixture moving apart.

The consequence is testable, and it holds: if the favourite's index falls by about as much as the underdog's rises, the two should cancel when you aggregate the whole match. They do. Match-level fixture quality versus match booking index: median r = +0.002 across the eleven leagues. Nothing. The gradient exists strictly within a match, not between matches.

The within-club version is the cleanest evidence, because club identity is differenced out. Hold the club fixed and split its own matches by opponent quality:

Opponent Booking index
Weakest third 0.937
Middle third 0.996
Strongest third 1.086

Higher against better opposition in 11 of 11 leagues, and for 76% of 326 clubs with enough matches to test. The same club, playing the same way, is booked differently depending on who is on the other side — because against a stronger opponent it is closer to being the underdog.

So the correct statement is not "dominant clubs are booked more", and it is not "cards per foul rise with the quality of the fixture" either. It is:

Your booking rate per foul depends on your role in the fixture, not on who you are.

A club's own quality determines its role most weeks. Strong clubs are usually favourites, so they are usually booked less per foul. They are not being singled out; they are occupying the position that gets booked less, and any club occupying it gets the same treatment.

Wrong (4): someone got there first

Dawson, Dobson, Goddard and Wilson published a gradient in cards by team strength in 2007 (JRSS-A 170(1)), on 2,660 Premier League matches, with a Nash-equilibrium model of aggression. Their reduced form is a subset of the one used here.

What may be new is conditioning on fouls — asking not "who gets more cards" but "who gets more cards for the same number of fouls". Rediscovering a known result on 22 times the data is worth doing. Presenting it as new is not.

Wrong (5): a number reached the page without a script behind it

The first version of this study published the own-strength correlation as +0.486, positive, and concluded from it that cards per foul rise with fixture quality for both teams. The true value is −0.173, negative in every one of the eleven leagues. The sign was inverted, and the conclusion was built on it.

Two things about how that happened are worth more than the correction itself.

It contradicted this study's own opening paragraph. Porto are a strong club and their index is 0.884, below 1.0. A negative correlation between strength and booking index is required by that sentence. The two numbers could not both be true, they sat four paragraphs apart, and re-reading the prose never caught it. Recomputing did, immediately.

The figures were right; the prose was not. scripts/build_discipline_story.py loads Portugal only, and says so in a comment. The correlations quoted alongside came from exploratory work that was never committed as code. So the study reported Portugal-only context numbers as though they covered all eleven leagues, and quoted a correlation no script in the repository could produce. METHODS.md already required that every published numeral trace to a run artifact. Nothing enforced it.

Hence scripts/discipline_numbers.py, which now owns every figure in this report, and numbers.json, which is committed beside it. The generalisable version: a number without an owner is a number nobody checked. Provenance is not a filing convention, it is the mechanism by which a sign error becomes findable.

So: was the club being favoured?

On this measure, no.

Portugal has no referee data in the standard source, so we assembled it — 2,737 match-official rows across ten seasons, 98.4% joined to our matches and validated on scorelines rather than names. Portuguese referees vary as much as English ones (booking multipliers 0.76 to 1.32 among those with 40+ matches), but that variation is not concentrated on anyone: every club's mean referee draw falls between 0.979 and 1.019.

Adjusting for era, match situation and the actual official on the pitch:

Club Booking index 95% interval
Sporting 1.183 [1.100, 1.271]
Benfica 1.001 [0.923, 1.084]
Porto 0.989 [0.916, 1.067]

One club of eighteen separates from expectation, and it is not the one in the original claim. Porto sit at 0.989 — indistinguishable from average.

Note what the correction in Wrong (3) does and does not change here. The raw gradient runs in the direction the original claim expected: favourites are booked less per foul. But it is a property of the role, not the club, and once the role is adjusted for, the club that was accused is ordinary. Getting the sign right made the finding more consistent with the claim's premise and no more supportive of its conclusion.

What this cannot say

It measures yellow cards relative to fouls committed and nothing else. It says nothing about penalties given or denied, offside calls, disallowed goals, red cards specifically, added time, or whether any individual decision was correct. If "favoured" means those things, this analysis is silent on it, and should not be quoted as though it were not.

The data lesson

A result is not ready because it is significant. It is ready when a serious attempt to destroy it has failed — including an attempt to destroy the arithmetic, not just the interpretation.

Four attacks were run on the interpretation. Three landed:

  • Aggregation — does the effect live at the level you are attributing it to? Computing the same metric for the other party takes twenty lines and reframed the entire finding.
  • Specification — what does the model assume that the data might not support? The assumptions that look like arithmetic are the dangerous ones. Expected = rate × fouls looks like a definition. It is a claim, and it is false.
  • Baseline sufficiency — what does a model with no team property in it already predict? This is what surfaced the match-level cancellation, and it is what turned "both teams" into "the two sides move apart".
  • Adjustment coarseness — this one did not land. Five strength bands under-corrected the strongest clubs, so it was redone as a continuous fit. The association strengthened.

A fifth attack was not run at all, and should have been first: does every number reproduce? It is the cheapest test available and the only one that catches a sign error, because a sign error survives every argument about interpretation perfectly intact.

The generalisable version: when a metric is an average over a group, ask whether the effect belongs to the group or to the situations the group is in. Cohorts, segments, cost centres, model slices — the arithmetic is identical, and so is the error. Then check that the number you are arguing about is the number your code produces.

Tri-anchor

Anchor Source
Data 58,150 matches, 11 leagues, 2000–2026, committed snapshot 2026-W35; plus 2,737 Portuguese match officials
Football Dawson et al. (2007) on referee inconsistency and the strength gradient; Phatak et al. (2021) on fouls-to-cards across leagues
Data science Efron & Morris (1975) on shrinkage; Benjamini & Hochberg (1995) on multiplicity

References

Open questions

Sporting. The one club still separating, at 1.183 across three managers and with no unusual referee draw. Nobody was arguing about Sporting, which is part of why it is interesting.

Absolute versus relative quality. Our strength measure is standardised within league, so a Celtic–St Mirren fixture scores as "high quality" on the same scale as Barcelona–Getafe. If the real driver is absolute quality, that would explain why the within-match gradient varies as much as it does between leagues. Testable with UEFA country coefficients, and next.

What we cannot resolve at all. Whether the same foul is punished more harshly when committed by the underdog, or whether underdogs simply commit more cardable fouls. Distinguishing those requires foul-level video coding. That is the honest limit here.

Reproduce this

uv run python scripts/discipline_numbers.py     # every number in this report
uv run python scripts/build_discipline_story.py # figures 1-4
uv run python scripts/referee_decomposition.py
uv run python scripts/manager_travel.py