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Relative risk answers “how much more likely.” It does not answer “how much of this problem is actually caused by the exposure” or “how much disease would disappear if the exposure were removed.” Those are the questions attributable risk and attributable fraction are built to answer, and they come in two pairs that get conflated constantly: an absolute-versus-proportional pair, and an individual-versus-population pair. This guide keeps all four measures straight, shows the arithmetic that connects them, and works one illustrative example through every step so the difference from relative risk is concrete rather than definitional.
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The four measures, and why they get confused
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All four measures start from the same two incidence numbers: the incidence of an outcome among people exposed to a risk factor (Ie), and the incidence among people not exposed (Iu). Everything below is derived from just those two numbers, plus, for the population-level pair, the prevalence of exposure in the wider population.
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- Attributable risk (AR) — absolute, individual/exposed-group level. The risk difference: Ie − Iu.
- Attributable fraction among the exposed (AR%, sometimes AFe) — proportional, individual/exposed-group level. The share of risk in the exposed group attributable to the exposure.
- Population attributable risk (PAR) — absolute, whole-population level. The risk difference between the total population and the unexposed group.
- Population attributable fraction (PAF) — proportional, whole-population level. The share of the outcome in the entire population attributable to the exposure.
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The pattern is symmetric: AR and PAR are both risk differences (subtraction, reported in the outcome’s own units — percentage points, or cases per however many people); AR% and PAF are both fractions of a risk that get attributed to the exposure (division, reported as a percentage with no units). The first pair asks “what happens to someone with this exposure.” The second pair asks “what happens to this population.” Confusing the two produces the most common misreading in this space: quoting an AR% figure as though it describes the whole population, when it only describes people who were actually exposed.
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Attributable risk (AR): the absolute measure
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AR = Ie − Iu
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Attributable risk is the risk difference — the excess incidence among the exposed that would not be there if their incidence matched the unexposed group’s. It is reported in the same units as the incidence itself (a rate difference, or percentage points), which is exactly what makes it the least distortable of the four measures: “the exposed group has 6 more cases per 100 people than the unexposed group” is a complete, concrete statement that doesn’t need a denominator supplied by the reader.
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AR is mathematically the same operation as the absolute risk reduction (ARR) used to report clinical-trial results — see Absolute Risk Reduction, Relative Risk Reduction, and Number Needed to Treat — but the two terms are used in different study contexts and shouldn’t be swapped casually. ARR describes a controlled intervention’s effect (treatment arm vs. control arm in a trial, where exposure is assigned). AR describes an observed association in etiologic/exposure epidemiology (exposed vs. unexposed in a cohort study, where exposure is not assigned). The arithmetic is identical; the causal footing under it is not — a trial’s randomization supports a causal reading of ARR far more directly than an observational cohort supports one for AR, which is why AR calculations lean on confounding adjustment in a way a well-randomized trial mostly doesn’t need to.
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AR can only be calculated directly from a design that yields true incidence in both groups — a cohort or comparable prospective design (see cohort study vs. case-control study design). A case-control study fixes the ratio of cases to controls by design, so Ie and Iu are not observable and AR cannot be computed directly from it — the odds ratio, not AR, is the native measure there.
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Attributable fraction among the exposed (AR%): the proportional measure
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AR% = (Ie − Iu) / Ie = (RR − 1) / RR
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Where relative risk (RR) is Ie / Iu. AR% answers a different question than AR: not “how much excess risk” but “of the risk that exposed people actually experience, what proportion is attributable to the exposure itself, versus the baseline risk they’d have had anyway.” It is a fraction of the exposed group’s own risk, which is the detail that gets lost when AR% is misquoted as if it applied to everyone.
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The two formulas are algebraically identical — one uses the two incidences directly, the other uses relative risk alone. That second form is useful because it means AR% can be recovered from a relative risk or, with the rare-disease approximation, an odds ratio, even when raw incidence figures aren’t reported. It is also, by construction, a statement conditional on the association being causal and unconfounded: AR% describes what the data attribute to the exposure, not what a randomized removal of the exposure is guaranteed to prevent, unless the underlying association has been shown (or can reasonably be assumed) to be causal.
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Scaling up: population attributable risk and population attributable fraction
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AR and AR% describe exposed people only. To describe the exposure’s impact on the whole population — the question a public-health decision usually needs answered — the calculation has to bring in one more number: Pe, the prevalence of the exposure in the total population (not just among cases).
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PAR = Itotal − Iu, where Itotal is the incidence in the whole population (exposed and unexposed combined). This is the population-level absolute measure — the excess cases per population-unit attributable to the exposure being present in the population at all, rather than everyone having the unexposed group’s risk.
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PAF = (Itotal − Iu) / Itotal — the population-level proportional measure. Where total-population incidence isn’t directly available, PAF is more commonly calculated from exposure prevalence and relative risk, a version generally attributed to Levin’s 1953 formulation:
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PAF = Pe(RR − 1) / [1 + Pe(RR − 1)]
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This form makes the key public-health point visible algebraically: PAF depends on both the strength of the association (RR) and how common the exposure is (Pe). A large RR attached to a rare exposure can still produce a small PAF, and a modest RR attached to a very common exposure can produce a large one — which is precisely why relative risk alone cannot answer a prioritization question, and PAF has to be calculated rather than eyeballed from RR.
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Two caveats worth stating plainly rather than glossing over: the relative risk that goes into this formula should be adjusted for known confounders (an unadjusted, confounded RR biases PAF the same way it biases the RR itself), and when more than one risk factor contributes to the same outcome, their individual PAFs generally do not sum to 100% — they can overlap (the same case can be “attributable” to more than one contributing exposure) or, on the other side, the residual can leave meaningful unexplained incidence even after every known exposure is accounted for. A PAF is a statement about one exposure in isolation, not a slice of a fixed pie.
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Worked example
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The figures below are an illustrative composite constructed to demonstrate the arithmetic cleanly — not a real study, real exposure, or real outcome. Do not cite these numbers as evidence of any actual association.
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| Population | Cases | Incidence | |
|---|---|---|---|
| Exposed | 1,000 | 100 | Ie = 100/1,000 = 10% |
| Unexposed | 1,000 | 40 | Iu = 40/1,000 = 4% |
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Assume this exposure affects 30% of the wider population (Pe = 0.30) and the exposed/unexposed incidences above are representative of exposed and unexposed people in that population.
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- Relative risk: RR = 10% ÷ 4% = 2.5 — exposed people have 2.5 times the risk of unexposed people. On its own, this is the number that gets quoted as “150% higher risk,” which sounds dramatic before any of the following is calculated.
- AR: 10% − 4% = 6 percentage points (60 excess cases per 1,000 exposed people, over and above their unexposed baseline).
- AR%: (10% − 4%) / 10% = 6/10 = 60%. Cross-check via RR: (2.5 − 1) / 2.5 = 1.5/2.5 = 60%. Of the risk an exposed person carries, 60% is attributable to the exposure; the remaining 40% is the baseline risk they’d have had regardless.
- Total population incidence: Itotal = (0.30 × 10%) + (0.70 × 4%) = 3.0% + 2.8% = 5.8%.
- PAR: 5.8% − 4% = 1.8 percentage points (18 excess cases per 1,000 people in the total population).
- PAF: 1.8% ÷ 5.8% ≈ 31.0%. Cross-check via Levin’s formula: [0.30 × (2.5 − 1)] / [1 + 0.30 × (2.5 − 1)] = 0.45 / 1.45 ≈ 31.0%. Both routes agree.
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This is the comparison that RR alone can’t give you: a 2.5-fold relative risk sounds like the dominant driver of this outcome, but because only 30% of the population carries the exposure, eliminating it would (assuming the association is fully causal) prevent an estimated 31% of cases in the population — meaning the other 69% of cases would still occur even in a world with no exposure at all. RR describes the strength of the association for someone who has the exposure; PAF describes how much of the population’s actual disease burden that exposure explains. A screening or policy decision aimed at population-level impact needs the second number, not the first.
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Common questions
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Is attributable risk the same thing as absolute risk reduction?
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The arithmetic is identical (a risk difference), but the terms belong to different study contexts. ARR describes an assigned intervention in a trial; AR describes an observed exposure-outcome association in a cohort. See ARR, RRR, and NNT for the trial-side version.
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Can I calculate attributable risk or PAF from a case-control study?
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Not directly. Case-control studies fix the case:control ratio by design, so true incidence — and therefore AR, AR%, PAR, and PAF as defined above — isn’t observable from the raw counts. Under the rare-disease assumption, the odds ratio approximates RR closely enough to substitute into the AR% and Levin’s-formula PAF equations, but that’s an approximation, not a direct calculation; see case-control study design for when the rare-disease assumption holds.
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Does a high PAF mean an exposure is the strongest risk factor for a disease?
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No — PAF is the product of association strength (RR) and how common the exposure is, not association strength alone. A weaker but widespread exposure can carry a higher PAF than a stronger but rare one. Comparing PAFs across exposures answers “which exposure explains more current disease burden,” which is a different question from “which exposure carries the highest relative risk for an individual.”
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Do PAFs for multiple risk factors of the same disease add up to 100%?
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Not in general. Individual PAFs can overlap (a case can be attributable to more than one contributing exposure at once), and summing them past 100% is a common analytical error. Treat each PAF as describing one exposure in isolation against the counterfactual of its removal, not as a slice of a fixed total.
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Why does the relative risk used in the PAF formula need to be adjusted for confounders?
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An unadjusted RR carries whatever bias the confounding introduces, and that bias propagates directly into the PAF calculated from it — see confounding variable. Using an RR (or odds ratio) adjusted for known confounders, from a stratified analysis or regression model, is standard practice before it goes into an AR%, PAR, or PAF calculation intended to inform a real decision.
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