Attributable Risk

Attributable risk (AR)—also termed the risk difference (RD), excess risk, or attributable incidence—is the fundamental additive measure of association and public health impact in epidemiology and biostatistics, defined as the absolute arithmetic difference in disease incidence (either cumulative incidence risk or person-time incidence rate) between an exposed population (I_e) and an unexposed reference population (I_u): AR = I_e − I_u. First applied quantitatively to compare occupational and environmental mortality excesses in the nineteenth century and systematically contrasted against multiplicative relative risk in the landmark British Doctors Study of smoking and lung vs. coronary mortality by Richard Doll and Austin Bradford Hill (1954–1964) alongside Brian MacMahon, Thomas F. Pugh, and Kenneth J. Rothman, attributable risk answers a distinct scientific question from the relative risk (RR = I_e / I_u). Whereas the multiplicative relative risk measures the etiologic strength of an association (how many times more likely an exposed person is to develop the disease), the additive attributable risk measures the absolute excess burden of cases added per 1,000 or 100,000 exposed individuals (or per 100,000 person-years) that can be causally attributed to the exposure above the baseline background incidence I_u. Consequently, attributable risk and its companion metrics—the attributable fraction among the exposed (AF_e) and the population attributable fraction (PAF)—serve as the primary foundation for public health resource allocation and clinical number needed to treat (NNT) and number needed to harm (NNH) calculations.

In a prospective cohort study or randomized clinical trial following an exposed cohort (E = 1) and an unexposed cohort (E = 0) over a defined follow-up period, let:

• I_e = a / N_e (or a / PT_e) denote the cumulative incidence proportion (risk) or incidence density rate in the exposed group.
• I_u = c / N_u (or c / PT_u) denote the baseline cumulative incidence proportion or incidence rate in the unexposed group, which represents the background incidence of the disease caused by all other genetic, environmental, and stochastic factors in the absence of exposure E.
• RR = I_e / I_u denote the Relative Risk (risk ratio or incidence rate ratio).

From these quantities, epidemiologists define three closely linked attributable risk metrics (along with their protective counterparts when an intervention reduces risk, I_e < I_u):

Asymptotic Variance and 95% Confidence Intervals for Attributable Risk

For cumulative incidence proportions I_e = a / N_e and I_u = c / N_u drawn from independent binomial samples, the standard error of the attributable risk (AR = I_e − I_u) is given by the normal (Wald) binomial variance sum:

SE(AR) = √[ I_e(1 − I_e) / N_e + I_u(1 − I_u) / N_u ]

and the 95% confidence interval is AR ± 1.96 × SE(AR) (or computed via Newcombe's hybrid score method for small sample sizes). For person-time incidence rates IR_e = a / PT_e and IR_u = c / PT_u under Poisson count assumptions, the standard error of the incidence rate difference (IRD = IR_e − IR_u) simplifies to:

SE(IRD) = √[ (a / PT_e²) + (c / PT_u²) ]

The most famous empirical demonstration of why epidemiologists must examine both the multiplicative relative risk (RR) and the additive attributable risk (AR) comes from Richard Doll and Austin Bradford Hill's British Doctors Study tracking 34,439 male British physicians by cigarette smoking status:

Observe the policy lesson embedded in the Doll–Hill table:

1. Etiologic Attribution in an Individual Patient (RR and AF_e): Because lung cancer is rare in lifelong non-smokers (I_u = 7 per 100,000 person-years), cigarette smoking elevates lung cancer mortality by nearly 24-fold (RR = 23.7), and 95.8% (AF_e) of all lung cancers diagnosed in heavy smokers are directly caused by their smoking. By contrast, coronary heart disease is already common in aging non-smokers (I_u = 422 per 100,000 person-years due to hypertension, hyperlipidemia, diabetes, and genetics), so smoking increases the coronary mortality rate by a modest 42% (RR = 1.42), meaning that 29.6% (AF_e) of myocardial infarctions among smokers are attributable to smoking.
2. Population Excess Mortality Burden (AR): Yet when a public health ministry asks "If we help 100,000 heavy smokers quit smoking for one year, will smoking cessation prevent more deaths from lung cancer or more deaths from coronary heart disease?", the answer is governed by the additive attributable risk (AR = I_e − I_u): smoking causes +177 excess coronary heart disease deaths per 100,000 person-years compared to +159 excess lung cancer deaths per 100,000 person-years! Because a modest relative risk (RR = 1.42) is multiplying a large baseline incidence (I_u = 422), smoking kills more total people via coronary heart disease than via lung cancer. Relying on relative risk ratios alone (RR = 23.7 vs. 1.42) would mislead policymakers into overlooking the single largest mortality toll of tobacco use.

In the potential outcomes (counterfactual) framework developed by Jerzy Neyman, Donald B. Rubin, James M. Robins, and Miguel A. Hernán, let Y^e=1 denote an individual's binary disease outcome if exposed (E = 1) and Y^e=0 denote that same individual's outcome if unexposed (E = 0). Under conditional exchangeability (no unmeasured confounding between exposed and unexposed groups, such as after randomization or Mantel–Haenszel / inverse-probability-of-treatment weighting):

AR = E[Y^e=1 | E = 1] − E[Y^e=0 | E = 1]

1. Etiologic Fraction vs. Excess Fraction (Greenland & Robins, 1988)

A subtle causal distinction demonstrated by Sander Greenland and James M. Robins (1988) is that the epidemiological attributable fraction among the exposed (AF_e = [I_e − I_u] / I_e) measures the excess fraction (the proportion of cases over a fixed follow-up interval T that would not have occurred by time T had exposure been absent), which can be smaller than the true biological etiologic fraction (the proportion of cases in which exposure played an active mechanistic causal role). Whenever an exposure accelerates the onset time of a disease that would eventually have occurred slightly later inside the same observation window T even without the exposure (a phenomenon connected to years of potential life lost and lead-time bias), those accelerated cases are causally triggered by the exposure (etiologic cases) without increasing the net cumulative count I_e at the end of window T.

2. Additive Interaction and Synergism (Rothman's RERI and AP)

As proven by Kenneth J. Rothman (1976, 1986) using his Sufficient-Component Cause ("Causal Pie") Model, biological independence of two risk factors A and B that never participate in the same causal mechanism implies additivity of attributable risks (rather than multiplication of relative risks):

• If exposure A alone adds excess risk AR₁₀ = I₁₀ − I₀₀ and exposure B alone adds excess risk AR₀₁ = I₀₁ − I₀₀, then in the absence of mechanistic biological interaction (synergism), the combined excess risk in doubly exposed individuals (A = 1, B = 1) equals the simple sum of their individual attributable risks:

I₁₁ − I₀₀ = (I₁₀ − I₀₀) + (I₀₁ − I₀₀)

• Whenever two exposures act as co-dependent components of the same causal mechanism (such as occupational asbestos exposure and cigarette smoking in bronchogenic carcinoma, analyzed by Hammond, Selikoff, and Seidman, 1979), the doubly exposed incidence vastly exceeds additive expectations, quantified by the Relative Excess Risk Due to Interaction (RERI) or Interaction Contrast (IC):

IC = (I₁₁ − I₀₀) − [(I₁₀ − I₀₀) + (I₀₁ − I₀₀)] = I₁₁ − I₁₀ − I₀₁ + I₀₀ > 0 Dividing by I₀₀ yields RERI = RR₁₁ − RR₁₀ − RR₀₁ + 1, and dividing by I₁₁ yields the Attributable Proportion Due to Interaction (AP = RERI / RR₁₁)—showing that nearly 80% of lung cancers in asbestos-exposed smokers are attributable strictly to the synergistic interaction between asbestos fibers and tobacco carcinogens.

In a traditional hospital- or population-based case-control study (which samples diseased cases and non-diseased controls at fixed investigator-chosen ratios rather than following a cohort over time), the absolute incidence rates I_e and I_u are unknown unless external population registry incidence data are linked. Consequently, the additive attributable risk (AR = I_e − I_u) cannot be calculated directly from an unlinked case-control table (and is further vulnerable to Berkson's bias and the healthy worker effect). However, when the rare-disease assumption holds (or when controls are sampled via incidence-density / risk-set sampling so the exposure odds ratio OR = ad / bc equals the incidence rate ratio RR), the dimensionless attributable fraction among the exposed (AF_e) can be estimated directly from the case-control odds ratio:

AF_e ≈ (OR − 1) / OR

providing a direct measure of the proportion of exposed cases attributable to the risk factor even when baseline population incidence is unavailable.

1. Doll, R., & Hill, A. B. (1964). "Mortality in relation to smoking: ten years' observations of British doctors (Part I)." British Medical Journal, 1(5395), 1399–1410. https://doi.org/10.1136/bmj.1.5395.1399
2. Rothman, K. J. (1976). "Causes." American Journal of Epidemiology, 104(6), 587–592. https://doi.org/10.1093/oxfordjournals.aje.a112335
3. Greenland, S., & Robins, J. M. (1988). "Conceptual problems in the definition and interpretation of attributable fractions." American Journal of Epidemiology, 128(6), 1185–1197. https://doi.org/10.1093/oxfordjournals.aje.a115073
4. Poole, C. (2015). "A history of the population attributable fraction and related measures." Annals of Epidemiology, 25(3), 147–154. https://doi.org/10.1016/j.annepidem.2014.11.015