Population Attributable Fraction

The population attributable fraction (PAF)—also termed the population attributable risk percent (*PAR*%) or etiologic fraction in the population—is a cornerstone metric of epidemiology, biostatistics, and global public health policy that quantifies the proportional reduction in total population disease incidence or mortality over a specified time interval that would be achieved under the counterfactual intervention in which exposure to a causal risk factor were eliminated across the entire population (or shifted to a Theoretical Minimum Risk Exposure Level, TMREL). First derived in 1953 by Morton L. Levin to calculate the proportion of all lung cancer cases in the general population attributable to cigarette smoking, and generalized to confounder-adjusted and multi-level exposure settings by Walter S. Miettinen (1974), Sander Greenland, Karsten Drescher, and Christopher J. L. Murray and Majid Ezzati for the Global Burden of Disease (GBD) Comparative Risk Assessment, the population attributable fraction synthesizes two distinct dimensions of an epidemiological threat into a single percentage: the etiologic strength of the risk factor (Relative Risk, RR) and the prevalence of the exposure in the population (P_e). Because a moderate risk factor that affects a large proportion of society (such as elevated systolic blood pressure or ambient PM_2.5 air pollution) generates a much larger PAF than a potent risk factor to which almost no one is exposed, PAF serves as the quantitative compass for prioritizing population-wide preventive interventions to reduce years of potential life lost and extend healthy life expectancy.

1. Direct Population Incidence Definition

Consider a total population of size N partitioned into an exposed fraction P_e = N_e / N (experiencing disease incidence I_e) and an unexposed fraction P_u = 1 − P_e = N_u / N (experiencing baseline background disease incidence I_u). The total population disease incidence (I_total) is the weighted average of the exposed and unexposed incidences:

I_total = P_e I_e + (1 − P_e) I_u = I_u + P_e (I_e − I_u)

The absolute Population Attributable Risk (PAR) is the excess disease incidence in the total population above what would occur if everyone had the unexposed baseline risk I_u:

PAR = I_total − I_u = P_e (I_e − I_u) = P_e × AR

where AR = I_e − I_u is the exposed attributable risk. Dividing PAR by the total population incidence I_total yields the Population Attributable Fraction (PAF):

PAF = (I_total − I_u) / I_total

2. Morton Levin's Classical Formula (1953) for Unconfounded Binary Exposures

In his 1953 monograph in Acta Unio Internationalis Contra Cancrum, Morton L. Levin substituted I_total = P_e I_e + (1 − P_e) I_u into the definition above and divided both numerator and denominator by the unexposed incidence I_u (setting the Relative Risk RR = I_e / I_u), yielding Levin's Formula:

PAF = P_e (RR − 1) / [ 1 + P_e (RR − 1) ]

Levin's formula demonstrates the dual dependence of PAF on both P_e (exposure prevalence in the general population) and RR (relative risk):

• If exposure is rare (P_e = 0.001, i.e., 0.1% of the population works with vinyl chloride monomer), even a huge relative risk (RR = 21.0) produces a small population attributable fraction: PAF = 0.001(20) / [1 + 0.02] = 1.96%.
• Conversely, if exposure is widespread (P_e = 0.50, i.e., 50% of adults have suboptimal blood pressure or dietary sodium intake), even a modest relative risk (RR = 2.0) produces a massive population attributable fraction: PAF = 0.50(1.0) / [1 + 0.50] = 33.3% of all strokes in the entire nation.

3. Walter Miettinen's Confounder-Adjusted Case-Fraction Formula (1974)

A critical statistical pitfall discovered by Walter S. Miettinen (1974) and Sander Greenland (1984) is that plugging a confounder-adjusted relative risk (RR_adj) directly into Levin's formula produces a mathematically biased estimate of PAF whenever confounding is present!

To derive an exact formula that remains valid in the presence of confounding, Miettinen observed that total population excess cases (N × PAR) are simply the sum of excess cases occurring among the exposed subpopulation. Therefore, the proportion of all population cases attributable to exposure equals the proportion of all diseased cases in the population who were exposed (P_c = Pr(E = 1 | D = 1)) multiplied by the attributable fraction among the exposed (AF_e = (RR_adj − 1) / RR_adj):

PAF = P_c × AF_e = P_c × [ (RR_adj − 1) / RR_adj ]

When exposure consists of K polychotomous categories (k = 0, 1, …, K, such as never-smokers, former smokers, light smokers, and heavy smokers) or is stratified across confounder strata s, Miettinen's formula generalizes to:

PAF = Σ_k=1^K P_c,k [ (RR_adj,k − 1) / RR_adj,k ] = 1 − Σ_k=0^K ( P_c,k / RR_adj,k )

Notice an additional practical advantage of Miettinen's formula: in a population-based case-control study, the exposure prevalence among cases (P_c) is observed directly in the case sample, and the adjusted odds ratio (OR_adj from multivariable logistic regression) estimates RR_adj, allowing epidemiologists to calculate confounder-adjusted PAF directly from case-control data (Bruzzi, Green, Byar, Brinton, & Schairer, 1985).

One of the most frequent misunderstandings in public health communication is the belief that the population attributable fractions of all causes of a disease (such as coronary heart disease or cancer) must add up to 100%. In reality, individual PAFs for a single disease routinely sum to 150%, 200%, or more, and there is no mathematical upper bound on the sum of individual cause-specific PAFs!

Why do PAFs sum to more than 100%? As explained by Kenneth J. Rothman (1976) and Walter Willett, diseases are caused by multi-factorial sufficient-component cause constellations ("causal pies") in which multiple component causes must act together in the same patient:

• Suppose a fatal myocardial infarction requires the simultaneous presence of both hypertension (Factor A) and hypercholesterolemia (Factor B) in the same patient.
• If public health clinicians eliminate hypertension (A), 100% of those joint myocardial infarctions are prevented (PAF_A = 100%).
• Alternatively, if clinicians eliminate hypercholesterolemia (B), the exact same 100% of joint myocardial infarctions are prevented (PAF_B = 100%).
• Thus, any disease case caused by the joint interaction of M risk factors is counted M times when each risk factor's PAF_i is calculated separately (PAF_A + PAF_B = 200%), even though eliminating both risk factors simultaneously cannot prevent more than 100% of cases.

To compute the combined Joint Population Attributable Fraction (PAF_joint) for M independent, non-overlapping risk factors acting multiplicatively without biological synergy, the Global Burden of Disease (GBD) Study (Ezzati et al., 2002; Murray et al., 2020) uses the multiplicative complement (prevented-fraction) product formula:

PAF_joint = 1 − Π_i=1^M (1 − PAF_i)

For example, if smoking has PAF₁ = 0.30 (leaving 70% of cases unprevented), hypertension has PAF₂ = 0.40 (removing 40% of the remaining 70%), and elevated LDL cholesterol has PAF₃ = 0.50 (removing 50% of the remainder), the naive sum is 30% + 40% + 50% = 120%, whereas the true joint PAF is:

PAF_joint = 1 − (1 − 0.30)(1 − 0.40)(1 − 0.50) = 1 − (0.70 × 0.60 × 0.50) = 1 − 0.21 = 0.79 (79.0%)

When sequential interventions are modeled along a causal mediation pathway (for example, high dietary sodium → elevated systolic blood pressure → hemorrhagic stroke), John Ferguson and Martin O'Donnell's "Sequential and Average Attributable Fractions" (Shapley-value decomposition) partition the joint PAF fairly across upstream behavioral determinants and downstream physiological mediators.

In the World Health Organization and Institute for Health Metrics and Evaluation (IHME) Global Burden of Disease (GBD) framework pioneered by Christopher Murray and Alan Lopez, many physiological and environmental exposures—such as systolic blood pressure (SBP), fasting plasma glucose, body-mass index (BMI), LDL cholesterol, and ambient particulate matter (PM_2.5)—cannot be reduced to "zero" because zero blood pressure or zero blood glucose is incompatible with human life. Instead, GBD defines for each continuous exposure x a Theoretical Minimum Risk Exposure Level (TMREL, P*(x)), which is the exposure distribution associated with the lowest population morbidity and mortality across epidemiological cohorts (for example, SBP of 110 to 115 mmHg, fasting plasma glucose of 4.8 to 5.4 mmol/L, BMI of 20 to 25 kg/m², and ambient PM_2.5 of 2.4 to 5.9 μg/m³). Multiplying each disease's age- and sex-specific Disability-Adjusted Life Years (DALYs = YPLL + YLD) by the continuous integral PAF reveals that high systolic blood pressure, tobacco smoking, ambient and household air pollution, high fasting plasma glucose, and high BMI account for the largest preventable fractions of global mortality across the epidemiological transition.

1. Levin, M. L. (1953). "The occurrence of lung cancer in man." Acta Unio Internationalis Contra Cancrum, 9(3), 531–541. https://doi.org/10.1016/j.annepidem.2014.11.015
2. Miettinen, O. S. (1974). "Proportion of disease caused or prevented by a given exposure, trait or intervention." American Journal of Epidemiology, 99(5), 325–332. https://doi.org/10.1093/oxfordjournals.aje.a121617
3. Bruzzi, P., Green, S. B., Byar, D. P., Brinton, L. A., & Schairer, C. (1985). "Estimating the population attributable risk for multiple risk factors using case-control data." American Journal of Epidemiology, 122(5), 904–914. https://doi.org/10.1093/oxfordjournals.aje.a114174
4. Rockhill, B., Newman, B., & Weinberg, C. (1998). "Use and misuse of population attributable fractions." American Journal of Public Health, 88(1), 15–19. https://doi.org/10.2105/AJPH.88.1.15