Population Attributable Fraction

Population Attributable Fraction in Prevention and Burden Estimates

The population attributable fraction (PAF) measures how much of the disease in a whole population would disappear if a harmful exposure were taken away, counting exposed and unexposed people together. Its size depends jointly on how strongly the exposure raises risk and on how many people are exposed, so a weak but common risk factor can account for more cases than a strong but rare one. In health economics the PAF links a relative risk and a survey prevalence to the cases, DALYs or NHS costs an exposure accounts for, and it marks the ceiling on what complete removal of an exposure could achieve. Brown and colleagues, for example, estimated that 37.7% of cancer cases in the UK in 2015 were attributable to known risk factors. The fraction among the exposed, and the bias of Levin's formula under confounding, are treated in depth on the attributable fraction page.

Three equivalent ways to write the population fraction

The PAF compares the disease actually observed with the disease expected had nobody been exposed. Khosravi and colleagues write the general form as $(O - E)/O$, where $O$ is the observed number of cases and $E$ the number expected with the exposure removed. In terms of risks the same idea reads as follows.

$$PAF = \frac{I_p - I_u}{I_p}$$

where $I_p$ is the incidence of disease over a defined period in the whole population and $I_u$ is the incidence among unexposed people. The numerator is the absolute excess of disease in the population, the population counterpart of the attributable risk among the exposed, so the PAF is that excess expressed as a share of all disease.

The population incidence is a prevalence-weighted average, $I_p = p_e I_e + (1 - p_e) I_u$. Substituting it and dividing through by $I_u$ gives Levin's formula.

$$PAF = \frac{p_e (RR - 1)}{1 + p_e (RR - 1)}$$

where $p_e$ is the proportion of the population exposed, $I_e$ is the incidence among exposed people and $RR = I_e / I_u$ is the relative risk. Khosravi and colleagues give Miettinen's case-based form as a third version.

$$PAF = p_c \, \frac{RR - 1}{RR}$$

where $p_c$ is the proportion of cases who were exposed. The three forms agree when the relative risk is free of confounding. When confounding is present, Ferguson and colleagues show that the case-based form with a confounding-adjusted relative risk stays unbiased, provided that relative risk is constant across confounder strata, while Levin's formula remains biased even with the adjusted value.

How prevalence and relative risk trade off

Levin's formula makes the joint role of prevalence and relative risk easy to see. The table applies it to illustrative combinations, assuming each relative risk is unconfounded.

An exposure carried by 5% of people with a fivefold risk gives the same PAF, 16.7%, as one carried by 20% of people with a doubled risk. In the UK cancer estimates of Brown and colleagues, tobacco smoking had the highest cancer PAF because it greatly increases risk and is linked with many cancer types, while overweight and obesity came second because they affect a high proportion of the population and are also linked with many cancer types. Smoking accounted for 15.1% and overweight and obesity for 6.3% of all UK cancer cases in 2015.

They also found that differences in PAFs between UK countries and between the sexes were primarily due to varying exposure prevalence and varying proportions of specific cancer types. A PAF therefore describes one population at one time and does not transfer to a population with a different exposure distribution.

Exposures measured at several levels

Burden studies usually measure a risk factor at several levels. Khosravi and colleagues give Walter's generalisation of Levin's formula, which Brown and colleagues used for most UK risk factors in the excess relative risk form described by Parkin and colleagues.

$$PAF = \frac{\sum_{i=1}^{I} p_i (RR_i - 1)}{1 + \sum_{i=1}^{I} p_i (RR_i - 1)}$$

where $p_i$ is the proportion of the population at exposure level $i$, $RR_i$ is the relative risk at level $i$ against the unexposed reference level 0, and $RR_i - 1$ is the excess relative risk at that level. Miettinen's generalisation uses the distribution of exposure among cases instead.

$$PAF = 1 - \sum_{i=0}^{I} \frac{p_{ci}}{RR_i}$$

where $p_{ci}$ is the proportion of cases at exposure level $i$, including the unexposed level, and $RR_0 = 1$. Khosravi and colleagues recommend this form when the relative risks are confounding-adjusted. For a continuous exposure, the comparative risk assessment framework of Murray and colleagues works with the population distribution of exposure and a relative risk at each exposure level, and compares the current distribution with a counterfactual one. The Global Burden of Disease (GBD) study takes as its counterfactual the theoretical minimum risk exposure level, the exposure associated with minimum risk.

Combining several risk factors for one disease

Fractions for different risk factors cannot simply be added. Rowe and colleagues explain that each PAF assumes its risk factor is the first to be eliminated, so the fractions describe mutually exclusive scenarios and can sum to more than one. Khosravi and colleagues give the product formula for a joint fraction.

$$PAF_{joint} = 1 - \prod_{k=1}^{K} (1 - PAF_k)$$

where $PAF_k$ is the population attributable fraction for risk factor $k$ and $K$ is the number of risk factors. The formula assumes no association and no interaction between the risk factors, which Khosravi and colleagues describe as often unrealistic in practice.

Brown and colleagues combined UK cancer fractions in sequence, applying each PAF only to the cases not already explained by earlier factors, and note that the order does not change the total. Sequential application is arithmetically the product formula. GBD 2019 uses a mediation matrix to try to correct the overestimation that assuming independence would give for combinations of risks.

Estimating the fraction and its uncertainty

A PAF needs an effect estimate and an exposure distribution, usually from different studies. Brown and colleagues drew relative risks from meta-analyses of cohort studies where possible and exposure prevalence from nationally representative surveys. They used exposure data from 2005 for 2015 cancer incidence, a ten-year lag that they judged almost certainly too short for tobacco smoking, although a 20-year lag raised the smoking PAF by only one percentage point.

With case-control data, Khosravi and colleagues note that an adjusted odds ratio can stand in for the relative risk in Miettinen's formula when the disease is rare. With individual-level data they describe model-based standardisation, also called the parametric g-formula, which predicts each person's risk with exposure set to zero and sums the predictions to give $E$. They add that valid 95% confidence intervals must reflect uncertainty in both the relative risk and the exposure prevalence, and can come from the delta method, robust standard errors or nonparametric bootstrapping.

Worked example with a graded exposure and a prevention programme

All numbers in this example are illustrative. A region has 1,000,000 adults, and a chronic disease has an annual risk of 2.0 per 1,000 among unexposed adults. A graded exposure has relative risks of 1.5 at the moderate level and 3.0 at the heavy level, assumed free of confounding. Each new case costs the NHS GBP 12,000 in its first year.

1. Count the excess cases. With nobody exposed there would be 2,000 cases, so 700 of the 2,700 cases are attributable to the exposure and the PAF is 700 / 2,700 = 0.259.

2. Apply the multilevel Levin formula. The excess terms are 0.30 x 0.5 = 0.15 and 0.10 x 2.0 = 0.20, summing to 0.35.

$$PAF = \frac{0.35}{1 + 0.35} = \frac{0.35}{1.35} = 0.259$$

where 0.35 is the prevalence-weighted sum of excess relative risks. Miettinen's form agrees, since dividing the case counts by their relative risks gives 1,200 + 600 + 200 = 2,000 expected cases.

3. See where the cases come from. The heavy level holds 10% of adults but 400 of the 700 attributable cases, a fraction of 0.20 / 1.35 = 0.148 against 0.15 / 1.35 = 0.111 for the moderate level.

4. Express the result in money terms. In the simplest form of attributable cost estimation, the attributable cost is 700 x GBP 12,000 = GBP 8.4 million of the GBP 32.4 million total.

5. Value a realistic programme. A programme moves half of the heavy group, 50,000 adults, to the moderate level. The potential impact fraction compares current and counterfactual exposure distributions.

$$PIF = \frac{\sum_i p_i RR_i - \sum_i p'_i RR_i}{\sum_i p_i RR_i} = \frac{1.35 - 1.275}{1.35} = 0.056$$

where $PIF$ is the potential impact fraction, $p_i$ is the current share at level $i$, $p'_i$ is the share after the programme and $RR_i$ is the relative risk at level $i$. Current exposure gives 0.60 x 1 + 0.30 x 1.5 + 0.10 x 3.0 = 1.35 and the programme gives 0.60 x 1 + 0.35 x 1.5 + 0.05 x 3.0 = 1.275. The programme prevents 0.075 / 1.35 x 2,700 = 150 cases a year (50,000 adults whose risk falls by 3.0 per 1,000), worth GBP 1.8 million a year once their risk has fallen. Khosravi and colleagues warn that this form shares Levin's bias under confounding, ruled out here by construction.

6. Add a second risk factor. Suppose an independent second factor, with no interaction, has a PAF of 0.20 for the same disease. Adding the fractions gives 0.459, or 1,240 cases. In sequence, the first factor explains 700 cases and the second 0.20 x 2,000 = 400 more, a joint fraction of 1,100 / 2,700 = 0.407, which is what the product formula gives. Taking the factors in the other order gives 540 and then 700 / 2,700 x 2,160 = 560, again 1,100.

7. Read the result. GBP 8.4 million a year is the most that complete removal could save in first-year treatment costs. The programme's GBP 1.8 million is a gross saving before its own costs, and the 150 cases averted, followed over time with their QALYs and costs, are what a cost-effectiveness analysis of the programme would use.

Where the population fraction fits among neighbouring measures

The population fraction is the fraction among the exposed scaled by the exposed share of cases. Khosravi and colleagues note that the PAF is useful for comparing strategies against different exposures for the same disease, but that comparing strategies across different diseases needs the numerator, which they call the attributable community risk. Prioritising prevention across diseases rests on attributable cases, DALYs or costs, not on shares.

A protective exposure needs a different measure. Khosravi and colleagues state that the PAF cannot be used for a preventive factor and describe the preventable fraction, obtained by reverse coding the exposure, and the prevented fraction $(E - O)/E$, the proportion of cases that have not occurred because of the exposure. For partial change, Morgenstern and Bursic proposed the potential impact fraction, the proportion of expected new cases that programmes of varying success might prevent.

Misreadings in prevention and burden work

Two errors recur when the population fraction is used in prevention and cost work. Others, such as confusing the excess fraction with the etiologic fraction, are covered on the attributable fraction page.

1. Treating the PAF as a programme effect. The fraction assumes complete removal, and partial reach is captured by the potential impact fraction, as step 5 shows.
2. Matching current exposure to current cases. Disease reflects past exposure, which is why Brown and colleagues used a lag and still expected the smoking fraction to be an underestimate as smoking prevalence falls.

Sources

• Brown KF, Rumgay H, Dunlop C, et al. The fraction of cancer attributable to modifiable risk factors in England, Wales, Scotland, Northern Ireland, and the United Kingdom in 2015. British Journal of Cancer. 2018;118(8):1130-1141. https://doi.org/10.1038/s41416-018-0029-6
• Ferguson J, Alvarez A, Mulligan M, Judge C, O'Donnell M. Bias assessment and correction for Levin's population attributable fraction in the presence of confounding. European Journal of Epidemiology. 2024;39(2):111-119. https://doi.org/10.1007/s10654-023-01063-8
• GBD 2019 Risk Factors Collaborators. Global burden of 87 risk factors in 204 countries and territories, 1990-2019: a systematic analysis for the Global Burden of Disease Study 2019. The Lancet. 2020;396(10258):1223-1249. https://doi.org/10.1016/S0140-6736(20)30752-2
• Khosravi A, Nazemipour M, Shinozaki T, Mansournia MA. Population attributable fraction in textbooks: time to revise. Global Epidemiology. 2021;3:100062. https://doi.org/10.1016/j.gloepi.2021.100062
• Morgenstern H, Bursic ES. A method for using epidemiologic data to estimate the potential impact of an intervention on the health status of a target population. Journal of Community Health. 1982;7(4):292-309. https://doi.org/10.1007/BF01318961
• Murray CJL, Ezzati M, Lopez AD, Rodgers A, Vander Hoorn S. Comparative quantification of health risks: conceptual framework and methodological issues. Population Health Metrics. 2003;1(1):1. https://doi.org/10.1186/1478-7954-1-1
• Rowe AK, Powell KE, Flanders WD. Why population attributable fractions can sum to more than one. American Journal of Preventive Medicine. 2004;26(3):243-249. https://doi.org/10.1016/j.amepre.2003.12.007