Attributable Fraction

Attributable Fraction Formulas, Confounding and Burden of Disease

An attributable fraction says what share of cases would disappear if an exposure were removed. Among exposed people it depends only on the relative risk; across a whole population, it also depends on how common the exposure is. In health economics it is the step that turns a relative risk into attributable deaths, DALYs or NHS spending. Of the Levin and Miettinen formulas for the population fraction, only Miettinen's survives confounding adjustment. The Global Burden of Disease (GBD) study and cost-of-illness studies both use the measure, and in an illustrative example the choice of formula changes an attributable cost by millions of pounds.

Two fractions, among the exposed and in the population

Both fractions take the form $(O - E)/O$, where $O$ is the observed number of cases and $E$ the number expected had nobody been exposed. They differ in whose cases are counted.

The fraction among the exposed uses exposed cases only and is calculated as follows.

$$AF_e = \frac{R_e - R_u}{R_e} = \frac{RR - 1}{RR}$$

where $AF_e$ is the attributable fraction among the exposed, $R_e$ is the risk of disease in exposed people, $R_u$ is the risk in unexposed people and $RR = R_e / R_u$ is the relative risk. A relative risk of 2 gives $AF_e = 0.5$, meaning that half of the cases among exposed people would not have occurred without the exposure.

The population fraction uses all cases and is calculated as follows.

$$PAF = \frac{R_p - R_u}{R_p}$$

where $PAF$ is the population attributable fraction and $R_p$ is the risk in the whole population, exposed and unexposed together. Steenland and Armstrong call the attributable fraction the primary measure of the burden due to a risk factor; multiplied by total cases it gives a "body count" of cases due to that factor. The population attributable fraction page covers the population measure separately; the difference $R_e - R_u$ is the attributable risk, an absolute risk difference.

Greenland and Robins separate the excess fraction, cases in excess of those expected without exposure by a given time, from the etiologic (aetiological) fraction, cases the exposure caused, including those it only brought forward. The two need not be close, and the etiologic fraction is not generally estimable without strong biological assumptions. The formulas on this page estimate excess fractions.

Levin's formula and Miettinen's formula

Published population fractions are usually computed from a relative risk and a prevalence. Two formulas do this, and they diverge once confounding enters.

Levin introduced the population fraction in 1953 to ask what proportion of lung cancer is attributable to smoking. His formula uses exposure prevalence in the population, as follows.

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

where $PAF_L$ is the Levin estimate, $p_e$ is the prevalence of exposure in the whole population and $RR$ is the relative risk. Khosravi and colleagues found it in 29 of 43 textbooks reviewed; only 6 of the 29 noted that it is biased when confounding adjustment is needed, even with an adjusted relative risk.

Miettinen's 1974 formula uses exposure prevalence among cases instead.

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

where $PAF_M$ is the Miettinen estimate, $p_c$ is the proportion of cases who were exposed and $RR$ is the confounding-adjusted relative risk. It is $AF_e$ multiplied by $p_c$, because removing the exposure cannot prevent cases among unexposed people. Ferguson and colleagues state that it equals the true population fraction whatever the degree of confounding, provided the causal relative risk is constant across confounder strata; otherwise the relative risk among the exposed is used. For a rare disease, an adjusted odds ratio, rate ratio or hazard ratio can stand in for the relative risk, which makes the formula usable with case-control data.

When only published summary estimates are available, Ferguson and colleagues show that $p_c$ can be recovered from the population prevalence and the unadjusted relative risk, as follows.

$$PAF_M = \frac{p_e \, RR_U}{1 + p_e (RR_U - 1)} \times \frac{RR_C - 1}{RR_C}$$

where $RR_U$ is the unadjusted (crude) relative risk and $RR_C$ is the causal, confounding-adjusted relative risk. When $RR_U = RR_C$ this reduces to Levin's formula. Darrow and Steenland showed by simulation that Levin's formula with an adjusted relative risk underestimates the fraction when the crude relative risk exceeds the adjusted one and overestimates it when the crude is smaller, with most bias when exposure is rare and the association weak. Ferguson and colleagues note that GBD uses Levin's form.

From fractions to attributable burden and costs

Multiplying an attributable fraction by a disease total turns a proportion into burden. In the comparative risk assessment framework of Murray and colleagues, the population fraction is the proportional reduction in disease or death if exposure fell to zero, other things equal, and attributable burden applies it to deaths, incidence or a summary measure such as disability-adjusted life years.

GBD 2019 applied this to 87 risk factors in 204 countries and territories. It replaced zero exposure with the theoretical minimum risk exposure level (TMREL), the exposure associated with minimum risk, and multiplied population fractions by deaths, years of life lost, years lived with disability and DALYs for every age, sex, location and year.

The counterfactual need not be the theoretical minimum. Morgenstern and Bursic proposed the potential impact fraction, the proportion of expected new cases a programme of given success might prevent, to assess the efficacy, effectiveness, adequacy and efficiency of prevention strategies. Murray and colleagues list theoretical, plausible, feasible (observed in some population) and cost-effective minimum counterfactuals, the last adding the cost of exposure reduction as a criterion. For exposure levels $i$, the general form is as follows.

$$PIF = \frac{\sum_i P_i RR_i - \sum_i P'_i RR_i}{\sum_i P_i RR_i}$$

where $PIF$ is the potential impact fraction, $P_i$ is the current proportion of the population at exposure level $i$, $P'_i$ is the counterfactual proportion and $RR_i$ is the relative risk at level $i$ against the reference level. Setting all of $P'$ at the reference level gives the population fraction. Khosravi and colleagues warn that this form shares the Levin formula's bias under confounding; when an intervention removes a fraction $r$ of exposure at random among exposed people, the impact fraction is simply $r \times PAF$.

In a cost-of-illness study, the same multiplication allocates spending. Scarborough and colleagues applied risk-factor-specific population fractions to disease-specific NHS costs for 2006-07 and estimated that ill health related to poor diet cost the NHS in the UK GBP 5.8 billion, overweight and obesity GBP 5.1 billion and physical inactivity GBP 0.9 billion. The attributable cost estimation page compares this top-down allocation with regression methods.

Assumptions behind a causal reading

An attributable fraction is a causal quantity, so every estimate carries the assumptions of causal inference. Khosravi and colleagues list four.

1. Causality. The exposure causes the disease. Only about half of the textbooks they reviewed stated this.
2. No confounding or other bias. Confounders are identified from a causal model, and the exposure is measured without bias.
3. Other risk factors unaffected. Removing the exposure leaves other risk factors unchanged, an assumption likely to be violated in practice.
4. A realistic intervention. Some intervention could remove the exposure.

Timing matters too. Greenland and Robins argue that when the question is when disease occurs, effects on incidence time may matter as much as any attributable fraction; in Khosravi and colleagues' extreme case, an exposure that only brings disease forward has an etiologic fraction of 100% but a population fraction of zero.

Worked example of confounding by age in 100,000 adults

The numbers below are illustrative. In 100,000 adults, exposure is more common in the older group, which also has a higher baseline risk, so age confounds the crude association. Within each age group the exposure doubles risk.

1. Relative risks. Crude risk is 860 in 26,000 (3.31%) if exposed and 670 in 74,000 (0.91%) if unexposed, a crude relative risk of 3.65. The age-adjusted relative risk is 2.0, the same in both groups.

2. The true fraction by counting. Without exposure, the 20,000 exposed older adults would have 20,000 x 0.02 = 400 cases and the 6,000 exposed younger adults 6,000 x 0.005 = 30, so 430 of the 1,530 cases would not occur. The population fraction is 430 / 1,530 = 0.281, and the fraction among the exposed is 430 / 860 = 0.5, matching $(2 - 1)/2$.

3. Miettinen's formula.

$$PAF_M = \frac{860}{1530} \times \frac{2 - 1}{2} = \frac{430}{1530} = 0.281$$

where 860 / 1,530 is $p_c$, the exposed share of cases, and 2 is the adjusted relative risk. The formula recovers the true value.

4. Levin's formula. With the adjusted relative risk, Levin's formula is applied as follows.

$$PAF_L = \frac{0.26 \times (2 - 1)}{1 + 0.26 \times (2 - 1)} = \frac{0.26}{1.26} = 0.206$$

where 0.26 is $p_e$, the population prevalence of exposure. With the crude relative risk of 3.65, Levin's formula gives 0.408, which equals the crude fraction $(R_p - R_u)/R_p$ and still carries the confounding by age. The crude relative risk exceeds the adjusted one, so the adjusted Levin estimate falls short of the truth, as Darrow and Steenland found.

5. Attributable cost. Suppose the NHS spends GBP 30.6 million a year treating the 1,530 cases, GBP 20,000 per case. The attributable cost is 0.281 x GBP 30.6 million = GBP 8.6 million (430 cases x GBP 20,000). Levin's formula would report GBP 6.3 million with the adjusted relative risk and GBP 12.5 million with the crude one.

6. A realistic programme. A programme that removed exposure from half of the exposed adults in each age group, at random, has a potential impact fraction of 0.5 x 430 / 1,530 = 0.141. This means 215 fewer cases and GBP 4.3 million less treatment spending a year, once risk in those who stop has fallen to the unexposed level. Only the GBP 4.3 million, not the GBP 8.6 million for complete removal, belongs in an evaluation of this programme.

Common misreadings in burden and cost work

Most errors read an attributable fraction as the answer to another question. Five recur.

• Adding fractions across risk factors. Rowe and colleagues explain that each population fraction assumes its risk factor is eliminated first, so the fractions describe mutually exclusive scenarios and can sum to more than one. GBD 2019 notes that its risk-specific burdens can exceed 100% for causes such as cardiovascular diseases, and estimates combinations with a mediation matrix instead. The Scarborough figures, each built on a risk-factor-specific fraction, should not be added together for the same reason.
• Levin's formula with an adjusted relative risk. Adjusting the relative risk does not remove confounding bias from Levin's formula, as step 4 above shows; Miettinen's formula, or Ferguson's re-expression from summary data, does.
• Population fraction as programme impact. The population fraction assumes complete removal of the exposure; a programme with partial reach prevents only the potential impact fraction, as step 6 shows. Khosravi and colleagues use "preventable fraction" for a protective exposure (the proportion of cases that would not occur if everyone were exposed) and "prevented fraction" for the proportion of expected cases that have not occurred because of the exposure.
• Fraction among the exposed as probability of causation. A fraction of 0.5 among the exposed does not give each exposed case a 50% chance of exposure-caused disease, because excess and etiologic fractions can differ.
• Attributable cost as cash saving. A cost apportioned by a fraction is current spending linked to the exposure; savings depend on an intervention's reach, the lag before risk falls and whether released resources are recovered.

Sources

• Darrow LA, Steenland K. Confounding and bias in the attributable fraction. Epidemiology. 2011;22(1):53-58. https://doi.org/10.1097/EDE.0b013e3181fce49b
• 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
• Greenland S, Robins JM. Conceptual problems in the definition and interpretation of attributable fractions. American Journal of Epidemiology. 1988;128(6):1185-1197. https://doi.org/10.1093/oxfordjournals.aje.a115073
• 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
• Scarborough P, Bhatnagar P, Wickramasinghe KK, Allender S, Foster C, Rayner M. The economic burden of ill health due to diet, physical inactivity, smoking, alcohol and obesity in the UK: an update to 2006-07 NHS costs. Journal of Public Health. 2011;33(4):527-535. https://doi.org/10.1093/pubmed/fdr033
• Steenland K, Armstrong B. An overview of methods for calculating the burden of disease due to specific risk factors. Epidemiology. 2006;17(5):512-519. https://doi.org/10.1097/01.ede.0000229155.05644.43