Attributable Risk

Attributable Risk, numbers needed to treat and the cost of an exposure

Attributable risk is found by subtracting the risk of an outcome in unexposed people from the risk in exposed people over the same period, which gives the extra risk carried by exposure. A one-year risk of 3.0% among exposed people against 1.8% among unexposed people gives an attributable risk of 1.2 percentage points, or 12 extra events per 1,000 exposed people a year, if the exposure causes the difference. In health economics this absolute excess is what a model turns into extra events, costs and lost quality-adjusted life years, so it sets the size of an economic result in a way a relative risk cannot. Some textbooks give the name to a proportion of cases instead, but here it means the difference in risk, which also leads to numbers needed to treat for benefit or harm and to the population attributable risk.

Excess risk on the absolute scale

Coggon, Rose and Barker, in their BMJ introduction to epidemiology, define attributable risk as the disease rate in exposed persons minus the rate in unexposed persons, and call it the measure of association most relevant to decisions for individuals. The same quantity is often labelled the risk difference or the excess risk; the CDC course on epidemiology in public health practice uses both labels. With risks measured over a stated period, it is calculated as follows.

$$AR = R_e - R_u = R_u \times (RR - 1)$$

where $AR$ is the attributable risk, $R_e$ is the risk of the outcome over the period among exposed people, $R_u$ is the risk over the same period among unexposed people, and $RR = R_e / R_u$ is the relative risk. Coggon, Rose and Barker give the second form, which is the relative excess scaled by the baseline risk.

That scaling is the reason the absolute and relative measures can rank exposures differently. Coggon, Rose and Barker compare a doubling in the risk of death from bronchial carcinoma with a doubling in the risk of death from oral cancer. The relative risk is two in both cases, but the attributable risk is lower for oral cancer because the disease is rarer. When the exposure is a treatment in a trial, the same difference is reported as an absolute risk reduction or absolute risk increase, terms the Cochrane Handbook uses for the risk difference.

From attributable risk to numbers needed to treat for benefit or harm

The reciprocal of an absolute difference in risk gives the expected number of people who must be exposed, or treated, for one additional event. It is the same calculation whether the event is a benefit or a harm. Altman describes the number needed to treat as the reciprocal of the absolute risk difference, and the Cochrane Handbook computes it from the absolute value of the risk difference, rounded up to the next whole number by convention, as follows.

$$NNT = \frac{1}{|AR|}$$

where $NNT$ is the number exposed or treated for one additional event and $|AR|$ is the attributable risk with any minus sign ignored. For a treatment that raises the risk of an adverse event, the result is the number needed to treat for one additional harmful outcome. The Cochrane Handbook recommends the labels NNTB and NNTH to show the direction of effect, and advises against "number needed to harm" because it can be read as the number of people who will be harmed.

Attributable fraction and population attributable risk

Two further measures start from the attributable risk. Dividing it by the risk in the exposed group gives the share of the exposed group's risk that the exposure explains. The CDC course calls this the attributable proportion, defined as follows.

$$AF_e = \frac{R_e - R_u}{R_e} = \frac{AR}{R_e}$$

where $AF_e$ is the attributable fraction among the exposed. That page covers its interpretation; the point here is that it is a proportion with no units, whereas the attributable risk is an absolute excess risk over a stated period.

The population attributable risk moves from the exposed group to the whole population. Coggon, Rose and Barker give it as the attributable risk multiplied by the prevalence of exposure, and describe it as a measure of the potential impact of control measures in a population, as follows.

$$PAR = AR \times p_e = R_{pop} - R_u$$

where $PAR$ is the population attributable risk, $p_e$ is the proportion of the population exposed, and $R_{pop} = p_e R_e + (1 - p_e) R_u$ is the risk in the whole population. The forms agree because $R_{pop} - R_u = p_e (R_e - R_u)$. Dividing the population attributable risk by $R_{pop}$ gives the population attributable fraction, the measure Levin introduced in 1953 to ask what proportion of lung cancer is attributable to cigarette smoking, according to the review of 43 textbooks by Khosravi and colleagues. The same review states that the Levin formula is valid only in the absence of confounding and effect modification.

How attributable risk enters an economic model

Decision models rarely take an attributable risk straight from a study. The NICE Decision Support Unit's Technical Support Document 5 (TSD 5) describes most cost-effectiveness analyses as having two components, a baseline model of the absolute natural history under standard treatment and a model for relative treatment effects, which are combined to give the natural history under the new treatment. Its example applies an odds ratio of 0.8 to a baseline probability of 0.25 and obtains 0.21 on treatment; the implied difference of about 0.04 is the treatment's attributable risk in that population, and it generates the events avoided in the model.

The Cochrane Handbook gives the reason. Relative effects tend to be substantially more stable across risk groups than absolute effects, and even when relative effects are similar across subgroups, absolute effects differ according to baseline risk. A model that carries a relative effect and a baseline risk for each subgroup therefore gives each subgroup its own attributable risk and economic result.

Once the attributable risk is known for a population and a period, the economic arithmetic is linear. The extra events equal the attributable risk multiplied by the number exposed, and each event carries its cost and its loss of QALYs, which feed the incremental cost-effectiveness ratio or a budget impact analysis; attributable cost estimation applies the same logic to costs linked to a condition or exposure. In a Markov model the risks must also match the cycle length. A one-year attributable risk suits an annual transition probability, and another cycle length needs each absolute risk converted through rate-to-probability conversion before the difference is taken; a difference between risks over unequal periods has no meaning.

Worked example of an adverse effect in two risk groups

All figures in this example are illustrative. A long-term drug is suspected of raising the risk of fracture needing hospital care, and a large cohort study, assumed free of confounding, finds a one-year risk of 3.0% among users and 1.8% among comparable non-users. A second group of patients, younger and with a one-year risk of 0.6% without the drug, is assumed to share the same relative risk.

1. Attributable risk. For the first group, 0.030 − 0.018 = 0.012, or 12 extra fractures per 1,000 users a year. The relative risk is 0.030 / 0.018 = 1.667, and the second formula gives the same answer, since 0.018 × 0.6667 = 0.012.

2. Number needed to treat for one additional harmful outcome. 1 / 0.012 = 83.3, rounded up to 84. On average, 84 people take the drug for a year for one extra fracture.

3. Attributable fraction among users. 0.012 / 0.030 = 0.40, so 40% of the fractures among users are attributable to the drug.

4. Population attributable risk. If 25% of the eligible population uses the drug, the population risk is 0.25 × 0.030 + 0.75 × 0.018 = 0.021. The population attributable risk is 0.012 × 0.25 = 0.003, the same as 0.021 − 0.018, or 3 fractures per 1,000 people a year. The population attributable fraction is 0.003 / 0.021 = 0.143.

5. Costs and QALYs. Assume each fracture costs GBP 6,000 to treat and costs 0.10 QALYs. Among 10,000 users, the excess is 0.012 × 10,000 = 120 fractures, costing 120 × 6,000 = 720,000 (GBP) and 120 × 0.10 = 12 QALYs, which is GBP 72 and 0.0012 QALYs per user per year.

6. The lower-risk group. With the same relative risk, the risk among users is 0.006 × 1.667 = 0.010, so the attributable risk is 0.010 − 0.006 = 0.004. Among 10,000 users this is 40 extra fractures, costing GBP 240,000 and 4 QALYs, and the number needed to treat for one additional harmful outcome is 1 / 0.004 = 250.

7. Interpretation. The relative risk is 1.667 in both groups, but the economic harm in the lower-risk group is one third as large, because the attributable risk is. Applying the first group's attributable risk of 0.012 to the second group would put the risk among users at 0.018 instead of 0.010 and triple the modelled harm. If the drug also prevents events, the decision compares the attributable risk of harm with the absolute risk reduction for benefit in the same group, each converted to costs and QALYs.

What attributable risk can and cannot support

The word "attributable" makes a causal claim, and the arithmetic does not test it. For the attributable proportion, which is built on the same difference, the CDC course states that the calculation assumes the unexposed group represents the baseline or expected risk, and that appropriate use depends on a single risk factor being responsible, so the measure may not be appropriate when several risk factors interact. Coggon, Rose and Barker warn that most epidemiological studies are observational and compare people who differ in many ways, which allows confounding. A crude difference from such a study is an association; reading it as an attributable risk requires the reasoning set out under causal inference and an adjusted comparison.

Several narrower errors recur in health economic work.

• Study design. Coggon, Rose and Barker note that relative risk can be estimated from case-control studies whereas attributable risk cannot; such studies sample on the outcome, so they do not measure the risk in each group. An attributable risk for a model then has to be built from a relative measure and an external baseline risk.
• Transport to another population. The Cochrane Handbook notes that a risk difference is naturally constrained, so applying a difference of −0.10 to a group with a baseline risk of 7% gives an impossible probability of −3%. Even when no impossible value appears, carrying a difference unchanged to a population with another baseline risk misstates the excess, as step 6 of the example shows.
• Delay mistaken for prevention. Khosravi and colleagues give the extreme case in which removing an exposure only delays the outcome, so that the fraction of cases caused by the exposure is then 100% while the population attributable fraction is zero. Counting events in a short window can overstate the gain from removing an exposure that mostly brings events forward; a lifetime model values timing through life years and discounting.
• Confused terminology. The textbook review by Khosravi and colleagues lists textbooks that use the labels "attributable risk" and "population attributable risk" for fractions of cases. Uter and Pfahlberg, who use "attributable risk" for Levin's population measure, found that 64.5% of 334 studies reporting it gave no exact definition of the version used. A reported figure should be checked as a difference in risk over a period or a proportion of cases before it enters a model.

Sources

• Coggon D, Rose G, Barker DJP. Epidemiology for the Uninitiated. 4th ed. BMJ Publishing Group; 1997. Chapter 3, Comparing disease rates. Online edition: https://www.bmj.com/about-bmj/resources-readers/publications/epidemiology-uninitiated/3-comparing-disease-rates.
• Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions, version 6.5. Cochrane; 2024. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06
• Schünemann HJ, Vist GE, Higgins JPT, Santesso N, Deeks JJ, Glasziou P, Akl EA, Guyatt GH. Chapter 15: Interpreting results and drawing conclusions. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions, version 6.5. Cochrane; 2024. https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-15
• Altman DG. Confidence intervals for the number needed to treat. BMJ. 1998;317(7168):1309-1312. https://doi.org/10.1136/bmj.317.7168.1309
• Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice, third edition (SS1978). Lesson 3: Measures of Risk, section 6. CDC; last reviewed 2012. https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/section6.html
• Dias S, Welton NJ, Sutton AJ, Ades AE. NICE DSU Technical Support Document 5: Evidence synthesis in the baseline natural history model. Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated April 2012. https://doi.org/10.15131/shef.data.33737536
• 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
• Uter W, Pfahlberg A. The application of methods to quantify attributable risk in medical practice. Statistical Methods in Medical Research. 2001;10(3):231-237. https://doi.org/10.1177/096228020101000305