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Population attributable fraction: names, types and issues with incorrect interpretation of relative risks
  1. Belen Zapata-Diomedi1,
  2. Jan J Barendregt2,3,
  3. J Lennert Veerman3
  1. 1 School of Public Health, The University of Queensland, Brisbane, Queensland, Australia
  2. 2 Epigear International, Sunrise Beach, Queensland, Australia
  3. 3 School of Public Health, The University of Queensland, Brisbane, Queensland, Australia
  1. Correspondence to Belen Zapata-Diomedi, School of Public Health, The University of Queensland, Level 2, Public Health Building, Herston, Brisbane, QLD 4006, Australia; b.zapatadiomedi{at}

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The development of the original population attributable fraction (PAF) dates back to 19531 and it has been widely used, misused and miscalled since then. We discuss two main issues here: use of appropriate terminology and calculations of the PAF, and of the potential impact fraction (PIF).

The PAF is the proportion of cases for an outcome of interest that can be attributed to a given risk factor among the entire population.2 Despite this clear description, it is not rare to find studies that call it population attributable risk (PAR). PAR is the difference in the rate or risk of disease for the population compared to the unexposed.Embedded Image 1 Embedded Image 2

PAR is only one of the other names being used; as noted previously,3 ,4 there is great vagueness in the use of terminology. Other common names used are ‘population attributable risk percent’, ‘excess fraction’, ‘etiological fraction’ and ‘attributable fraction’.

Equation two can also be expressed as:Embedded Image 3

Where p is the prevalence of the risk factor and RR is the relative risk of incidence of the disease of the exposed over the non-exposed. When multiple categories of exposure to the risk factor are evaluated, the following formula applies.Embedded Image 4

Where i is the level of exposure and n the number of exposure categories. Rockhill et al 5 presented a table with commonly applied formulas to calculate the PAF, depending on whether there is need to adjust for confounding or not.

Formula 3 and 4 are useful for assessing the impact of only one risk factor, however, the PAF of a group of risk factors for the same disease can be assessed as follows:Embedded Image 5

Where r is the risk factor index. It is not correct to add up the various single risk factor PAFs; doing so may result in sums greater than one.5

The equations above compare current risk factor exposure to a counterfactual zero exposure. When the counterfactual is not zero, a broader measure should be used, the PIF. This situation is not unusual, for instance, for the case of measuring the PAF of high blood pressure, a non-zero theoretical minimum6 is used as the counterfactual as zero blood pressure is not a valid one. The PIF is the proportional change in disease incidence when exposure to a risk factor is changed7 and its equation is as follows:Embedded Image 6

And when assessing a risk factor with multiple categories:Embedded Image 7

Where p is the prevalence of the risk factor and p′ the counterfactual, RR is the relative risk of the exposed compared to the reference level of exposure. From equation 6 and 7 it is possible to see that the PAF is a special case of the PIF when the counterfactual exposure to the risk factor p′ is 0. Barendregt and Veerman7 discussed three alternative methods to calculate the PIF, and named the traditional approach as in equation 7 ‘proportional shift’ where the risk factor exposure and relative risk are both categorical. An alternative to the proportional shift, called ‘distribution shift’, assumes the risk factor and relative risk have continuous distributions. And a third case for categorical exposure in which the RR of the category changes rather than the prevalence (‘RR shift’). The distribution shift method should be applied for cases in which risk factors are better described with a continuous rather than discrete distribution. For instance, traditionally body mass index (BMI) is assessed in terms of categories (normal, overweight and obese) with an associated relative risk for each category, however, BMI is better measured as a continuous distribution.8 Barendregt and Veerman7 demonstrated that the use of categorical distributions for continuously distributed risk factors, such as BMI, overestimates or underestimates the PIF due to the introduction of artefactual non-linearities. In the latest Global Burden of Disease study9 several risk factors are continuously distributed with corresponding continuous relative risks (per unit change), instead of categories. Examples are BMI and blood pressure. Inappropriate interpretation of continuous risk factors and corresponding ‘per unit’` relative risks leads to wrong conclusions. An example of this is explained in a recent letter to the editor by Zapata Diomedi and Barendregt.10

The field is still developing. Now if only epidemiologists could agree on terminology, and apply methods and interpret results correctly, that would constitute real progress.



  • Contributors BZ-D drafted the manuscript with the assistance of JJB. JJB and JLV commented on the manuscript and edited manuscript. BZ-D finalised the manuscript for submission.

  • Competing interests None declared.

  • Provenance and peer review Not commissioned; externally peer reviewed.