To the Editor:
Ding and VanderWeele’s article on sensitivity analysis for uncontrolled confounding published in this journal was a major step forward for quantitative bias analysis.1 Given an observed effect estimate, the authors define a bound (B) for the impact of uncontrolled confounding. We make the observation that, while not equivalent in terms of subject matter, there is a simple algebraic equivalence between the calculations of B and the familiar epidemiologic measure of population attributable fraction (PAF), which measures “the fraction of all cases of a disease or other adverse condition in a population that is attributable to a specific exposure.”2
As B is a relative risk (RR) and PAF an absolute proportion (P) we use the association between these measures (assuming RR ≥ 1; if RR < 1 the inverse can be used).
B is calculated using:
where RReu is the relative relationship between the exposure and uncontrolled confounding and RRud is the relative relationship between unobserved confounding and the outcome.1
PAF is calculated as:2,3
where RRE, is the risk of disease in the exposed relative to the risk in those unexposed and PT is the proportion of those with the disease who are exposed. Given (2) and (4a) can be rewritten as:
where RPT is the total number (or total proportion) of cases relative to the number (or proportion) of cases in the nonexposed.
Substituting RReu for RPT and RRud for RRE (or vice versa), (4b) can be used to calculate the “attributable proportion (4) of B (APB),” which can, in turn, be used to calculate B using (1). Similarly, (3) can be used to calculate a relative PAF for conversion to the absolute PAF using (2).
To illustrate we use hypothetical data where the 1-year risk of lung cancer among smokers is 39/30,000 = 0.0013 and among nonsmokers is 6/60,000 = 0.0001.4 These data give RRE = 13 and RPT, the ratio of total to nonexposed cases, 45/6 = 7.5. Using (4b) gives a relative PAF of five, which corresponds to an absolute PAF of (5-1)/5 = 0.8 or 80%. Setting RPeu = 13 and RRud = 7.5 in (3) gives the same result. So 80% of lung cancer in this population is due to smoking. Equivalently, on a relative scale, the lung cancer risk in this population is five times greater compared to one where no one smokes.
The same values can also be applied to the uncontrolled confounding paradigm. For a known exposure-outcome association consider an unobserved confounding factor that has an association with the outcome of 13 (RRud) and is 7.5 times more common in those with the exposure of interest (RReu). Using (3), B = 5. B is a measure of the impact of uncontrolled confounding, so dividing the observed exposure-outcome RR by B gives an estimate of the association after adjustment for the unmeasured confounder. Equivalently, the “attributable fraction of B,” from (4b), gives the same result as an 80% relative reduction.
Although this equivalence is of interest algebraically, there may also be practical implications. The E-value is the minimum value of both RReu and RRud (assuming they are equal) required to reduce an observed RR to exactly one.5 Importantly, the E-value is calculated just from RR:
For example, for an observed RR of three to be completely explained by confounding would require both RReu and RRud to be at least 5.45, which, in many circumstances, is unlikely.
Similarly, in PAF terms, for a known population risk of disease relative to those unexposed to a risk factor, (5) could be used to calculate the minimum values of RPT and RRE needed for the exposure to completely explain the excess of disease in the population and, therefore, the potential impact on population health of removing or eliminating the exposure.
Frank Popham and Elise Whitley
MRC/CSO Social and Public Health Sciences Unit, University of Glasgow, Glasgow, Scotland
1. Ding P, VanderWeele TJ. Sensitivity analysis without assumptions. Epidemiology. 2016;27:368–377.
2. Mansournia MA, Altman DG. Population attributable fraction. BMJ. 2018;360:k757.
3. Rockhill B, Newman B, Weinberg C. Use and misuse of population attributable fractions. Am J Public Health. 1998;88:15–19.
4. Kirkwood BR, Sterne J. Essential Medical Statistics. 2003.2nd ed. Malden, MA: Blackwell Science.
5. VanderWeele TJ, Ding P. Sensitivity analysis in observational research: introducing the E-value. Ann Intern Med. 2017;167:268–274.