To the Editor:
In epidemiology, the attributable fraction (AF; also known as the attributable risk1 and excess fraction2) is used widely to assess the effect of exposure on a disease. It was introduced by Levin3 and is defined as the fraction of disease events that would be eliminated if the exposure were eliminated. We present a simple formula for sensitivity analysis of unmeasured confounding in estimation of the AF, for a dichotomous and non–time-varying exposure with constant unmeasured confounding across all levels of measured confounders. (A more general version, using the parametric g-formula, has been presented previously.4)
We use the following notation. Let A denote the exposure status of a person and Y the observed outcome for that person. Suppose that A and Y are dichotomous. Let X denote a measured confounder or a set of measured confounders. Let Ya denote the potential outcome of Y for an individual if the exposure A, perhaps contrary to fact, had been set to the value a. Then, AF is defined as
where Pr(Y = 1) is the disease prevalence and Pr(Y0 = 1) is the exposure-free prevalence, ie, the hypothetical probability of disease in the same population, but with all exposure eliminated.1
To propose a sensitivity analysis formula for unmeasured confounding, we apply the sensitivity parameter, which was originally introduced by Brumback et al5 to estimate causal differences, with the following formula:
which does not vary between the strata of X. This δ corresponds to the confounding risk difference with the exposed group as the target population.6,7
The true AF is expressed as the difference between the initial estimate adjusted only for X and a function of δ, under certain assumptions. This result is as follows:
where AFO is the initial estimate adjusted only for X, and has the following formula:
The derivation of this result is given in the eAppendix (http://links.lww.com/EDE/A537). Result 1 shows that the AF estimate adjusted only for measured confounders will give a good approximation to the true AF if Pr(A = 1) is small and Pr(Y = 1) is large, even when unmeasured confounders exist.
Result 1 suggests that a sensitivity analysis can be conducted easily. The sensitivity parameter δ is set by the investigator according to what is thought to be plausible. The analysis can be varied over a range of plausible values to examine how conclusions vary according to various parameter values. We can readily display the results of the sensitivity analysis graphically, where the horizontal axis represents the sensitivity parameter and the vertical axis represents the true AF. To obtain the confidence interval (CI) of the true AF for fixed values of δ, we must evaluate the variance of the second term in Result 1, in addition to that of AFO. This is somewhat troublesome. A simple method to obtain the CI is to regard the estimates of Pr(A = 1) and Pr(Y = 1) as fixed values; then, the variance of the second term in Result 1 is 0. Although this rough method yields a smaller CI than the true CI, the difference is trivial when the investigator wants to examine small values of |δ|.
The simple sensitivity-analysis formula presented here does not assume any particular method, model, or functional form to yield the initial estimate adjusted for measured confounders. Therefore, we can immediately use a number of recent and attractive methods, such as doubly robust estimation,8 developed with the R program.
Division of Biostatistics
Clinical Research Center
Kinki University School of Medicine
1. Benichou J. Attributable risk. In: Armitage P, Colton T eds. Encyclopedia of Biostatistics. 2nd ed. Chichester, United Kingdom: John Wiley & Sons; 2005.
2. Rothman KJ, Greenland S, Lash TL. Modern Epidemiology. 3rd ed. Philadelphia: Lippincott Williams & Wilkins; 2008.
3. Levin ML. The occurrence of lung cancer in man. Acta Unio Int Contra Cancrum. 1953;9:531–541.
4. Taubman SL, Robins JM, Mittleman MA, Hernán MA. Intervening on risk factors for coronary heart disease: an application of the parametric g-formula. Int J Epidemiol. 2009;38:1599–1611.
5. Brumback BA, Hernán MA, Haneuse SJ, Robins JM. Sensitivity analyses for unmeasured confounding assuming a marginal structural model for repeated measures. Stat Med. 2004;23:749–767.
6. Arah OA, Chiba Y, Greenland S. Bias formulas for external adjustment and sensitivity analysis of unmeasured confounders. Ann Epidemiol. 2008;18:637–646.
7. VanderWeele TJ, Arah OA. Bias formulas for sensitivity analysis of unmeasured confounding for general outcomes, treatments, and confounders. Epidemiology. 2011;22:42–52.
8. Sjölander A, Vansteelandt S. Doubly robust estimation of attributable fractions. Biostatistics. 2011;12:112–121.