Attributable risk

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Population attributable fraction (PAF) of an exposure agent is the fraction of disease that would disappear if the exposure to that agent would disappear.

Question

How to calculate population attributable fraction?

Rationale

Based on WHO PAF is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PAF = \frac{\Sigma_{i=1}^n P_i RR_i - \Sigma_{i=1}^n P'_i RR_i}{\Sigma_{i=1}^n P_i RR_i}}

where i is a certain exposure level, P is the fraction of population in that exposure level, RR is the relative risk at that exposure level, and P' is the fraction of population in a counterfactual ideal situation (where the exposure is typically lower).

Based on this, we can limit our examination to a situation where there are only two population groups, one exposed to background level (with relative risk 1) and the other exposed to a higher level (with relative risk RR). In the counterfactual situation nobody is exposed. Thus, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PAF = \frac{(P RR + (1-P)*1) - (0*RR + 1*1)}{P RR + (1-P)*1}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PAF = \frac{P RR - P}{P RR + 1 - P}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PAF = \frac{P(RR - 1)}{P(RR -1) + 1}}

This equation is used in e.g. Health impact assessment.