Difference between revisions of "Attributable risk"
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==Answer== | ==Answer== | ||
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| + | Probably this is the most useful form of population attributable fraction (PAF or AF<sub>p</sub>) for impact assessment: | ||
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| + | <math>AF_p = \Sigma_i p_{c,i} \frac{p_i(RR_i - 1)}{p_i(RR_i - 1) + 1},</math> | ||
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| + | where | ||
| + | * p<sub>c,i</sub> is the proportion of '''cases''' falling in subgroup i (so that Σ<sub>i</sub>p<sub>c,i</sub> = 1), | ||
| + | * p<sub>i</sub> is the fraction of '''exposed''' people within subgroup i (and 1-p<sub>i</sub> is the fraction of unexposed), | ||
| + | * RR<sub>i</sub> is the risk ratio for subgroup i due to the subgroup-specific exposure level (assuming that everyone in that subgroup is exposed to that level or none). {{comment|# |Does the equation hold if exposure level varies between groups?|--[[User:Jouni|Jouni]] ([[User talk:Jouni|talk]]) 15:43, 25 April 2014 (EEST)}} | ||
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| + | This page does not contain [[R]] code. Instead, it is written as part of the model in [[Health impact assessment]]. | ||
==Rationale== | ==Rationale== | ||
Revision as of 12:43, 25 April 2014
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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?
Answer
Probably this is the most useful form of population attributable fraction (PAF or AFp) for impact assessment:
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF_p = \Sigma_i p_{c,i} \frac{p_i(RR_i - 1)}{p_i(RR_i - 1) + 1},
where
- pc,i is the proportion of cases falling in subgroup i (so that Σipc,i = 1),
- pi is the fraction of exposed people within subgroup i (and 1-pi is the fraction of unexposed),
- RRi is the risk ratio for subgroup i due to the subgroup-specific exposure level (assuming that everyone in that subgroup is exposed to that level or none). --# : Does the equation hold if exposure level varies between groups? --Jouni (talk) 15:43, 25 April 2014 (EEST)
This page does not contain R code. Instead, it is written as part of the model in Health impact assessment.
Rationale
WHO approach
[1] PAF is
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{(P RR + (1-P)*1) - (0*RR + 1*1)}{P RR + (1-P)*1}
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{P RR - P}{P RR + 1 - P}
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{P(RR - 1)}{P(RR -1) + 1}
See also Attributable risk, although it is a stub.
This equation is used in e.g. Health impact assessment.
Rothman approach
Modern Epidemiology [2] is the authoritative source of epidemiology. They first define attributable fraction AF for a cohort of people (pages 295-297). It is the fraction of cases among the exposed that would not have occurred if the exposure would not have taken place:
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF = \frac{RR - 1}{RR},
where RR is the causal risk ratio.
The population attributable fraction AFp is that fraction among the whole cohort:
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF_p = \frac{N_1 (R_1 - R_0)}{N_1 R_1 + N_0 R_0} = \frac{N_1 (R_1 - R_0)/R_0}{N_1 R_1/R_0 + N_0 R_0/R_0} = \frac{N_1 (RR - 1)}{N_1 RR + N_0}
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): = \frac{ \frac{N_1 (RR - 1)}{N_1 + N_0} }{ \frac{N_1 RR + N_0}{N_1 + N_0}} = \frac{ p (RR - 1) }{ \frac{N_1 RR - N_1 + (N_1 + N_0)}{N_1 + N_0}} = \frac{p (RR - 1)}{p RR - p + 1} = \frac{p (RR - 1)}{p (RR - 1) + 1},
where
- N1 and N0 are the numbers of exposed and unexposed people, respectively,
- R1 and R0 are the risks of disease in the exposed and unexposed group, respectively, and RR = R1 / R0,
- p is the fraction of exposed people among the whole cohort.
Note that there is a typo in the Modern Epidemiology book: the denominator should be p(RR-1)+1, not p(RR-1)-1.
Population attributable fraction can be calculated as a weighted average based on subgroup data:
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF_p = \Sigma_i p_i AF_{pi},
where
- pi is the proportion of cases falling in stratum (subgroup) i,
- AFpi is the population attributable fraction calculated for the subgroup.
Specifically, we can divide the cohort into subgroups based on exposure (in the simplest case exposed and unexposed), so we get
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF_p = p_c \frac{1(RR - 1)}{1(RR - 1) + 1} + (1 - p_c) \frac{0(RR - 1)}{0(RR - 1) +1} = p_c \frac{RR - 1}{RR},
where pc is the proportion of cases in the exposed group among all cases; this is the same as exposure prevalence among cases.
