Attributable risk

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Attributable risk is a fraction of total risk that can be attributed to a particular cause. There are a few different ways to calculate it. Population attributable fraction of an exposure agent is the fraction of disease that would disappear if the exposure to that agent would disappear in a population. Etiologic fraction is the fraction of cases that have occurred earlier than they would have occurred (if at all) without exposure. Etiologic fracion cannot typically be calculated based on risk ratio (RR) alone, but it requires knowledge about biological mechanisms.

Question

How to calculate attributable risk? What different approaches there are, and what are their differences in interpretation and use?

Answer

Risk ratio (RR)

risk among the exposed divided by the risk among the non-exposed

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 RR = \frac{R_1}{R_0}.}

Attributable fraction

the fraction of cases among the exposed that would not have occurred if the exposure would not have taken place:

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 AF = \frac{RR - 1}{RR}}

Population attributable fraction

the fraction of cases among the whole population that would not have occurred if the exposure would not have taken place. The most useful formulas are

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 = 1 - \frac{1}{\sum_{i=0}^k p_i (RR_i)}}

for use with several population subgroups (typically with different exposure levels). Not valid when confounding exists. Subscript i refers to the ith subgroup. p_i = proportion of source population in ith subgroup.

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 = 1- \sum_{i=0}^k \frac{pd_i}{RR_i} = \sum_{i=1}^kW_i\frac{p_i(RR_i-1)}{1+p_i(RR_i-1)}}

which produces valid estimates when confounding exists but with a problem that often pd_i = proportion of cases in ith subgroup of all cases is not known. W_i is the proportion of cases in subgroup i and p_i is the proportion of the population in subgroup i that is exposed. ⇤--#: . Is pd_i = W_i? Or is W the proportion of cases of all people in subgroup i? --Jouni (talk) 13:42, 7 April 2016 (UTC) (type: truth; paradigms: science: attack)

Etiologic fraction

Fraction of cases among the exposed that would have occurred later (if at all) if the exposure would not have taken place. It cannot be calculated without understanding of the biological mechanism, but it is always between

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 \frac{RR-1}{RR^{RR/(RR-1)}}} and 1.

Rationale

Attributable fraction

Rockhill et al. give an extensive description about different ways to calculate PAF and assumptions needed in each approach.

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 \frac{IP_1 - IP_0}{IP_1}}

is empirical approximation of

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 \frac{P(D) - \sum_C P(D|C, \bar{E}) P(C)}{P(D)}}

where IP₁ = cumulative proportion of total population developing disease over specified interval; IP_O = cumulative proportion of unexposed persons who develop disease over interval. Valid only when no confounding of exposure(s) of interest exists. If disease is rare over time interval, ratio of average incidence rates I_O/I₁ approximates ratio of cumulative incidence proportions, and thus formula can be written as (I₁ - I₀)I₁. Both formulations found in many widely used epidemiology textbooks.

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 \frac{p_e(RR-1)}{p_e(RR-1)+1}}

Transformation of formula 1. Not valid when there is confounding of exposure-disease association. p_e = proportion of source population exposed to the factor of interest. RR may be ratio of two cumulative incidence proportions (risk ratio), two (average) incidence rates (rate ratio), or an approximation of one of these ratios. Found in many widely used epidemiology texts, but often with no warning about invalidness when confounding exists.

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 \frac{\sum_{i=0}^k p_i (RR_i - 1)}{1 + \sum_{i=0}^k p_i (RR_i - 1)} = 1 - \frac{1}{\sum_{i=0}^k p_i (RR_i)}}

Extension of formula 2 for use with multicategory exposures. Not valid when confounding exists. Subscript i refers to the ith exposure level. p_i = proportion of source population in ith exposure level, RR_j = relative risk comparing ith exposure level with unexposed group (i = 0). Derived by Walter; given in Kleinbaum et al. but not in other widely used epidemiology texts.

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 pd(\frac{RR-1}{RR})}

Alternative expression. Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used. pd = proportion of cases exposed to risk factor. In Kleinbaum et al. and Schlesselman.

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 \sum_{i=0}^k pd_i (\frac{RR_i - 1}{RR_i}) = 1- \sum_{i=0}^k \frac{pd_i}{RR_i}}

Extension of formula 4 for use with multicategory exposures. Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used. pd_i = proportion of cases falling into ith exposure level; RR_i = relative risk comparing ith exposure level with unexposed group (i = 0). See Bruzzi et al. and Miettinen for discussion and derivations; in Kleinbaum et al. and Schlesselman.

Probably this is the most useful form of population attributable fraction (PAF or AF_p) for impact assessment:

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 AF_p = \Sigma_i p_{ci} \frac{p_i(RR_i - 1)}{p_i(RR_i - 1) + 1},}

where

• p_ci is the proportion of cases falling in subgroup i (so that Σ_ip_ci = 1),
• p_i is the fraction of exposed people within subgroup i (and 1-p_i is the fraction of unexposed),
• RR_i 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) (type: truth; paradigms: science: comment)

p_ci can be calculated for each subgroup with the following equation if the background risk of disease is equal in all subgroups (and thus cancels out):

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 p_{ci} = \frac{N_i \Pi_j RR_{i,j}}{\Sigma_i N_i \Pi_j RR_{i,j}},}

where

• N_i is the number of people in each subgroup i,
• RR_i,j is the risk ratio in subgroup i due to pollutant j (accounting for the estimated exposure in the subgroup). Note that this assumes that multiplicative assumption holds between different pollutant effects.

This page does not contain R code. Instead, it is written as part of the model in Health impact assessment.

Etiologic fraction

Etiologic fraction is defined as the fraction of cases that is advanced in time because of exposure. It can also be called probability of causation, which has importance in court. Its exact value cannot be estimated directly from risk ratio (RR) because some knowledge is needed about biological mechanisms (more precisely: timing of disease). In any case, the etiologic fraction always lies between f and 1, when f 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 \frac{RR - 1}{RR^{RR/(RR-1)}}.}

The code below calculates the attributable fraction and lower and upper bounds of the etiological fraction for user-defined RRs.

Comparison

What is (are) the relative risk(s), i.e. RR?: + Show code- Hide code library(OpasnetUtils) AF <- function(x) {return(data.frame(RR = x, AF = (x-1)/x, EF_lower = (x-1)/x^(x/(x-1)), EF_upper = 1))} oprint(AF(RR))

Impact of confounders

The problem with the two PAF equations (see [[#Answer|]]) is that the former has easier-to-collect input, but it is not valid if there is confounding. It is still often mistakenly used. The latter equation would produce an unbiased estimate, but the data needed is harder to collect. Darrow and Steenland have studied the impact of confounding on the bias in attributable fraction. This is their summary:

The impact of confounding on the bias in attributable fraction.

Bias in attributable fraction	Confounding in RR	Confounding in inputs
AF bias (-), calculated AF is smaller than true AF	Conf RR (+), crude RR is larger than adjusted (true) RR	Confounder is positively associated with exposure and disease (++)
		Confounder is negatively associated with exposure and disease (--)
AF bias (+), calculated AF is larger than true AF	Conf RR (-), crude RR is smaller than adjusted (true) RR	Confounder is negatively associated with exposure and positively with disease (-+)
		Confounder is positively associated with exposure and negatively with disease (+-)

WHO approach

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}}

See also Attributable risk, although it is a stub.

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

Rothman approach

Modern Epidemiology 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 (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 AF = \frac{RR - 1}{RR},}

where RR is the causal risk ratio.

The population attributable fraction AF_p is that fraction among the whole cohort:

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 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 (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 = \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 (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 AF_p = \Sigma_i p_{ci} AF_{pi},}

where

• p_ci is the proportion of cases falling in stratum (subgroup) i,
• AF_pi 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 (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 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 p_c is the proportion of cases in the exposed group among all cases; this is the same as exposure prevalence among cases.

p_c can be calculated by first calculating number of cases in each subgroup:

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 cases_i = N_i * background * \Pi_j e^{ln(ERF_{j}) exposure_{i,j}},}

where

• cases_i is the number of cases in subgroup i,
• N_i is the number of people in subgroup i,
• background is the background risk of the disease in the unexposed; we assume that it is the same in all subgroups,
• ERF_j is the risk ratio for unit exposure for each pollutant j (if the exposure response function ERF assumes another form than relative risk, i.e. exponential, then another equations must be used),
• exposure_i,j is the amount of exposure in a subgroup i to pollutant j.

Therefore,

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 p_{ci} = \frac{cases_i}{\Sigma_i cases_i} = \frac{N_i * background * \Pi e^{ln(ERF_{j}) exposure_{i,j}}}{background \Sigma N_i \Pi e^{ln(ERF_{j}) exposure_{i,j}}}}

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 p_{ci} = \frac{N_i \Pi_j RR_{i,j}}{\Sigma_i N_i \Pi_j RR_{i,j}},}

where RR_i,j = exp(ln(ERF_j) exposure_i,j).

In addition, if only fraction p of the population is exposed, for the whole population 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 RR = \frac{p * N * background * RR_{exposed} + (1-p) * N * background * RR_{unexposed}}{N * background * RR_{unexposed}}}

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 = \frac{p e^{ln(ERF)exposure} + (1-p)1}{1} = p e^{ln(ERF)exposure} -p + 1}

Calculations

⇤--#: . UPDATE AF TO REFLECT THE CURRENT IMPLEMENTATION OF ERF Exposure-response function --Jouni (talk) 05:20, 13 June 2015 (UTC) (type: truth; paradigms: science: attack)

+ Show code- Hide code library(OpasnetUtils) # UPDATE AF TO REFLECT THE CURRENT IMPLEMENTATION OF ERF [[Exposure-response function]] ### ESTIMATES OF ATTRIBUTABLE CASES BASED ON ATTRIBUTABLE FRACTION # We estimate the number of cases and their attributable causes based on [[Population attributable fraction]]. AF <- Ovariable("AF", # Cases attributed to specific (combinations of) causal exposures. dependencies = data.frame(Name = c( "ERF", # Exposure-response function "exposure", # Total exposure to an agent or pollutant "frexposed", # fraction of population that is exposed "bgexposure" # Background exposure to an agent (a level below which you cannot get in practice) )), formula = function(...) { # First calculate risk ratio and remove redundant columns because they cause harm when operated with itself. RR <- frexposed * exp(log(ERF) * (exposure - bgexposure)) - frexposed + 1 PAF <- (RR - 1) / unkeep(RR, sources = TRUE, prevresults = TRUE) # pollutants is a vector of pollutants considered. pollutants <- as.character(unique(exposure@output$Pollutant)) expname <- paste(exposure@name, "Result", sep = "") out <- 1 for(i in 1:length(pollutants)) { # Attributable fraction of a particular pollutant is combined with all pollutant AFs. # The combination has 2^n rows (n = number of pollutants). Pollutant is either + or - depending on # whether it caused the disease or not. temp <- Ovariable("temp", data = data.frame( Pollutant = pollutants[i], Temp1 = c(paste(pollutants[i], "-", sep = ""), paste(pollutants[i], "+", sep = "")), Result = c(-1, 1) # Non-causes are temporarily marked with negative numbers. )) temp <- temp * PAF # Non-causes are given the remainder (1-AF) of temporary attributable fraction AF. result(temp) <- ifelse(result(temp) > 0, result(temp), 1 + result(temp)) # Causes with 0 AF are marked 1. This must be corrected. result(temp) <- ifelse(result(temp) == 1 & grepl("\\+", temp@output$Temp1), 0, result(temp)) #If exists, the exposureResult is renamed so that it can be kept without side effects. #These should not be marginals but there seems to be problems in this respect. if(expname != "Result"){ colnames(temp@output)[colnames(temp@output) == expname] <- paste("expo", pollutants[i], sep = "") } out <- out * temp out <- unkeep(out, cols = "Pollutant", sources = TRUE, prevresults = TRUE) # Combine and rename columns. if(i == 1) { colnames(out@output)[colnames(out@output) == "Temp1"] <- "Causes" } else { out@output$Causes <- paste(out@output$Causes, out@output$Temp1) out@output$Temp1 <- NULL } } return(out) } ) objects.store(AF) cat("Ovariable AF stored.\n")