View source for Attributable risk

[[Category:Health impact]]
{{method|moderator=Jouni}}
'''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==

=== Attributable fraction ===

Also known as population attributable fraction (PAF). {{comment|# |What is the actual difference?|--[[User:Jouni|Jouni]] ([[User talk:Jouni|talk]]) 08:38, 7 April 2016 (UTC)}} Rockhill et al.<ref name="rockhill"/> give an extensive description about different ways to calculate PAF and assumptions needed in each approach.

<math>\frac{IP_1 - IP_0}{IP_1}</math>

is empirical approximation of 

<math>\frac{P(D) - \sum_C P(D|C, \bar{E}) P(C)}{P(D)}</math>

where IP<sub>1</sub> = cumulative proportion of total population developing disease over specified interval; IP<sub>O</sub> = 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<sub>O</sub>/I<sub>1</sub> approximates ratio of cumulative incidence proportions, and thus formula can be written as (I<sub>1</sub> - I<sub>0</sub>)I<sub>1</sub>. Both formulations found in many widely used epidemiology textbooks.

<math>\frac{p_e(RR-1)}{p_e(RR-1)+1}</math>

Transformation of formula 1. Not valid when there is confounding of exposure-disease association. p<sub>e</sub> = 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.

<math>\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)}</math>

Extension of formula 2 for use with multicategory exposures. Not valid when confounding exists. Subscript i refers to the ith exposure level. p<sub>i</sub> = proportion of source population in ith exposure level, RR<sub>j</sub> = relative risk comparing ith exposure level with unexposed group (i = 0). Derived by Walter<ref name="walter">Walter SD. The estimation and interpretation of attributable fraction in health research. Biometrics. 1976;32:829-849.</ref>; given in Kleinbaum et al.<ref name="kleinbaum">Kleinbaum DG, Kupper LL, Morgenstem H. Epidemiologic Research. Belmont, Calif: Lifetime Learning Publications; 1982:163.</ref> but not in other widely used epidemiology texts.

<math>pd(\frac{RR-1}{RR})</math>

Alternative expression. Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used.<ref name="miettinen">Miettinen 0. Proportion of disease caused or
prevented by a given exposure, trait, or intervention. Am JEpidemiol. 1974;99:325-332.</ref> pd = proportion of cases exposed to risk factor. In Kleinbaum et al.<ref name="kleinbaum"/> and Schlesselman.<ref name="schlesselman">Schlesselman JJ. Case-Control Studies: Design, Conduct, Analysis. New York, NY: Oxford University Press Inc; 1982.</ref>

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

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<sub>i</sub> = proportion of cases falling into ith exposure level; RR<sub>i</sub> = relative risk comparing ith exposure level with unexposed group (i = 0). See Bruzzi et al. <ref name="bruzzi">Bruzzi P, Green SB, Byar DP, Brinton LA, Schairer C. Estimating the population attributable risk for multiple risk factors using case-control data. Am J Epidemiol. 1985; 122: 904-914.</ref> and Miettinen<ref name="miettinen"/> for discussion and derivations; in Kleinbaum et al.<ref name="kleinbaum"/> and Schlesselman.<ref name="schlesselman"/>

Probably this is the most useful form of population attributable fraction (PAF or AF<sub>p</sub>) for impact assessment:

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

where 
* p<sub>ci</sub> is the proportion of '''cases''' falling in subgroup i (so that &Sigma;<sub>i</sub>p<sub>ci</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). 
<ref name="rockhill">Rockhill B, Newman B, Weinberg C. use and misuse of population attributable fractions. American Journal of Public Health 1998: 88 (1) 15-19.[http://www.ncbi.nlm.nih.gov/pubmed/9584027]</ref>{{comment|# |Does the equation hold if exposure level varies between groups?|--[[User:Jouni|Jouni]] ([[User talk:Jouni|talk]]) 15:43, 25 April 2014 (EEST)}}

p<sub>ci</sub> 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):

<math>p_{ci} = \frac{N_i \Pi_j RR_{i,j}}{\Sigma_i N_i \Pi_j RR_{i,j}},</math>

where
* N<sub>i</sub> is the number of people in each subgroup i,
* RR<sub>i,j</sub> 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.<ref name="robins">Robins JM, Greenland S. Estimability and estimation of excess and etiologic fractions. Statistics in Medicine 1989 (8) 845-859.</ref>  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

<math>\frac{RR - 1}{RR^{RR/(RR-1)}}.</math>

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

=== Comparison ===

<rcode label="Compare attributable and etiologic fractions" embed=1 variables="name:RR|description:What is (are) the relative risk(s), i.e. RR?|default:c(1, 1.02, 1.3, 1,5, 2, 3)">
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))
</rcode>


==Rationale==

=== WHO approach ===

<ref>WHO: Health statistics and health information systems. [http://www.who.int/healthinfo/global_burden_disease/metrics_paf/en/index.html]. Accessed 16 Nov 2013.</ref>
PAF is

<math>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}</math>

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

<math>PAF = \frac{(P RR + (1-P)*1) - (0*RR + 1*1)}{P RR + (1-P)*1}</math>

<math>PAF = \frac{P RR - P}{P RR + 1 - P}</math>

<math>PAF = \frac{P(RR - 1)}{P(RR -1) + 1}</math>

See also [[:en:Attributable risk|Attributable risk]], although it is a stub.

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

=== Rothman approach ===

Modern Epidemiology
<ref>Kenneth J. Rothman, Sander Greenland, Timothy L. Lash: Modern Epidemiology. Lippincott Williams & Wilkins, 2008. 758 pages.</ref>
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:

<math>AF = \frac{RR - 1}{RR},</math>

where RR is the causal risk ratio.

The ''population attributable fraction AF<sub>p</sub>'' is that fraction among the whole cohort:

<math>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}</math> 

<math>= \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},</math> 

where
* N<sub>1</sub> and N<sub>0</sub> are the numbers of exposed and unexposed people, respectively,
* R<sub>1</sub> and R<sub>0</sub> are the risks of disease in the exposed and unexposed group, respectively, and RR = R<sub>1</sub> / R<sub>0</sub>,
* 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:

<math>AF_p = \Sigma_i p_{ci} AF_{pi},</math>

where
* p<sub>ci</sub> is the proportion of '''cases''' falling in stratum (subgroup) i,
* AF<sub>pi</sub> 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

<math>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},</math>

where p<sub>c</sub> is the proportion of cases in the exposed group among all cases; this is the same as exposure prevalence among cases.

p<sub>c</sub> can be calculated by first calculating number of cases in each subgroup:

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

where
* cases<sub>i</sub> is the number of cases in subgroup i,
* N<sub>i</sub> 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<sub>j</sub> 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<sub>i,j</sub> is the amount of exposure in a subgroup i to pollutant j.

Therefore, 

<math>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}}}</math>

<math>p_{ci} = \frac{N_i \Pi_j RR_{i,j}}{\Sigma_i N_i \Pi_j RR_{i,j}},</math>

where RR<sub>i,j</sub> = exp(ln(ERF<sub>j</sub>) exposure<sub>i,j</sub>).

In addition, if only fraction p of the population is exposed, for the whole population we get

<math>RR = \frac{p * N * background * RR_{exposed} + (1-p) * N * background * RR_{unexposed}}{N * background * RR_{unexposed}}</math>

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

=== Calculations ===

{{attack|# |UPDATE AF TO REFLECT THE CURRENT IMPLEMENTATION OF ERF [[Exposure-response function]]|--[[User:Jouni|Jouni]] ([[User talk:Jouni|talk]]) 05:20, 13 June 2015 (UTC)}}

<rcode name="AF" label="Initiate AF (only for developers)" embed=1>
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")

</rcode>

==References==

<references/>