Attributable risk: Difference between revisions
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=== Estimating etiologic fraction === | === Estimating etiologic fraction === | ||
'''Etiologic fraction''' (EF) tells which fraction of the population dies earlier because of the exposure considered, compared with the situation that they would not be exposed. Often that is calculated is the ''attributable fraction AF'': | |||
<math> AF = \frac{RR - 1}{RR},</math> | |||
where RR is the relative risk: risk of exposed people divided by the risk of non-exposed. Robins and Greenland<ref name="robins"/> studied the estimability of etiologic fraction. They concluded that observations are not enough to conclude about the precise value of EF, because irrespective of observation, the same amount of observed life years lost may be due to many people losing a short time each, or due to a few losing a long time each. The upper limit in theory is always 1, and the lower bound they estimated by this equation (equation 9 in the article): | |||
<math>\frac{\int_G [f_1(u) - f_0(u)]du}{1 - S_1(t)},</math> | |||
where 1 means the exposed group, 0 means the non-exposed group, f is the proportion of population dying particular time points, S is the survival function, t is the length of the observation time, u the observation time and G is the set of all u < t such that f<sub>1</sub>(u) > f<sub>0</sub>(u). | |||
From this, they derive the following equation (number 11 in the article): | |||
<math>\frac{RR-1}{RR^{RR/(RR-1)}}</math> | |||
It is not clear how they got from the first equation to the second equation, especially because the second gives values that are typically less than half of that if the first one. Both are used in the code below. In addition, the ''true etiologic fraction'' is calculated for this simulated population, because in the simulation we assume that we know exactly what happens to each individual in each scenario and how much their lengths of lives change. | |||
<rcode label="Test different etiologic fractions" embed=1 graphics=1 variables=" | <rcode label="Test different etiologic fractions" embed=1 graphics=1 variables=" | ||
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BS <- 24 | BS <- 24 | ||
ggplot(lif@output, aes(x = Id, y = lifResult, colour = Scenario))+geom_point()+ | ggplot(lif@output, aes(x = Id, y = lifResult, colour = Scenario))+geom_point()+ | ||
coord_cartesian(ylim=c(0,10)) + theme_gray(base_size = BS) + labs(title = "Life expectancies of 180 individuals", y = "Age at death", x = "Individual") | |||
ggplot(fr@output, aes(x = Time, y = frResult, colour = Scenario, group = Scenario))+geom_line() + | ggplot(fr@output, aes(x = Time, y = frResult, colour = Scenario, group = Scenario))+geom_line() + | ||
theme_gray(base_size = BS) + labs(title = "fraction of people dying at different time groups") | theme_gray(base_size = BS) + labs(title = "fraction of people dying at different time groups") | ||
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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 total 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. pi = proportion of total 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{p_{ci}}{RR_i} = \sum_i p_{ci} \frac{p_{ei}(RR_i - 1)}{p_{ei}(RR_i - 1) + 1}}
- which produces valid estimates when confounding exists but with a problem that parameters are often not known. pci is the proportion of cases falling in subgroup i (so that Σipci = 1), pei is the proportion of exposed people within subgroup i (and 1-pi is the fraction of unexposed)
- 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
Definitions of terms
There are several different kinds of proportions that sound alike but are not. Therefore, we explain the specific meaning of several terms.
- Total population (N)
- The number of people in the total population considered, including cases, non-cases, exposed and non-exposed.
- Classifications
- There are three classifications, and every person in the total population belongs to exactly one group in each classification.
- Disease (D): classes case (C) and non-case (nc)
- Exposure (E): classes exposed (1) and non-exposed (0)
- Population subgroup (S): classes i = 1, 2, ..., k (typically based on different exposure levels)
- Attributable fraction (AF)
- The proportion of cases caused by exposure among all cases (in the subgroup)
- Proportion exposed (pe, pei)
- proportion of exposed among the total population or within subgroup i: pe = N(E=1)/N, pei = N(E=1,S=i)/N(S=i)
- Proportion of population (pi)
- proportion of population in subgroups i among the total population: N(S=i)/N
- Proportion of cases (pci)
- proportion of cases in subgroups i among the total cases: N(D=c,S=i)/N(D=c)
Etiologic fraction
Etiologic fraction is defined as the fraction of cases that is advanced in time because of exposure.[1]R↻ It can also be called probability of causation, which has importance in court. It can also be used to calculate premature cases, but that word is ambiguous and sometimes attributable fraction is used instead.R↻ Therefore, it is important to explicitly explain what is meant by the work premature.
The exact value of etiologic fraction 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.
Attributable fraction
Rockhill et al.[2] give an extensive description about different ways to calculate attributable fraction (AF) and population attributable fraction (PAF) and assumptions needed in each approach. Modern Epidemiology [3] 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.R↻
| # | Formula | Description |
|---|---|---|
| 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 AF = \frac{IP_1 - IP_0}{IP_1} = \frac{RR-1}{RR}} | is empirical approximation of [2]
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; IP0 = 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 I0/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. |
| 2 | 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.[2] Not valid when there is confounding of exposure-disease association. pe = proportion of total 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. |
| 3 | 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. pi = proportion of total population in ith exposure level, RRj = relative risk comparing ith exposure level with unexposed group (i = 0). Derived by Walter[4]; given in Kleinbaum et al.[5] but not in other widely used epidemiology texts. |
| 4 | 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 p_{ci} \frac{p_{ei}(RR_i - 1)}{p_{ei}(RR_i - 1) + 1}} | A useful formulation where [6]
|
| 5 | 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_c(\frac{RR-1}{RR})} | Alternative expression of formula 3.[2] Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used.[7] pc = proportion of cases exposed to risk factor. In Kleinbaum et al.[5] and Schlesselman.[8] |
| 6 | 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 p_{ci} (\frac{RR_i - 1}{RR_i}) = 1- \sum_{i=0}^k \frac{p_{ci}}{RR_i}} | Extension of formula 5 for use with multicategory exposures.[2] Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used. pci = proportion of cases falling into ith exposure level; RRi = relative risk comparing ith exposure level with unexposed group (i = 0). See Bruzzi et al. [9] and Miettinen[7] for discussion and derivations; in Kleinbaum et al.[5] and Schlesselman.[8] |
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[6] have studied the impact of confounding on the bias in attributable fraction. This is their summary:
| 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 (+-) |
Calculations
With this code, you can compare attributable fraction and lower and upper bounds of etiological fraction.
⇤--#: . 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)
A previous version of code looked at RRs of all exposure agents and summed PAFs up.
Derivation of PAF
⇤--#: . Do we need this section? --Jouni (talk) 20:53, 7 April 2016 (UTC) (type: truth; paradigms: science: attack)
←--#: . I think that we do. It clearly shows how to add exposure to PAF calclualtions and how we are using it HIA. --Arja (talk) 07:31, 8 April 2016 (UTC) (type: truth; paradigms: science: defence)
The population attributable fraction PAF 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 PAF = \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 PAF = \Sigma_i p_{ci} PAF_{i},}
where
- pci is the proportion of cases falling in stratum (subgroup) i,
- PAFi 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 PAF = 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.
WHO approach
PAF is [10]
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{\sum_{i=0}^k P_i RR_i - \Sigma_{i=0}^k P'_i RR_i}{\Sigma_{i=0}^k 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.
Constant background assumption
⇤--#: . Is this section necessary? --Jouni (talk) 20:53, 7 April 2016 (UTC) (type: truth; paradigms: science: attack)
pci 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
- Ni is the number of people in each subgroup i,
- RRi,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.
pc 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
- casesi is the number of cases in subgroup i,
- Ni 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,
- ERFj 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),
- exposurei,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 RRi,j = exp(ln(ERFj) exposurei,j).
In addition, if only fraction p of the population is exposed, for the total 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}
Estimating etiologic fraction
Etiologic fraction (EF) tells which fraction of the population dies earlier because of the exposure considered, compared with the situation that they would not be exposed. Often that is calculated is the attributable fraction AF:
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 relative risk: risk of exposed people divided by the risk of non-exposed. Robins and Greenland[1] studied the estimability of etiologic fraction. They concluded that observations are not enough to conclude about the precise value of EF, because irrespective of observation, the same amount of observed life years lost may be due to many people losing a short time each, or due to a few losing a long time each. The upper limit in theory is always 1, and the lower bound they estimated by this equation (equation 9 in the article):
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{\int_G [f_1(u) - f_0(u)]du}{1 - S_1(t)},}
where 1 means the exposed group, 0 means the non-exposed group, f is the proportion of population dying particular time points, S is the survival function, t is the length of the observation time, u the observation time and G is the set of all u < t such that f1(u) > f0(u).
From this, they derive the following equation (number 11 in the article):
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)}}}
It is not clear how they got from the first equation to the second equation, especially because the second gives values that are typically less than half of that if the first one. Both are used in the code below. In addition, the true etiologic fraction is calculated for this simulated population, because in the simulation we assume that we know exactly what happens to each individual in each scenario and how much their lengths of lives change.
See also
References
- ↑ 1.0 1.1 Robins JM, Greenland S. Estimability and estimation of excess and etiologic fractions. Statistics in Medicine 1989 (8) 845-859.
- ↑ 2.0 2.1 2.2 2.3 2.4 Rockhill B, Newman B, Weinberg C. use and misuse of population attributable fractions. American Journal of Public Health 1998: 88 (1) 15-19.[1]
- ↑ Kenneth J. Rothman, Sander Greenland, Timothy L. Lash: Modern Epidemiology. Lippincott Williams & Wilkins, 2008. 758 pages.
- ↑ Walter SD. The estimation and interpretation of attributable fraction in health research. Biometrics. 1976;32:829-849.
- ↑ 5.0 5.1 5.2 Kleinbaum DG, Kupper LL, Morgenstem H. Epidemiologic Research. Belmont, Calif: Lifetime Learning Publications; 1982:163.
- ↑ 6.0 6.1 6.2 Darrow LA, Steenland NK. Confounding and bias in the attributable fraction. Epidemiology 2011: 22 (1): 53-58. [2] doi:10.1097/EDE.0b013e3181fce49b
- ↑ 7.0 7.1 Miettinen 0. Proportion of disease caused or prevented by a given exposure, trait, or intervention. Am JEpidemiol. 1974;99:325-332.
- ↑ 8.0 8.1 Schlesselman JJ. Case-Control Studies: Design, Conduct, Analysis. New York, NY: Oxford University Press Inc; 1982.
- ↑ 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.
- ↑ WHO: Health statistics and health information systems. [3]. Accessed 16 Nov 2013.