Difference between revisions of "Attributable risk"

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(equations corrected and made uniform. Constant background assumption removed)
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::<math>PAF = 1 - \frac{1}{\sum_{i=0}^k p_i (RR_i)}</math>
 
::<math>PAF = 1 - \frac{1}{\sum_{i=0}^k p_i (RR_i)}</math>
 
::for use with several population subgroups (typically with different exposure levels). Not valid when confounding exists. Subscript i refers to the ith subgroup. p<sub>i</sub> = proportion of '''total population''' in ith subgroup.
 
::for use with several population subgroups (typically with different exposure levels). Not valid when confounding exists. Subscript i refers to the ith subgroup. p<sub>i</sub> = proportion of '''total population''' in ith subgroup.
−
::<math>PAF = 1- \sum_{i=0}^k \frac{p_{di}}{RR_i} = \sum_i p_{di} \frac{p_e^i(RR_i - 1)}{p_e^i(RR_i - 1) + 1}</math>
+
::<math>PAF = 1- \sum_{i=0}^k \frac{p_{di}}{RR_i} = \sum_i p_{di} \frac{p_{ie}(RR_i - 1)}{p_{ie}(RR_i - 1) + 1}</math>
 
::which produces valid estimates when confounding exists but with a problem that parameters are often not known. p<sub>di</sub> is the proportion of '''cases''' falling in subgroup i (so that &Sigma;<sub>i</sub>p<sub>di</sub> = 1), p<sub>ie</sub> is the proportion of '''exposed''' people within subgroup i (and 1-p<sub>ie</sub> is the fraction of unexposed)
 
::which produces valid estimates when confounding exists but with a problem that parameters are often not known. p<sub>di</sub> is the proportion of '''cases''' falling in subgroup i (so that &Sigma;<sub>i</sub>p<sub>di</sub> = 1), p<sub>ie</sub> is the proportion of '''exposed''' people within subgroup i (and 1-p<sub>ie</sub> 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 there are equations for several different models:
 
; 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 there are equations for several different models:
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; Etiologic fraction (EF): The proportion of exposed cases that would have occurred later (if at all) without exposure.
 
; Etiologic fraction (EF): The proportion of exposed cases that would have occurred later (if at all) without exposure.
 
; Population attributable fraction (PAF): The proportion of all cases (exposed and unexposed) that would not have occurred without exposure. PAF<sub>i</sub> is that of subgroup i.
 
; Population attributable fraction (PAF): The proportion of all cases (exposed and unexposed) that would not have occurred without exposure. PAF<sub>i</sub> is that of subgroup i.
−
; Risk of disease (hazard rates): 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>,
+
; Risk of disease (hazard rates): 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>. RR<sub>i</sub> = relative risk comparing ith exposure level with unexposed group (i = 0).
−
; Proportion exposed (p<sub>e</sub>, p<sub>ie</sub>, p<sub>ed</sub>): proportion of exposed among the total population or within subgroup i or among cases (we use subscript d as diseased rather than c as cases to distinguish it from subscript e): p<sub>e</sub> = N(E=1)/N, p<sub>ie</sub> = N(E=1,S=i)/N(S=i), p<sub>ed</sub> = N(E=1,D=c)/N(D=c)
+
; Proportion '''exposed''' (p<sub>e</sub>, p<sub>ie</sub>, p<sub>ed</sub>): proportion of exposed among the total population or within subgroup i or within cases (we use subscript d as diseased rather than c as cases to distinguish it from subscript e): p<sub>e</sub> = N(E=1)/N, p<sub>ie</sub> = N(E=1,S=i)/N(S=i), p<sub>ed</sub> = N(E=1,D=c)/N(D=c)
−
; Proportion of population (p<sub>i</sub>): proportion of population in subgroups i among the total population: N(S=i)/N. p'<sub>i</sub> is the fraction of population in a counterfactual ideal situation (where the exposure is typically lower).
+
; Proportion of '''population''' (p<sub>i</sub>): proportion of population in subgroups i among the total population: N(S=i)/N. p'<sub>i</sub> is the fraction of population in a counterfactual ideal situation (where the exposure is typically lower).
−
; Proportion of cases of the disease (p<sub>di</sub>): proportion of cases in subgroups i among the total cases: N(D=c,S=i)/N(D=c).
+
; Proportion of '''cases''' of the disease (p<sub>di</sub>): proportion of cases in subgroups i among the total cases: N(D=c,S=i)/N(D=c) (so that &Sigma;<sub>i</sub>p<sub>di</sub> = 1).
  
 
=== Attributable fraction ===
 
=== Attributable fraction ===
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|2
 
|2
 
|<math>\frac{p_e(RR-1)}{p_e(RR-1)+1}</math>
 
|<math>\frac{p_e(RR-1)}{p_e(RR-1)+1}</math>
−
|Transformation of formula 1.<ref name="rockhill"/> Not valid when there is confounding of exposure-disease association. p<sub>e</sub> = 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.
+
|Transformation of formula 1.<ref name="rockhill"/> Not valid when there is confounding of exposure-disease association. 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
 
|3
 
|<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>
 
|<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 total 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.
+
|Extension of formula 2 for use with multicategory exposures. Not valid when confounding exists. Subscript i refers to the ith exposure level. 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.
 
|----
 
|----
 
|4
 
|4
 
|<math>\sum_i p_{di} \frac{p_{ie}(RR_i - 1)}{p_{ie}(RR_i - 1) + 1}</math>
 
|<math>\sum_i p_{di} \frac{p_{ie}(RR_i - 1)}{p_{ie}(RR_i - 1) + 1}</math>
−
|A useful formulation where <ref name="darrow"/>
+
|A useful formulation from<ref name="darrow"/>. Note that RR<sub>i</sub> is the risk ratio for subgroup i due to the subgroup-specific exposure level and assums that everyone in that subgroup is exposed to that level or none.  
−
* p<sub>di</sub> is the proportion of '''cases''' falling in subgroup i (so that &Sigma;<sub>i</sub>p<sub>di</sub> = 1),
 
−
* p<sub>ie</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).  
 
 
|----
 
|----
 
|5
 
|5
−
|<math>p_ed(\frac{RR-1}{RR})</math>
+
|<math>p_{ed}(\frac{RR-1}{RR})</math>
−
|Alternative expression of formula 3.<ref name="rockhill"/> 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> p<sub>c</sub> = 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>
+
|Alternative expression of formula 3.<ref name="rockhill"/> 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> 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>
 
|----
 
|----
 
|6
 
|6
 
|<math>\sum_{i=0}^k p_{di} (\frac{RR_i - 1}{RR_i}) = 1- \sum_{i=0}^k \frac{p_{di}}{RR_i}</math>
 
|<math>\sum_{i=0}^k p_{di} (\frac{RR_i - 1}{RR_i}) = 1- \sum_{i=0}^k \frac{p_{di}}{RR_i}</math>
−
|Extension of formula 5 for use with multicategory exposures.<ref name="rockhill"/> Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used. p<sub>di</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"/>
+
|Extension of formula 5 for use with multicategory exposures.<ref name="rockhill"/> Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used. 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"/>
 
|}
 
|}
  
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Specifically, we can divide the cohort into subgroups based on exposure (in the simplest case exposed and unexposed), so we get
 
Specifically, we can divide the cohort into subgroups based on exposure (in the simplest case exposed and unexposed), so we get
  
−
<math>PAF = p_{ec} \frac{1(RR - 1)}{1(RR - 1) + 1} + (1 - p_{ec}) \frac{0(RR - 1)}{0(RR - 1) +1}
+
<math>PAF = p_{ed} \frac{1(RR - 1)}{1(RR - 1) + 1} + (1 - p_{ed}) \frac{0(RR - 1)}{0(RR - 1) +1}
  
−
= p_{ec} \frac{RR - 1}{RR},</math>
+
= p_{ed} \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.
 
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.
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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):
 
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>
+
<math>\int_G [f_1(u) - f_0(u)]\mathrm{d}u / [1 - S_1(t)],</math>
  
 
where 1 means the exposed group, 0 means the non-exposed group, f is the proportion of population dying at particular time points, S is the survival function (and thus f(u) = -dS(u)/du), 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).  
 
where 1 means the exposed group, 0 means the non-exposed group, f is the proportion of population dying at particular time points, S is the survival function (and thus f(u) = -dS(u)/du), 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).  
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* [[Health impact assessment]]
 
* [[Health impact assessment]]
−
* [[Disease burden]]
 
 
* [[:en:Attributable risk|Attributable risk]] in Wikipedia.
 
* [[:en:Attributable risk|Attributable risk]] in Wikipedia.
  

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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 are there, 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): RR = \frac{R_1}{R_0}.
Attributable fraction
(aka excess fraction) 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}
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 (Missing <code>texvc</code> executable. Please see math/README to configure.): 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = 1- \sum_{i=0}^k \frac{p_{di}}{RR_i} = \sum_i p_{di} \frac{p_{ie}(RR_i - 1)}{p_{ie}(RR_i - 1) + 1}
which produces valid estimates when confounding exists but with a problem that parameters are often not known. pdi is the proportion of cases falling in subgroup i (so that Σipdi = 1), pie is the proportion of exposed people within subgroup i (and 1-pie 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 there are equations for several different models:
Equation Model and description
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF = 1 Rank-preserving model. The rank of individual deaths is not affected by exposure, i.e. everyone dies in the same order as without exposure, just sooner.
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF = \frac{RR - 1}{RR} Competing causes model. The hazard rate (occurrence rate of cases in time) in the exposed population is relative to that of the non-exposed population: h1(t) = RR h0(t). The ratio is constant although the hazard rates are functions of time t.
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF_l = \frac{RR-1}{RR^{RR/(RR-1)}} Exponential survival model. When the survival times in the population follow exponential distribution, the lowest possible EF can be calculated from this equation. However, the exponential survival model says nothing about which individuals are affected and lose how much life years, and therefore in this model the actual EF may be between the lower bound and 1.

With this code, you can compare attributable fraction and lower (assuming exponential survival distribution) and upper bounds of etiological fraction.

What is (are) the relative risk(s), i.e. RR?:

+ Show code

This code creates a simulated population of 180 individuals and calculates their survival and attributable and etiologic fractions in different mechanistic settings.

What scenarios do you want to see?:
BAU (business as usual, no exposure
Orderly (everyone dies one year earlier because of exposure)
Relative (everyone dies 20 % faster because of exposure)
Maxloss (those who live longest are killed by exposure as early as possible)
Competcause (20 % of people died earlier than others)
EF0.25 (25 % random people die at 30 % of their expected life)
BAU_expon (Exponential distribution for deaths)
Rel_expon (RR 1.2 to reduce eveyone's life)
Rel_exMaxloss (Lost life comes to those who would have lived the longest)

Which of the scenarios is used as the reference (BAU)?:

+ Show code

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.

Number of people (N)
The number of people in the total population considered, including cases, non-cases, exposed and non-exposed. N1 and No are the numbers of exposed and unexposed people in the population, respectively.
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 = 0, 1, 2, ..., k (typically based on different exposure levels)
  • Confounders (C): other factors correlating with exposure and disease and thus potentially causing bias in estimates unless measured and adjusted for.
Attributable fraction (AF)
The proportion of exposed cases that would not have occurred without exposure.
Etiologic fraction (EF)
The proportion of exposed cases that would have occurred later (if at all) without exposure.
Population attributable fraction (PAF)
The proportion of all cases (exposed and unexposed) that would not have occurred without exposure. PAFi is that of subgroup i.
Risk of disease (hazard rates)
R1 and R0 are the risks of disease in the exposed and unexposed group, respectively, and RR = R1 / R0. RRi = relative risk comparing ith exposure level with unexposed group (i = 0).
Proportion exposed (pe, pie, ped)
proportion of exposed among the total population or within subgroup i or within cases (we use subscript d as diseased rather than c as cases to distinguish it from subscript e): pe = N(E=1)/N, pie = N(E=1,S=i)/N(S=i), ped = N(E=1,D=c)/N(D=c)
Proportion of population (pi)
proportion of population in subgroups i among the total population: N(S=i)/N. p'i is the fraction of population in a counterfactual ideal situation (where the exposure is typically lower).
Proportion of cases of the disease (pdi)
proportion of cases in subgroups i among the total cases: N(D=c,S=i)/N(D=c) (so that Σipdi = 1).

Attributable fraction

Rockhill et al.[1] 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 [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.R↻

Impact of confounders

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Darrow and Steenland[3] studied the direction and magnitude of bias in attributable fraction with different confounding situations. For details, see Attributable risk#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[3] 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 (+-)

Population attributable fraction

The population attributable fraction PAF is the fraction of all cases (exposed and unexposed) that would not have occurred if the exposure had been absent.

Different ways to calculate population attributable fraction PAF.
# Formula Description
1 Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{IP_1 - IP_0}{IP_1} is empirical approximation of [1]

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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, C means other confounders, and E is exposure and a bar above E means no exposure. 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{p_e(RR-1)}{p_e(RR-1)+1} Transformation of formula 1.[1] Not valid when there is confounding of exposure-disease association. 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): \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. Derived by Walter[4]; given in Kleinbaum et al.[5] but not in other widely used epidemiology texts.
4 Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sum_i p_{di} \frac{p_{ie}(RR_i - 1)}{p_{ie}(RR_i - 1) + 1} A useful formulation from[3]. Note that RRi is the risk ratio for subgroup i due to the subgroup-specific exposure level and assums that everyone in that subgroup is exposed to that level or none.
5 Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): p_{ed}(\frac{RR-1}{RR}) Alternative expression of formula 3.[1] Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used.[6] In Kleinbaum et al.[5] and Schlesselman.[7]
6 Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sum_{i=0}^k p_{di} (\frac{RR_i - 1}{RR_i}) = 1- \sum_{i=0}^k \frac{p_{di}}{RR_i} Extension of formula 5 for use with multicategory exposures.[1] Produces internally valid estimate when confounding exists and when, as a result, adjusted relative risks must be used. See Bruzzi et al. [8] and Miettinen[6] for discussion and derivations; in Kleinbaum et al.[5] and Schlesselman.[7]

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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 (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_e (RR - 1) }{ \frac{N_1 RR - N_1 + (N_1 + N_0)}{N_1 + N_0}} = \frac{p_e (RR - 1)}{p_e RR - p_e + 1} = \frac{p_e (RR - 1)}{p_e (RR - 1) + 1}.

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.): PAF = \Sigma_i p_{di} PAF_{i}.

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.): PAF = p_{ed} \frac{1(RR - 1)}{1(RR - 1) + 1} + (1 - p_{ed}) \frac{0(RR - 1)}{0(RR - 1) +1} = p_{ed} \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

According to WHO, PAF is [9]

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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}.

We can see that this reduces to PAF equation 2 when we 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. in this specific case, pi = pe. Thus, we get

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{(p_e RR + (1-p_e)*1) - (0*RR + 1*1)}{p_e RR + (1-p_e)*1}

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{p_e RR - p_e}{p_i RR + 1 - p_e}

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{p_e(RR - 1)}{p_e(RR -1) + 1}

--#: Constant background assumption section was archived because it was only relevant for a previous HIA model version. --Jouni (talk) 13:17, 25 April 2016 (UTC)

Etiologic fraction

Etiologic fraction ((EF)is defined as the fraction of cases that is advanced in time because of exposure.[10]R↻ In other words, those cases would have occurred later (if at all), if there had not been exposure. EF can also be called probability of causation, which has importance in court. It can also be used to calculate premature cases, but that term is ambiguous and sometimes it is used to mean cases that have been substantially advanced in time, in contrast to the harvesting effect where an exposure kills people that would have died anyway within a few days. There has been a heated discussion about harvesting effect related to fine particles. Therefore, sometimes attributable fraction is used instead to calculate what they call premature mortality.R↻ Therefore, it is important to explicitly explain what is meant by the word premature.

Robins and Greenland[10] 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): \int_G [f_1(u) - f_0(u)]\mathrm{d}u / [1 - S_1(t)],

where 1 means the exposed group, 0 means the non-exposed group, f is the proportion of population dying at particular time points, S is the survival function (and thus f(u) = -dS(u)/du), 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).

Although the exact value of etiologic fraction cannot be estimated directly from risk ratio (RR), different models offer equations to estimate EF. It is just important to understand, discuss, and communicate, which of the models most closely represents the actual situation observed. Three models are explained here.R↻

Rank-preserving model says that everyone dies at the same rank order as without exposure, but that the deaths occur earlier. If the exposed population loses life years compared with unexposed population, it is in theory always possible that everyone dies a bit earlier and thus EF = 1.

Competing causes model is the most commonly assumed model, but unfortunately this assumption is often not realised or understood, and therefore it is common to see misinterpreted results from this model. (Of course this is may be true for any model, but it is emphasised here because of the popularity of the model.) The model says that the exposure of interest and other causes of death are constantly competing, and that the impact of the exposure is relative to the other competing causes. In other words, the hazard rate in the exposed population is h1(t) = RR h0(t). Hazard rates are functions of time, and may become very high in very old populations. In any case, the proportional impact of the exposure stays constant.

In the case of competing cause model, EF equals attributable fraction AF:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF = AF = \frac{RR - 1}{RR},

Exponential survival model assumes that the hazard rate is constant and the deaths occur following the exponential distribution. Although this model has very elegant formulas, it is typically far from plausible, as the differences in survival may be very large. E.g. with average life expectancy of 70 years, 10 % of the population would die before 8 years of age, while 10 % would live beyond 160 years. In situations where exponential survival model can be used, the lower bound of EF (equation 9[10]) is as low as

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): EF_l = \frac{RR - 1}{RR^{RR/(RR-1)}}.

For an illustration of the behaviour of EF, see the code "Test different etiologic fractions" in the Answer. Also 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.

Calculations

⇤# : UPDATE AF TO REFLECT THE CURRENT IMPLEMENTATION OF ERF Exposure-response function --Jouni (talk) 05:20, 13 June 2015 (UTC)

+ Show code

A previous version of code looked at RRs of all exposure agents and summed PAFs up.


Some interesting model runs:

+ Show code

See also

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Rockhill B, Newman B, Weinberg C. use and misuse of population attributable fractions. American Journal of Public Health 1998: 88 (1) 15-19.[1]
  2. ↑ Kenneth J. Rothman, Sander Greenland, Timothy L. Lash: Modern Epidemiology. Lippincott Williams & Wilkins, 2008. 758 pages.
  3. ↑ 3.0 3.1 3.2 Darrow LA, Steenland NK. Confounding and bias in the attributable fraction. Epidemiology 2011: 22 (1): 53-58. [2] doi:10.1097/EDE.0b013e3181fce49b
  4. ↑ Walter SD. The estimation and interpretation of attributable fraction in health research. Biometrics. 1976;32:829-849.
  5. ↑ 5.0 5.1 5.2 Kleinbaum DG, Kupper LL, Morgenstem H. Epidemiologic Research. Belmont, Calif: Lifetime Learning Publications; 1982:163.
  6. ↑ 6.0 6.1 Miettinen 0. Proportion of disease caused or prevented by a given exposure, trait, or intervention. Am JEpidemiol. 1974;99:325-332.
  7. ↑ 7.0 7.1 Schlesselman JJ. Case-Control Studies: Design, Conduct, Analysis. New York, NY: Oxford University Press Inc; 1982.
  8. ↑ 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.
  9. ↑ WHO: Health statistics and health information systems. [3]. Accessed 16 Nov 2013.
  10. ↑ 10.0 10.1 10.2 Robins JM, Greenland S. Estimability and estimation of excess and etiologic fractions. Statistics in Medicine 1989 (8) 845-859.
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