Difference between revisions of "Attributable risk"

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=== Attributable fraction ===
 
=== Attributable fraction ===
  
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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)}}
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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.
  
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<math>\frac{IP_t - IP_0}{IP_t}</math>
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<math>\frac{IP_1 - IP_0}{IP_1}</math>
  
 
is empirical approximation of  
 
is empirical approximation of  
  
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<math>\frac{P(D) - \Sum_C P(D|C, \bar{E}) P(C)}{P(D)}</math>
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<math>\frac{P(D) - \sum_C P(D|C, \bar{E}) P(C)}{P(D)}</math>
  
−
where IP<sub>t</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>t</sub> approximates ratio of cumulative incidence proportions, and thus formula can be written as (I<sub>t</sub> - I<sub>0</sub>)I<sub>t</sub>. Both formulations found in many widely used epidemiology textbooks.
+
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>
 
<math>\frac{p_e(RR-1)}{p_e(RR-1)+1}</math>
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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.
 
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.
  
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<math>1 - \frac{1}{\Sum_{i=0}^k p_i (RR_i)}
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<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.
 
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.
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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>
 
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>
  
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<math>\Sum_{i=0}^k pd_i (\frac{RR_i - 1}{RR_i} = 1- \Sum_{i=0}^k \frac{pd_i}{RR_i}</math>
+
<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"/>
 
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:
 
Probably this is the most useful form of population attributable fraction (PAF or AF<sub>p</sub>) for impact assessment:
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=== Etiologic fraction ===
 
=== Etiologic fraction ===
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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 ===
 
=== Comparison ===

Revision as of 10:09, 7 April 2016


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). --# : What is the actual difference? --Jouni (talk) 08:38, 7 April 2016 (UTC) Rockhill et al.[1] give an extensive description about different ways to calculate PAF and assumptions needed in each approach.

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

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; IPO = 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 IO/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 (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. Not valid when there is confounding of exposure-disease association. pe = 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 (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. pi = proportion of source population in ith exposure level, RRj = relative risk comparing ith exposure level with unexposed group (i = 0). Derived by Walter[2]; given in Kleinbaum et al.[3] but not in other widely used epidemiology texts.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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.[4] pd = proportion of cases exposed to risk factor. In Kleinbaum et al.[3] and Schlesselman.[5]

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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. pdi = proportion of cases falling into ith exposure level; RRi = relative risk comparing ith exposure level with unexposed group (i = 0). See Bruzzi et al. [6] and Miettinen[4] for discussion and derivations; in Kleinbaum et al.[3] and Schlesselman.[5]

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF_p = \Sigma_i p_{ci} \frac{p_i(RR_i - 1)}{p_i(RR_i - 1) + 1},

where

  • pci is the proportion of cases falling in subgroup i (so that Σipci = 1),
  • pi is the fraction of exposed people within subgroup i (and 1-pi is the fraction of unexposed),
  • RRi is the risk ratio for subgroup i due to the subgroup-specific exposure level (assuming that everyone in that subgroup is exposed to that level or none).

[1]--# : Does the equation hold if exposure level varies between groups? --Jouni (talk) 15:43, 25 April 2014 (EEST)

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

Etiologic fraction

Etiologic fraction is defined as the fraction of cases that is advanced in time because of exposure.[7] 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): \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


Rationale

WHO approach

[8] PAF is

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): PAF = \frac{\Sigma_{i=1}^n P_i RR_i - \Sigma_{i=1}^n P'_i RR_i}{\Sigma_{i=1}^n P_i RR_i}

where i is a certain exposure level, P is the fraction of population in that exposure level, RR is the relative risk at that exposure level, and P' is the fraction of population in a counterfactual ideal situation (where the exposure is typically lower).

Based on this, we can limit our examination to a situation where there are only two population groups, one exposed to background level (with relative risk 1) and the other exposed to a higher level (with relative risk RR). In the counterfactual situation nobody is exposed. Thus, we get

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

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

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

See also Attributable risk, although it is a stub.

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

Rothman approach

Modern Epidemiology [9] is the authoritative source of epidemiology. They first define attributable fraction AF for a cohort of people (pages 295-297). It is the fraction of cases among the exposed that would not have occurred if the exposure would not have taken place:

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

where RR is the causal risk ratio.

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF_p = \frac{N_1 (R_1 - R_0)}{N_1 R_1 + N_0 R_0} = \frac{N_1 (R_1 - R_0)/R_0}{N_1 R_1/R_0 + N_0 R_0/R_0} = \frac{N_1 (RR - 1)}{N_1 RR + N_0}

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

where

  • N1 and N0 are the numbers of exposed and unexposed people, respectively,
  • R1 and R0 are the risks of disease in the exposed and unexposed group, respectively, and RR = R1 / R0,
  • p is the fraction of exposed people among the whole cohort.

Note that there is a typo in the Modern Epidemiology book: the denominator should be p(RR-1)+1, not p(RR-1)-1.

Population attributable fraction can be calculated as a weighted average based on subgroup data:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): AF_p = \Sigma_i p_{ci} AF_{pi},

where

  • pci is the proportion of cases falling in stratum (subgroup) i,
  • AFpi is the population attributable fraction calculated for the subgroup.

Specifically, we can divide the cohort into subgroups based on exposure (in the simplest case exposed and unexposed), so we get

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

where pc is the proportion of cases in the exposed group among all cases; this is the same as exposure prevalence among cases.

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): 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 (Missing <code>texvc</code> executable. Please see math/README to configure.): 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 whole population we get

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): RR = \frac{p * N * background * RR_{exposed} + (1-p) * N * background * RR_{unexposed}}{N * background * RR_{unexposed}}

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

+ Show code

References

  1. ↑ 1.0 1.1 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. ↑ Walter SD. The estimation and interpretation of attributable fraction in health research. Biometrics. 1976;32:829-849.
  3. ↑ 3.0 3.1 3.2 Kleinbaum DG, Kupper LL, Morgenstem H. Epidemiologic Research. Belmont, Calif: Lifetime Learning Publications; 1982:163.
  4. ↑ 4.0 4.1 Miettinen 0. Proportion of disease caused or prevented by a given exposure, trait, or intervention. Am JEpidemiol. 1974;99:325-332.
  5. ↑ 5.0 5.1 Schlesselman JJ. Case-Control Studies: Design, Conduct, Analysis. New York, NY: Oxford University Press Inc; 1982.
  6. ↑ 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.
  7. ↑ Robins JM, Greenland S. Estimability and estimation of excess and etiologic fractions. Statistics in Medicine 1989 (8) 845-859.
  8. ↑ WHO: Health statistics and health information systems. [2]. Accessed 16 Nov 2013.
  9. ↑ Kenneth J. Rothman, Sander Greenland, Timothy L. Lash: Modern Epidemiology. Lippincott Williams & Wilkins, 2008. 758 pages.
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