Attributable risk
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Attributable risk is a fraction of total risk that can be attributed to a particular cause. There are a few different ways to calculate it. Population attributable fraction of an exposure agent is the fraction of disease that would disappear if the exposure to that agent would disappear in a population. Etiologic fraction is the fraction of cases that have occurred earlier than they would have occurred (if at all) without exposure. Etiologic fracion cannot typically be calculated based on risk ratio (RR) alone, but it requires knowledge about biological mechanisms.
Question
How to calculate attributable risk? What different approaches 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 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 RR = \frac{R_1}{R_0}.}
 Excess fraction
 (sometimes called 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 XF = \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 i^{th} subgroup. p_{i} = proportion of total population in i^{th} 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_{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. p_{di} is the proportion of cases falling in subgroup i (so that Σ_{i}p_{di} = 1), p_{ie} is the proportion of exposed people within subgroup i (and 1p_{ie} is the fraction of unexposed)
 Etiologic fraction
 Fraction of cases among the exposed that would have occurred later (if at all) if the exposure had not taken place. It cannot be calculated without understanding of the biological mechanism, but there are equations for several specific cases. If survival functions are known, the lower limit of EF can be calculated:
 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 \int_G [f_1(u)  f_0(u)]\mathrm{d}u / [1  S_1(t)],}
 where 1 means the exposed group, 0 means the unexposed 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_{1}(u) > f_{0}(u).
 In a specific case where the survival distribution is exponential, the following formula can be used for the lowest possible EF. 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.
 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 EF_l = \frac{RR  1}{RR^{RR/(RR1)}}.}
 Finally, it should be remembered that if the rank preserving assumption holds (i.e. the rank of individual deaths is not affected by exposure: everyone dies in the same order as without exposure, just sooner), the EF can be as high as 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 EF_u = 1}
With this code, you can compare excess fraction and lower (assuming exponential survival distribution) and upper bounds of etiological fraction.
This code creates a simulated population of 200 individuals that are now 60 years of age. It calculates their survival and excess and etiologic fractions in different mechanistic settings. Relative risk of 1.2 and a constant hazard rate will be applied in all scenarios.
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, noncases, exposed and unexposed. N_{1} and N_{o} 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 noncase (nc)
 Exposure (E): classes exposed (1) and unexposed (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.
 Excess fraction (XF)
 The proportion of exposed cases that would not have occurred without exposure on population level.
 Etiologic fraction (EF)
 The proportion of exposed cases that would have occurred later (if at all) without exposure on individual level.
 Hazard fraction (HF)
 The proportion of hazard rate that would not be there without exposure, HF = [h_{1}(t)  h_{0}(t)]/h_{1}(t) = [R(t)  1]/R(t), where h(t) is hazard rate at time t and R(t) = h_{1}(t)/h_{0}(t).
 Attributable fraction (AF)
 An ambiguous term that has been used for excess fraction, etiologic fraction and hazard fraction without being specific. Therefore, its use is not recommended.
 Population attributable fraction (PAF)
 The proportion of all cases (exposed and unexposed) that would not have occurred without exposure on population level. PAF_{i} is PAF of subgroup i.
 Risk of disease (hazard rates)
 R_{1} and R_{0} are the risks of disease in the exposed and unexposed group, respectively, and RR = R_{1} / R_{0}. RR_{i} = relative risk comparing i^{th} exposure level with unexposed group (i = 0). Note that often texts are not clear when they talk about risk proportion = number of cases / number of population and thus risk ratio; and when about hazard rates = number of cases / observation time and thus rate ratio. RR may mean either one. If occurrence of cases is small, risk ratio and rate ratio approach each other, because then cases hardly shorten the observation time in the population.
 Proportion exposed (p_{e}, p_{ie}, p_{ed})
 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_{e} = N(E=1)/N, p_{ie} = N(E=1,S=i)/N(S=i), p_{ed} = N(E=1,D=c)/N(D=c)
 Proportion of population (p_{i})
 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 (p_{di})
 proportion of cases in subgroups i among the total cases: N(D=c,S=i)/N(D=c) (so that Σ_{i}p_{di} = 1).
Excess fraction
Rockhill et al.^{[1]} give an extensive description about different ways to calculate excess fraction (XF) and population attributable fraction (PAF) and assumptions needed in each approach. Modern Epidemiology ^{[2]} is the authoritative source of epidemiology. They first define excess fraction XF for a cohort of people (pages 295297). It is the fraction of cases among the exposed that would not have occurred if the exposure would not have taken place.R↻ However, both sources use the term attributable fraction rather than excess fraction.
Impact of confounders
The problem with the two PAF equations (see Answer) is that the former has easiertocollect 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:
Bias in excess 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.
#  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 \frac{IP_t  IP_0}{IP_t} \approx \frac{I_t  I_0}{I_t}}  is empirical approximation of ^{[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 \frac{P(D)  \sum_C P(DC, \bar{E}) P(C)}{P(D)}} where IP_{1} = cumulative proportion of total population developing disease over specified interval; IP_{0} = 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 I_{0}/I_{t} approximates ratio of cumulative incidence proportions, and thus formula can be written as (I_{t}  I_{0})I_{t}. Both formulations found in many widely used epidemiology textbooks. ⇤#: . Is there an error in the text about the approximation? Jouni (talk) 10:05, 28 June 2016 (UTC) (type: truth; paradigms: science: attack) 
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(RR1)}{p_e(RR1)+1}}  Transformation of formula 1.^{[1]} Not valid when there is confounding of exposuredisease 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 (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 i^{th} exposure level. 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_{di} \frac{p_{ie}(RR_i  1)}{p_{ie}(RR_i  1) + 1}}  A useful formulation from^{[3]}. Note that RR_{i} is the risk ratio for subgroup i due to the subgroupspecific exposure level and assumes that everyone in that subgroup is exposed to that level or none. 
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_{ed}(\frac{RR1}{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 (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_{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 (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_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(RR1)+1, not p(RR1)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_{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 (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_{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 p_{c} 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 (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}.}
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, p_{i} = p_{e}. 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_e RR + (1p_e)*1)  (0*RR + 1*1)}{p_e RR + (1p_e)*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_e RR  p_e}{p_i RR + 1  p_e}}
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_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) (type: truth; paradigms: science: comment)
Etiologic fraction
Etiologic fraction (EF) is defined as the fraction of cases that are 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 excess fraction is used instead to calculate what they call premature mortality, but unfortunately that practice causes even more confusion.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 (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 \int_G [f_1(u)  f_0(u)]\mathrm{d}u / [1  S_1(t)],}
where 1 means the exposed group, 0 means the unexposed 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_{1}(u) > f_{0}(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↻
Rankpreserving 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
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 EF_u = 1.}
Competing causes model is the most commonly assumed model, but often people do not realise that they make such an assumption. 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 h_{1}(t) = RR h_{0}(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 where competing causes model and independence assumtption applies, lower end of EF range is often close to the excess fraction XF. (But it can be lower, as the next example with a skewed exponential distribution demonstrates.)
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 EF_l = XF = \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 (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 EF_l = \frac{RR  1}{RR^{RR/(RR1)}}.}
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. By testing with several inputs, we can see the following pattern (table).
Survival distribution  Scenario  Excess fraction XF  True etiologic fraction  EF_low from Eq 9  EF_low from Eq 11 

Uniform  Competing causes, minimize EF[4]  0.17  0.17  0.17  0.07 
Competing causes, preserve rank order[5]  0.17  1.00  0.17  0.07  
Exponential  Competing causes, minimize EF[6]  0.17  0.07  0.07  0.07 
Competing causes, preserve rank order[7]  0.17  1.00  0.07  0.07 
As we can see from the table, true etiologic fraction can vary substantially  in theory. High values assume that most people are affected by a small life loss. This might be true with causes that worsen general health, thus killing the person a bit earlier than what would have happened if the person had been in a hardier state.
When we compare equations 9 and 11, we can see that the former never performs worse than the latter. This is simply because equation 11 was derived from equation 9 by making an additional assmuption that the survival distribution is exponential. Indeed, in such a case they produce identical values but in other cases equation 11 underestimates EF compared with equation 9. A practical conclusion is that if survival curves for exposed and unexposed groups are available, equation 9 rather than equation 11 should always be used. Even excess fraction is usually a better estimate than an estimate from equation 11, with the exception of exponential survival distribution.
Calculations
⇤#: . UPDATE AF TO REFLECT THE CURRENT IMPLEMENTATION OF ERF Exposureresponse 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.
Some interesting model runs:
 Population with exponentially distributed lifetimes
 Life loss to a fraction of people across the whole population
 Give all life loss to the people living the longest
Demonstration of hazard fractions, survival, and age at death
 Figure for manuscript Morfeld, Erren, Hammit etc.
See also
 These Wikipedia pages should be updated and made coherent. Also they should distinguish etiological fraction, excess fraction, and attributable fraction and describe their important conceptual differences.
 Health impact assessment
 Jacqueline C. M. Witteman, Ralph B. D'Agostino, Theo Stijnen, William B. Kannel, Janet C. Cobb, Maria A. J. de Ridder, Albert Hofman and James M. Robins. Gestimation of Causal Effects: Isolated Systolic Hypertension and Cardiovascular Death in the Framingham Heart Study. Am. J. Epidemiol. (1998) 148 (4): 390401. [8]
References
 ↑ ^{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) 1519.[1]
 ↑ Kenneth J. Rothman, Sander Greenland, Timothy L. Lash: Modern Epidemiology. Lippincott Williams & Wilkins, 2008. 758 pages.
 ↑ ^{3.0} ^{3.1} ^{3.2} Darrow LA, Steenland NK. Confounding and bias in the attributable fraction. Epidemiology 2011: 22 (1): 5358. [2] doi:10.1097/EDE.0b013e3181fce49b
 ↑ Walter SD. The estimation and interpretation of attributable fraction in health research. Biometrics. 1976;32:829849.
 ↑ ^{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} Miettinen 0. Proportion of disease caused or prevented by a given exposure, trait, or intervention. Am JEpidemiol. 1974;99:325332.
 ↑ ^{7.0} ^{7.1} Schlesselman JJ. CaseControl 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 casecontrol data. Am J Epidemiol. 1985; 122: 904914.
 ↑ WHO: Health statistics and health information systems. [3]. Accessed 16 Nov 2013.
 ↑ ^{10.0} ^{10.1} ^{10.2} ^{10.3} Robins JM, Greenland S. Estimability and estimation of excess and etiologic fractions. Statistics in Medicine 1989 (8) 845859.