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AMERICAN

Journal of EpidemiologyFormed, AMERICAN JOURNAL OF HYGIENE

© 1988 by The Johns Hopkins University School of Hygiene and Public Health

VOL. 128 DECEMBER 1988 NO. 6

Reviews and CommentaryCONCEPTUAL PROBLEMS IN THE DEFINITION ANDINTERPRETATION OF ATTRIBUTABLE FRACTIONS

SANDER GREENLAND1 AND JAMES M. ROBINS2

The concept of attributable fraction (1-3) has grown in importance as epidemiolo-gists and epidemiologic data have played alarger role in interventions, regulations,and lawsuits concerning hazardous expo-sures. For example, in a lawsuit, the courtmay wish to determine the likelihood thata particular case's illness was caused by theexposure at issue, and the attributable frac-tion has been interpreted as just this like-lihood (e.g., see ref. 4, p. 164).

While the concept is known by manynames (including attributable risk (5), eti-ologic fraction (4, 6, 7), and attributableproportion (8)), we would think this varietywould cause no problem as long as theconceptual and algebraic formulations wereunambiguous. Unfortunately, at least threedistinct concepts have been variously iden-tified as the attributable fraction, althoughthese concepts have usually not been dis-tinguished in the literature. Furthermore,certain equations used to relate attributable

1 Division of Epidemiology, UCLA School of PublicHealth, Los Angeles, CA 90024-1772.

2 Occupational Health Program, Harvard School ofPublic Health, Boston, MA.

The authors thank Drs. Norman Breslow, HarveyCheckoway, Douglas Crawford-Brown, Ralph Frer-ichs, Jennifer Kelsey, Hal Morgenstern, Neil Pearce,Charles Poole, and Kenneth Rothman for their helpfulcomments.

fractions to incidence and relative risk failto hold in many circumstances. These prob-lems are of some importance because of therecent appearance of attributable fractionconcepts in legislation (9,10). We will showthat the conceptual problems appear toarise from a failure of some definitions totake account of time of incidence whenevaluating the role of the study exposure indisease etiology. These conceptual prob-lems are distinct from study validity issues(such as misclassification, selection bias, orsampling error) and thus constitute an ad-ditional obstacle to valid estimation of ex-posure effects.

EXCESS VERSUS ETIOLOGIC FRACTIONS

Suppose we are asked to estimate thefraction of leukemia cases attributable toexposure within a cohort of former militarypersonnel who had been exposed to radia-tion from a nuclear weapons test. It is notclear from this question whether a case"attributable to exposure" is 1) a case forwhich exposure played an etiologic role,that is, for which exposure was a contribu-tory cause of the outcome (an "etiologiccase"), or 2) a case that would not haveoccurred had exposure not occurred (an"excess case").

All excess cases are etiologic cases, butnot vice versa. We will illustrate this point

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1186 GREENLAND AND ROBINS

and show that the distinction of these casescan be of critical importance: From thestandpoint of both law and biology it canbe essential to measure the fraction of allcases that are etiologic cases. Unfortu-nately, in traditional definitions, the attrib-utable fraction measures only the fractionof all cases that are excess cases, and thiscan be much smaller than the fraction ofcases that are etiologically attributable toexposure.

To illustrate these points mathemati-cally, suppose that incidence is evaluatedover a specified risk period or time interval(0, t) after exposure at time zero (this in-terval may vary across individuals, al-though we will treat it as constant in thefollowing development). In the leukemiaexample, t might be "20 years from the dateof the test." Furthermore, suppose that ex-posure action follows a deterministicmodel, so that there are only three types ofexposed subjects who become cases duringthe interval:

Type 0: The exposure had no impactwhatsoever on the case's incidence time.

Type 1: The exposure made the case'sincidence time earlier than it would havebeen in the absence of exposure (so expo-sure played a role in the etiology of thiscase), but had exposure never occurred (orhad its effect been blocked), this subjectwould still have become a case by t, al-though later in the interval.

Type 2: Had exposure never occurred,the subject would not have become a caseby t because, in the absence of exposure,disease would have occurred after t, or notat all.

Let the number or set of each of thesetypes be denoted Ao, A\, and A2, respec-tively, with A+ = Ao + Ai + A2, and let Mequal the total number of cases understudy. (M sometimes equals A+, as in tra-ditional standardized morbidity ratio(SMR) studies in which only an exposedcohort is studied, and M sometimes equalsA+ plus unexposed cases, as in a case-control study.) We think it clear that a caseof type 0 is not "attributable" to exposure

and a case of type 2 is. Furthermore, type2 cases correspond exactly with excesscases, as defined earlier. Type 2 cases arealso etiologic cases, as defined earlier.

What about type 1 cases? Like type 2cases, they are etiologic cases, since expo-sure played a role in the etiology of theirdisease. They are not, however, excesscases, because they still would have becomecases by time t had exposure not occurred.The issue of whether to count these casesas attributable to exposure is importantbecause, as we will show, their number maybe large relative to excess (type 2) cases.

Some textbooks can be interpreted toimply that only excess cases contribute tothe attributable fraction, so that the lattershould be algebraically equivalent to A2/M(which we will call the excess fraction).Consider the following definitions of attrib-utable fraction (the first two of which as-sume M = A+): "the proportion of the casesof disease occurring among exposed personswhich is in excess in comparison with thenonexposed" (5, p. 74); "[the attributablefraction] conveys a sense of how much ofthe disease in an exposed population canbe prevented by blocking the effect of ex-posure or eliminating the exposure" (8, pp.38-9); "the proportion of disease in thetarget population that would not have oc-curred had the factor been absent" (6, p.44); "the proportion of the disease occur-rence that would potentially be eliminatedif exposure to the risk factor were pre-vented" (3, pp. 39-40).

Since type 1 cases become cases by theend of the risk period whether or not theyare exposed, they cannot be counted amongthe proportion of disease that would nothave occurred had exposure been absent,prevented, or eliminated. Thus, it seems tous that such cases would not be counted bythe above definitions. Nevertheless, it ispossible that within the risk period, a type1 case may have suffered a considerableloss of healthy, productive life because ofexposure's effect.

Some textbooks could be interpreted toimply that all etiologic cases—both type 1

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DEFINITIONS OF ATTRIBUTABLE FRACTIONS 1187

and type 2—should contribute to the at-tributable fraction, so that the latter shouldbe algebraically equivalent {Ax + A2)/M(which we will call the etiologic fraction).Consider the following definitions of attrib-utable fraction (all of which assume M isrestricted to exposed cases only, i.e., M =A+): "the proportion of exposed cases thatare due to the risk factor" (4, pp. 163-4);"the proportion of the actual cases in theindex [exposed] domain that are caused bythe cause at issue [if the cause is neverpreventive]" (7, p. 255); "[the attributablefraction] can be interpreted as the propor-tion of exposed cases for whom the diseaseis attributable to exposure" (8, p. 38).

Since type 1 cases are caused by theexposure (i.e., exposure is a contributory orcomponent cause of their disease), theywould be counted among the proportion ofcases "due to" or "attributable to" exposureif "due to" or "attributable to" is given theinterpretation of "caused by." Kleinbaumet al. (4) give another definition, of consid-erable legal interest, to the effect that theattributable fraction " . . . may also be in-terpreted as the probability that a ran-domly selected case from the populationdeveloped the disease as a result of the riskfactor" (4, p. 160). This "probability-of-causation" definition appears to us to cor-respond to (Ai + A2)/M, since the latter isthe probability that a randomly selectedcase had exposure as a contributory cause.

Algebraic definitions

Several textbooks also offer algebraicdefinitions of attributable fractions, and insome situations, the defining formulas arenot equivalent to possible interpretationsof the verbal definitions. For example,Miettinen (7, pp. 254-5) defines the attrib-utable fraction among the exposed as (Ox —E\)/Ou where O\ is the observed number ofexposed cases, that is, O1 = A+, and Ei isthe number of exposed cases that wouldhave occurred had the exposed populationnot experienced the exposure effect. Be-cause type 1 (as well as type 0) cases wouldhave become cases by t even if the exposure

effect was absent, it is apparent that Ex =Ao + Ai and so Or - Ex = A2, the numberof excess cases only. Thus, Miettinen's for-mula equals the excess fraction. On theother hand, Breslow and Day (5, p. 74) andRothman (8, p. 38) algebraically define theattributable fraction in terms of incidencedensities. As we discuss later, these defini-tions are not in general equivalent to eitherthe excess fraction A2/M or the etiologicfraction (Ax + A2)/M, although under cer-tain biologic models they will be equivalentto the latter quantity (11).

RELATIONS BETWEEN QUANTITIES

We recognize that several of the passagesquoted above may have more than one pos-sible interpretation. Nevertheless, it is ap-parent that there are several different con-cepts of attributable fraction in use. Thisobservation raises two questions: 1) Howfar apart will the quantities correspondingto the different concepts be? 2) Which ofthese quantities are estimated by the at-tributable fraction estimates offered in theliterature?

The answers to both questions hinge onthe observation that the quantities Ao andAi are not empirically distinguishable with-out strong biologic assumptions; only thetotal Ao + Ai can be estimated without suchassumptions, even if there is no bias in thestudy. To see this, consider again the leu-kemia illustration. Let t = 20 years, with atotal of 24 exposed cases occurring by t, nocases occurring in the first five years afterexposure, and six cases occurring in the lastfive years before t.

Example 1

Suppose the effect of exposure had beento "age" everyone five years with respect totheir leukemia risk, that is, the effect ofexposure was to make leukemia occur fiveyears sooner among those persons destinedto contract leukemia (in the absence ofother causes of death). Then the six sub-jects who became cases in the last five yearsbefore t would have remained leukemia-freeup to t had exposure not occurred, while

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the remaining cases still would have con-tracted leukemia by t. Hence A2 = 6, andthe excess fraction (up to t) among theexposed would be 6/24 = 0.25. But, underthis "uniform aging" biologic model, theexposure was a contributory cause in everyone of the 24 cases that occurred, so thatAi + A2 = 24 and the etiologic fractionamong the exposed is 24/24 = 1.0.

Example 2

Suppose the effect of exposure had beento produce leukemogenic marrow-cell mu-tations in six of the exposed subjects, withleukemia arising from these mutationswithin the 20 years, but had no effect onleukemia risk in the remaining subjects.Suppose also that 18 leukemias etiologi-cally unrelated to exposure ("spontaneouscases") occurred in the remainder. Then A^= 0, A2 = 6, and so the excess and etiologicfractions would be identical, that is, 6/24 =0.25.

In both examples, the exposure producedsix cases in excess of the 18 that would haveoccurred had exposure or its effect beenabsent or prevented, so that the true excessfraction was 6/24 = 0.25. But the etiologicfraction was four times higher in the firstexample than in the second, and four timeshigher than the excess fraction in the firstexample. Given a perfect unexposed com-parison group for this study, we would beable to accurately estimate the number ofleukemia cases to expect among the ex-posed if exposure had been absent. But thisinformation would only allow us to estimatethe excess fraction. The residual number18 = Ao + Ai could not be further parti-tioned without assumptions about the bio-logic process leading from exposure to dis-ease.

The following example, while somewhatabsurd in its extremity, shows a rare situ-ation in which biologic knowledge is sostrong that both fractions can be preciselycomputed. It also illustrates how, for inev-itable outcomes, the excess fraction willapproach zero as follow-up time t becomes

large, while in general the etiologic fractionwill not do so.

Example 3

Consider the 1860 United States birthcohort, with overt (nervous-system) rabiesas the exposure, death as the outcome, andt = 120 years from infection. Then theetiologic fraction among the exposed is one,since (as far as is known) overt rabiescaused death in all its victims before theadvent of modern life-support systems.Nevertheless, all these victims would havedied within 120 years anyway—so that theexcess mortality produced by rabies (oranything) by 120 years of follow-up is zero.Thus, the excess fraction is zero.

Examples 1-3 demonstrate that the ex-cess fraction A2/M and the etiologic frac-tion (Ai + A2)/M may be arbitrarily farfrom one another. More generally, the ex-cess fraction, A2/M, can never exceed theetiologic fraction, (Ai + A2)/M, and mustbe strictly less than the latter if Aj > 0 (aswill be the case if exposure is not a neces-sary cause and t is large enough). It followsthat an unbiased estimate of the excessfraction will often be a null-biased estimateof the etiologic fraction. As is apparentfrom example 1, this bias can be dramaticin realistic cases, and increases with follow-up time t. Parallel results can be obtainedunder a stochastic model for individual ef-fects (11).

As noted before, strong biologic assump-tions may be needed to determine the eti-ologic fraction. If the exposure is neverpreventive, the necessary and sufficientcondition for the etiologic fraction to equalthe excess fraction is that Ax = 0, that is,that no cases caused by exposure wouldbecome cases in the absence of exposure.There are several biologic conditions underwhich this will be so. For example, in asituation in which an exposure is a neces-sary cause of the outcome (as in manyfoodborne disease outbreaks), Ao = A\ = 0for that exposure, and so both fractions willbe one.

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Relations to incidence time

Whether an etiologic case is an excesscase depends on how much exposure ad-vanced the time of disease incidence. Forexample, an etiologic case occurring atfollow-up year 10 will not be an excess caseby year 25 unless exposure advances inci-dence time by more than 15 years. Thus,the excess fraction directly depends on theamount by which exposure advances inci-dence time, albeit in a crude fashion.

In contrast, an etiologic case remains anetiologic case regardless of the degree towhich exposure advances incidence time.To take an extreme example, suppose in astudy of the first battle of the Somme (in1915), we wish to determine the fraction ofdeaths caused by machine-gun hits (expo-sure is thus being hit by a machine-gunbullet). Consider a soldier hit in the headand killed instantly by a machine gun just10 seconds before an artillery shell ex-ploded in his trench. The cause of the sol-dier's death was a machine-gun hit, and sothe soldier is an etiologic case. This is soregardless of whether, had the machine gunmissed, the soldier would have been killedby the artillery burst 10 seconds later orthe soldier would have survived the artilleryburst and died 70 years later.

The preceding example shows that theetiologic fraction (which is the fraction ofcases for whom exposure advanced the timeof incidence) is insensitive to how muchexposure advances the time of incidence.This remains so even if one inteprets theetiologic fraction as the "probability ofcausation": in the preceding example, theprobability that the soldier was killed by amachine-gun hit is one, regardless of howlong the soldier would have survived hadthe machine gun missed. As will be dis-cussed later, such insensitivity renders theetiologic fraction and probability of causa-tion inappropriate for certain applications.

INCIDENCE FRACTIONS

One often sees the attributable fractiondefined as the fraction of the incidence rate

"attributable" to exposure, that is, the ex-cess incidence rate in the exposed expressedas a proportion of the total incidence ratein the exposed. As is apparent from theliterature (2-8), there are several differentways to define "incidence rate." Each pos-sible definition of incidence rate leads to adifferent quantity for the attributable frac-tion. For some definitions, the quantity isnot equivalent to either the excess or theetiologic fraction as defined above.

Incidence-proportion fractions

Consider first the definition in which the"incidence rate" is the proportion of aclosed (i.e., uncensored) cohort that con-tracts a disease over a specified time inter-val, that is, "incidence rate" is taken to bethe incidence proportion (7) (i.e., averagerisk (4) or cumulative incidence (4, 8)).Given an exposed cohort of size JVi, theincidence proportion (IP) over the intervalis IPi = A+/Nu whereas the proportion thatwould have contracted the disease had ex-posure been absent is IP0 = (Ao + Ai)/Nx.It follows that the incidence-proportion dif-ference expressed as a fraction of the ex-posed incidence proportion is

IPi - IPo A+/N,. - (Ao + A1)/N1

IP i

A+ — Ao - A*

A+

The latter term is simply the excess frac-tion among the exposed. Thus, defining theattributable fraction as the fraction of theincidence proportion "attributable" to ex-posure is algebraically equivalent to theexcess fraction definition given earlier.Note, however, that the definition in termsof incidence proportions is restricted toclosed cohorts (since the incidence propor-tion must be defined in reference to a closedcohort), whereas the excess fraction is de-fined for any population.

Incidence-density fractions

Consider next definitions in which the"incidence rate" is the instantaneous inci-

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dence density (ID) (hazard rate or person-time rate), so that the attributable fraction(at time u) is defined as (IDi - ID0)/ID!,where IDi and ID0 are the incidence den-sities (at time u) when exposure is presentand absent (cf. references 4, 5, and 8).

As has been noted elsewhere (12-16), itis possible for an exposure which onlycauses and never prevents disease to haveIDi < ID0 over certain time intervals fol-lowing exposure (the "crossing hazards"phenomenon). The instantaneousincidence-density ratio IDi/ID0 will be lessthan one and the quantity (IDi - ID0)/IDiwill be negative over such intervals. Sinceneither the excess nor the etiologic frac-tions can be negative for purely causal ex-posures, such examples show that (IDi —ID0)/IDi is not equivalent to either frac-tion. The following example shows that,even if there are no competing risks andIDi, IDo, and their ratio are constant overtime, the quantity (IDj — ID0)/IDi may stillbe far from either fraction.

Example 4

Suppose at time zero, we randomly sam-ple a large number of exposed persons (in-dexed by 0 and proceed to follow them.Assume that each person i would have hada death time Doi if unexposed, but woulddie instead at time Du = DOi/2 when ex-posed. Finally, assume that the Doi are ex-ponentially distributed with expectation T;as a consequence, the Du will be exponen-tially distributed with expectation T/2 (thisis a simple special case of the accelerated-life model given by Cox and Oakes (17,equation 5.4)). Since exposure cuts every-one's lifetime in half, the etiologic fractionis 1. However, the expected death rates IDiand IDo in the presence and absence ofexposure will be 2/T and 1/T, respectively,so that IDj/IDo = 2 and (ID! - ID0)/ID, =0.5, much less than the true etiologic frac-tion. Furthermore, the incidence propor-tions under exposure and nonexposure attime T will be

1 - exp[-(2/T)/T] = 1 - e~2 and

1 - exp[-(l/T)/T] = 1 - e~\

so that at time T the excess fraction will be

[(1 - e"2) - (1 - e-l)]/(\ - e~2) = 0.27,

much less than (IDi - IDOVIDL Note thatthese results hold if ID! and ID0 are inter-preted as either instantaneous or average(interval) incidence densities. (In this ex-ample, (IDi — ID0)/IDi does equal the pro-portionate reduction in life expectancy dueto exposure. This relation is, however, aconsequence of the constancy of the deathrates, and does not hold in general.)

The quantity (IDi - ID0)/IDi has beentermed the "assigned shares" in the riskassessment literature (15, 16); because itcan take on negative values, we propose toinstead call it the incidence-density frac-tion. This fraction has no general relationto excess and etiologic fractions in that itmay fall above, between, or below the otherfractions. Nevertheless, it does have sys-tematic relations to the other fractions un-der certain biologic models. For example,under certain models, (ID! - ID0)/IDi willequal the etiologic fraction (16, 18), and,under a broader class of models, it is betterthan the excess fraction as a lower boundfor the etiologic fraction (18). Thus, theincidence-density fraction may be useful inthe estimation of the etiologic fraction,provided one does not lose sight of theassumptions required for such use.

If the incidence-density fraction iscomputed using average instead of instan-taneous densities, it can, in special circum-stances, approximate the excess fraction.Consider again a closed cohort of initialsize iVi with average incidence density IDiif exposed and ID0 if unexposed, and inci-dence proportion IP! if exposed and IP0 ifunexposed. Let IDR be the incidence-density ratio IDi/IDo, and let IPR be theincidence-proportion ratio IPi/IP0. If thedisease is rare over the study interval, IDRwill approximate IPR (3, 4), so that (IDi -IDo)/IDi = (IDR - 1)/IDR will approxi-

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mate the excess fraction (IP! - IP0)/IPi =(IPR - 1)/IPR for the cohort. Further-more, since the ratio of the average densi-ties (IDR) will exceed IPR in a closed co-hort if IPR > 1, (IDR - 1)/IDR can serveas an upper bound for the excess fractionin the cohort, even if the disease is not rare.

Attributable fractions and relative risks

One often sees expressions for computingan attributable fraction (AF) from someform of relative risk (RR) (i.e., a risk, rate,or odds ratio), for example AFe = (RR —1)/RR, where AFe is the attributable frac-tion among the exposed, or AFP = PC(RR- 1)/RR = PcAFe, where AFP is the popu-lation attributable fraction and Pc is theexposure rate among cases (2-8). The re-sults given earlier show that if RR is inter-preted as the incidence-density ratio, thesecomputing formulas are not always validfor estimating either the excess or the eti-ologic fraction, whereas if RR is as theincidence-proportion ratio, the computingformulas will be valid for estimating theexcess fraction in a closed cohort. It followsthat if RR is replaced by an odds ratio, thecomputing formulas will validly approxi-mate the excess fraction only insofar as theodds ratio approximates the incidence-proportion ratio.

RELEVANCE OF THE MEASURES

In the preceding sections, we have arguedfor the need to distinguish three conceptsof attributable fraction: the excess fraction,the etiologic fraction, and the incidence-density fraction. In this section, we wouldlike to examine the relevant domain ofapplication of these quantities in publichealth.

If disease status as of time t is the onlyrelevant aspect of an application, the excessfraction is the relevant measure. Consider,for example, the issue of the effect of oxy-tocin use on intrapartum death rates: Froma public health perspective, the outcome ofinterest would be whether a death occurredby time t (end of delivery), not when the

death occurred, and so the excess fraction(or its preventive analogue) would be therelevant measure. The instances in whichthe treatment delayed or accelerated aninevitable death would be of interest instudying the mechanism of treatment ac-tion, but would not count for or against theeffectiveness of the treatment in preventingor causing intrapartum deaths. In manyother planning and policy questions, theexcess caseload that exposure would pro-duce over an interval must be estimated,and here again the excess fraction is therelevant parameter.

In many situations, when the disease oc-curs is (or should be) of as much or morepublic health (and legal) concern thanwhether it occurs by some time t. For adisease inevitable by t (as in example 3, inwhich the disease is death and t = 120years), time of occurrence is the only rele-vant issue. The excess fraction does notcapture this beyond a simple dichotomy,and it is an inadequate measure if time ofoccurrence in the interval (0, t) is impor-tant.

Unfortunately, even if we know exactlywhat the etiologic fraction is, it is not nec-essarily a useful measure of the effect ofexposure on disease occurrence. To see this,compare the impact of the genetic condi-tions that produce Tay-Sachs disease andHuntington's chorea. Both conditions leadto premature death, and both may be con-sidered to have etiologic fractions for death(among the exposed) that approach one.Nevertheless, persons who develop Tay-Sachs disease die in early childhood,whereas persons with the gene for Hun-tington's chorea usually survive well intoadulthood and can lead rich, if shortened,lives. The etiologic fraction is not sensitiveto this distinction.

Interestingly, in the preceding example,the excess fraction at age 20 years wouldclearly distinguish between the two condi-tions (since it would be near one for Tay-Sachs and near zero for Huntington's cho-rea), as would the incidence-density frac-

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tion in early childhood. More generally,however, we would suggest turning atten-tion to direct measures of exposure effecton incidence time whenever the latter isimportant. For example, one could examineexpected years of life lost (mean reductionin life expectancy).

Consider again examples 1 and 2: Onecan estimate the average number of yearsof leukemia-free life lost by exposed cases,provided one can construct a reasonableestimate of the leukemia-free survival curvein the absence of exposure (19). The latterconstruction reduces to the common meth-odological problem of finding a good com-parison group for the exposed population,and thus (unlike the etiologic fraction) isapproachable by standard epidemiologicmethods.

For public health purposes, impact mea-sures such as reduction in life expectancymake use of incidence-time information ig-nored by the excess fraction, while avoidingthe strong biologic assumptions usually re-quired to estimate the etiologic fraction andsome of the problems (such as crossinghazards) that can occur with incidence-density fractions. The greater emphasis onattributable fractions in the epidemiologicliterature may be in part due to their sim-plicity, and in part due to the fact that ofthe possible measures, only the excess andincidence-density fractions are directly es-timable from case-control data without re-strictive assumptions (18, 19). These arenot, however, sufficient reasons for neglect-ing other measures of impact.

Although in the previous example yearsof life lost is a more relevant measure ofexposure impact than the etiologic fraction,this will not always be so, since relevancewill often strongly depend on social andethical issues. For example, a large etiologicfraction for homelessness as a risk factorfor death would be of social concern, evenif removing the exposure (homelessness)would result in only slight additional sur-vival time for some persons (e.g., providingdormitories would prevent deaths due tofreezing, even though some rescued persons

might soon die of effects of chronic alco-holism). Consideration of other examples,in which years of life lost or the excessfraction would be considered more relevant,shows that no single measure can be re-garded as universally preferable.

IMPLICATIONS FOR INDIVIDUAL

COMPENSATION

Although no single measure is univer-sally preferable, the etiologic fraction hasbecome established in current legal think-ing regarding compensation for harmful ex-posure, usually under the heading of "prob-ability of causation" (15, 16). Unfortu-nately, as we have shown, one cannotestimate the etiologic fraction without re-sorting to very strong biologic assumptions;this fact can have dramatic implications forpersonal-injury suits.

Because of the inability to identifyexposure-induced cases, Hatch (9) has pro-posed that monetary awards for personal-injury suits be made in the following man-ner: First, the dollar amount V appropriateto compensate a single exposure-inducedcase is determined; then, each exposed caseis awarded the (exposed) attributable frac-tion of this amount, AF6 • V. If only excess(A2) cases are considered relevant (as in theperinatal example), one could simply sub-stitute an estimate of the excess fractionfor AFe. But if all persons who contractexposure-induced disease are consideredexposure victims (as in the leukemia ex-ample), the exposed etiologic fractionshould be used for AFe, for only the etio-logic fraction is interpretable as the pro-portion of exposed cases with exposure-induced disease. Thus, the dilemma is notresolved: Assuming the model in example1, it would be reasonable to claim thatexposure harmed all the exposed cases;after all, if not for exposure, all the caseswould have had more years of healthy lifethan they did. Nevertheless, the very samedata (24 exposed cases observed when 18should be expected under nonexposure) arecompatible with the model in example 2, inwhich exposure harmed only one fourth of

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the exposed cases. Note that larger num-bers would do nothing to resolve this di-lemma.

The same problem arises for legislationmandating full compensation in individualdamage suits if and only if the probabilitythat the plaintiffs disease was induced byexposure exceeds 50 per cent (10). For arandomly sampled case, this probability is(Ai + A2)/A+, the etiologic fraction. Use ofan excess fraction to estimate this proba-bility would yield an estimate biasedagainst the plaintiff; use of the incidence-density fraction would also yield a biasedestimate except in special cases (16, 18).

Multiple factors

The existence of biologic interactionsraises difficult issues for the use of attrib-utable fractions in compensation (20). Forexample, if some persons develop lung can-cer solely because of their exposure to as-bestos and smoking, such persons will con-tribute to the excess and etiologic fractionsfor both asbestos and smoking. As is wellknown (8), in such situations the attribut-able fractions for asbestos and smoking ascauses of lung cancer among the jointlyexposed can (and in fact do) sum to morethan one. If a jointly exposed case receivescompensation from each party responsiblefor each exposure, and the compensationfrom each is determined as an attributablefraction times the total loss incurred by thecase, the total of all awards could exceedthe total loss.

One theoretical solution to this problemis as follows: Suppose that two factors xand y are at issue, and let AFX, AF^, andAFX>, be the proportion of cases exposed toboth factors for whom disease was attrib-utable to x but not y, to y but not x, and toboth x and y, respectively. Here, "diseaseattributable to" a factor or factors can meaneither an excess case (i.e., disease would nothave occurred without the factor(s)) or acase with an etiology involving the fac-tors). Next, suppose it is decided that, forcases with disease attributable to both xand y, the party responsible for x should

pay a proportion Px of the total compensa-tion and the party responsible for y shouldpay the remainder. Finally, let V be thevaluation for the total loss one case incursfrom the disease. If cases with disease at-tributable to x and y cannot be distin-guished from the other jointly exposedcases, a jointly exposed case could receive(AFX + Px AF^) V from the party responsi-ble for x and [AFy + (1 - PJAF^] V fromthe party responsible for y; the total com-pensation for a jointly exposed case wouldthen be (AF* + AFy + AF^) V.

As in the univariate situation, AFX) AFy,and AFxy may be estimated using standardepidemiologic methods if they represent ex-cess fractions, whereas their estimation willrequire strong biologic assumptions if theyrepresent etiologic fractions (18, 20). Inparticular, expressions for AFX, AFy, andAFxy in terms of incidence densities (e.g.,as in reference 21) are valid only undercertain models.

In contrast to estimation of attributablefractions, determination of Px is a legalrather than a scientific problem. Note that,among jointly attributable cases, fully 100per cent of their disease can be causallyattributed to either exposure consideredsingly. In a causal sense, both factors maybe viewed as equally responsible for suchcases, in that both factors are necessarycauses of such cases. This observation doesnot, however, imply that the two responsi-ble parties should pay an equal share of thecompensation for such cases. For example,in current practice, lung cancer victims whoare exposed to asbestos and who are smok-ers usually receive full compensation fromthe party responsible for asbestos exposure;in effect, then, the courts hold the latterparty responsible for all jointly exposedcases.

Years of life lost

The above compensation rules take noaccount of the amount of healthy life lostby an exposed person. For example, therules make no explicit distinction betweenan exposure-induced case that occurs at age

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25 years and one that occurs at age 85 years.If expected years of life lost are consideredrelevant, one could estimate expected yearsof life lost separately for cases that occurat different ages, using the survival distri-butions in exposed and unexposed popula-tions, and provide compensation in propor-tion to expected years lost. Unfortunately,unlike the overall expected years of life lost(but like the etiologic fraction), the ex-pected years of life lost among cases thatoccur at a particular time is not estimablewithout strong biologic assumptions (19).

FURTHER ISSUES

Severity of outcome

In the above development, we have as-sumed that the chief manifestation of ex-posure playing an etiologic role in an out-come event is that the outcome event oc-curs earlier than it would have in theabsence of exposure. Thus, cases who suf-fered alteration of severity of outcomewithout alteration of time of outcome wouldnot be "etiologic cases" in the above sense.Severity of outcome is often not an issue(e.g., in mortality studies), but it can be ofcrucial importance in some contexts (e.g.,compensation for pneumoconiosis). Sever-ity issues can be dealt with in terms oftransition times to different degrees of se-verity. Here, we note only that, if onewished to include as "etiologic" all caseswith altered severity, the gap between theetiologic and excess fractions could belarger or smaller than those illustratedhere, depending on the situation.

Attributable fractions and susceptibleproportions

Recently, Khoury et al. (22) have soughtto revive interest in the concept of theproportion of persons susceptible toexposure-induced disease. Under a deter-ministic model for exposure effects, an ex-posed person is susceptible to exposure-induced disease by time t if, in the absenceof competing mechanisms, a mechanisminvolving exposure would induce disease in

the person by time t. In terms of a sufficientcomponent cause model (8), a person issusceptible to exposure-induced disease if asufficient cause involving exposure wouldbe completed by time t if exposure is pres-ent and competing events do not occur.(This definition of susceptibility should notbe confused with the definition used byMiettinen (7) and elsewhere by us (23), inwhich a susceptible is a person who wouldbe an excess case in the presence of expo-sure.) Among the exposed, this class ofsusceptibles includes but is potentiallylarger than the class of etiologic cases (Ar

+ A2), for it includes certain cases whoseetiology did not involve exposure (i.e., itincludes certain type 0 cases, as well as alltype 1 and type 2 cases). Specifically, theclass of susceptibles includes those type 0cases who would (by time t) have con-tracted disease from a sufficient cause in-volving exposure if sufficient causes notinvolving exposure had been absent. Aswith the etiologic fraction, estimation ofthe susceptible proportion requires strongbiologic assumptions, although broad upperand lower bounds can be estimated for thequantity (22).

Attributable fractions and cofactorsAttributable fractions, like relative risks,

are highly dependent on the prevalence ofcofactors of exposure. (By "cofactors," wemean factors that enhance (causal cofac-tors) or reduce (preventive cofactors) ex-posure effects on risk.) For example, thegene for phenylketonuria can lead to severemental retardation only when dietaryphenylalanine is above a certain level (34).It follows that both the excess and etiologicfractions for the phenylketonuria gene as acause of mental retardation will dependdirectly on the distribution of dietary phen-ylalanine levels in the study population,and will approach zero in a population withuniformly low phenylalanine diets. In asimilar fashion, attributable fractions andrelative risks depend on the incidence ofcompeting causes (i.e., causal mechanismsthat do not involve exposure).

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As has been noted elsewhere (8), thedependency of epidemiologic measures oncofactor distributions points out the needto avoid considering such measures as bio-logic constants. Rather, epidemiologicmeasures are characteristics of particularpopulations under particular conditions(analogous to the way in which measuressuch as relative weight and daily caloricintake are characteristics of particular peo-ple under particular conditions, and so arenot biologic constants). This should espe-cially be borne in mind if attributable frac-tions are used to decide compensation, foruse of estimates from the literature mustassume that the population of potentialplaintiffs experienced effects similar tothose seen in study populations.

Preventive factorsResults for prevented fractions (2, 4, 8)

parallel to those given here for attributablefractions may be obtained by noting thatpreventive action for a factor is logicallyequivalent to the factor's absence acting asa cause (24). In particular, two types ofprevented fractions should be distinguishedwhen considering a purely preventive ex-posure. Let Ao be the number of exposedcases who experienced no preventive action(delay in disease occurrence) from expo-sure; let Ax be the number of exposed caseswho experienced some preventive actionfrom exposure; let A2 be the number ofexposed persons in the study populationwho did not become cases but would haveif exposure had been absent; let A+ = Ao +Ax + A2. Then A2/A+ is the actual caseloadreduction produced by exposure, and is thepreventive analog of the excess fraction.

On the other hand, (Ai + A2)/A+ is thefraction of potential and actual cases whoexperienced some preventive action fromexposure, and is the preventive analog ofthe etiologic fraction. Like the etiologicfraction, it is not identifiable withoutstrong biologic assumptions; in particular,formulas for estimating this fraction fromincidence densities (e.g., reference 4, p. 166)will be valid only under certain models.

Statistical methods

Beginning with the work of Walter (25),there has been extensive development ofestimation techniques for attributable frac-tions, especially adjusted estimators (e.g.,see references 26-30). The results pre-sented here show that these estimatorsshould be interpreted with caution. De-pending on the sampling design of thestudy, the estimated parameter may beeither an excess fraction or an incidence-density fraction. Furthermore, although ad-justment may produce valid estimates ofthe latter two measures, in general, oneshould not expect it to produce a validestimate of the etiologic fraction (18).

Terminology

The number of terms for attributablefractions is perhaps the largest of any con-cept in epidemiology. Two traditions areextant. One tradition continues to employLevin's original term "attributable risk"(31), ignoring the fact that this term refersto the risk difference in several widely usedtextbooks (e.g., references 32 and 33), andthat the quantity at issue is not itself a risk.For these reasons, a second tradition aroseof amending the term "attributable risk" to"attributable risk proportion" (32), "attrib-utable proportion" (8), "etiologic fraction"(4, 6, 7), or the hybrid term "attributablefraction" (1-3). Neither tradition is ade-quate to cope with the fact that at leastthree different quantities should be distin-guished: the fractional excess caseload pro-duced by exposure, A2/M, which we havelabelled the "excess fraction"; the fractionof cases for whom exposure played a role inthe etiology of their disease, {Ax + A2)/M,which we have termed the "etiologic frac-tion"; and the incidence-density differenceexpressed as a fraction of the exposed in-cidence density, (IDt — ID0)/IDi, which wehave termed the "incidence-density frac-tion." The incidence-density fraction canbe further subdivided into two types, ac-cording to whether instantaneous or aver-age densities are used. We have used the

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term "attributable fraction" to refer to thefamily formed by these concepts. We canonly hope that our proposed terminologyhelps resolve the conceptual confusion sur-rounding attributable fractions.

SUMMARY

We have argued that the concept of at-tributable fraction requires separation intothe concepts of excess fraction, etiologicfraction, and incidence-density fraction.These quantities do not necessarily approx-imate one another, and the etiologic frac-tion is not generally estimable withoutstrong biologic assumptions. For these rea-sons, care is needed in deciding which (ifany) of the concepts is appropriate for aparticular application. It appears that theexcess fraction (like incidence proportion)will be most relevant in situations thatrequire only consideration of whether dis-ease occurs by a particular time. In situa-tions that require consideration of whendisease occurs, direct measures of effect onincidence time may be as relevant as ormore relevant than any attributable frac-tion.

To avoid technical complications, wehave not discussed additional problems ofcausal attribution that can arise when ex-posure has multiple levels or is sustainedover time, and the estimation problems thatcan arise when considering case-controlstudies, competing risks, or differentialcensoring. For more detailed discussions ofsuch problems and proposed solutions, seereferences 11-20.

REFERENCES

1. Ouellet BL, Romeder J-M, Lance J-M. Prematuremortality attributable to smoking and hazardousdrinking in Canada. Am J Epidemiol 1979; 109:451-63.

2. Last JM, ed. A dictionary of epidemiology. NewYork: Oxford University Press, 1983.

3. Kelsey JL, Thompson WD, Evans AS. Methodsin observational epidemiology. New York: OxfordUniversity Press, 1986.

4. Kleinbaum DG, Kupper LL, Morgenstern H. Ep-idemiologic research: principles and quantitativemethods. Belmont, CA: Lifetime Learning Publi-cations, 1982.

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studies. (IARC scientific publication no. 32).Lyon: IARC, 1980.

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19. Robins JM, Greenland S. Estimability and esti-mation of years of life lost due to hazardous ex-posure. Technical Report no. 5, OccupationalHealth Program, Harvard School of PublicHealth, Boston, 1988.

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21. Walker AM. Proportion of disease attributable tothe combined effects of two factors. Int J Epide-miol 1981;10:81-5.

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31. Levin ML. The occurrence of lung cancer in man.Acta Union International Contra Cancrum 1953;9:531-41.

32. Mausner JS, Bahn AK. Epidemiology. Philadel-phia, PA: WB Saunders, 1974.

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