In this paper, we use Rothman's definition of a cause - a component is considered to be a cause if it was necessary for the disease to occur at the moment it did in at least one person in the population [22]. In other words, a component is causal if it participated in the first fulfilled SC of disease for at least one person, whether or not this person would have become diseased at a later time by acquiring another SC. The following discussion assumes that our goal is to determine whether a particular component of interest (i.e. the exposure) meets this definition of a cause, and is restricted to the causal effect of the exposure in a group of people actually exposed. Ideally, we would like to compare the disease experience of each individual in a group of exposed people to the group consisting of their counterfactuals. This comparison between the risk of disease in the group of exposed individuals and the risk of disease in the same group of individuals under the condition of non-exposure is referred to as a causal contrast, which is the true but unobservable representation of the causal effect of the exposure in a given population at a given time (note: this is one of several causal contrasts) [23].
Causal contrasts are composed of two parts. The first part of our contrast of interest is the proportion of diseased people (i.e., risk of disease) among the exposed at the end of a specified time period. The second part is the proportion of diseased people among the exposed had they not been exposed (i.e., the counterfactual risk of disease). This causal contrast may refer to the full etiologic effect, or a subset of the full etiologic effect, i.e. the excess effect [3].
Distinction between etiologic and excess causation under the SCC model
Greenland and Robins differentiated between two types of diseased individuals that are germane to our work - etiologic cases and excess cases [3]. An etiologic case is any diseased individual for whom the exposure of interest was a cause [1–3]. An excess case is any diseased individual for whom the disease would not have occurred by the end of the study period in the absence of the exposure. The difference may appear subtle. For an etiologic case, the exposure was necessary for disease to occur at the moment it occurred. For an excess case, the exposure was necessary for disease to occur at all. All excess cases are etiologic cases, but an etiologic case is only an excess case if the exposure was necessary for the disease to occur by the end of the study period [1–3].
We can express Greenland's and Robin's analytic distinction between etiologic and excess cases in terms of the SCC model. An etiologic case is an individual who developed the disease from a SC that included the exposure of interest. An excess case is a type of etiologic case who does not later, over the course of the study, acquire any SC that excludes the exposure of interest (i.e. a non-redundant etiologic case). We can therefore split etiologic cases into excess and redundant cases. An etiologic redundant case is another type of etiologic case who later, over the course of the study, acquires at least one SC that excludes the exposure of interest. All etiologic cases are either excess or etiologic redundant cases. In contrast to an etiologic case, a non-causal case is an individual who developed the disease from a SC that excludes the exposure. The relationship among etiologic, excess, etiologic redundant and non-causal cases is illustrated in Figure 3.
The etiologic fraction, the proportion of cases for whom the exposure was a cause, is equal to the number of etiologic cases divided by the total number of diseased people at the end of the study [1–3]. We can subdivide the etiologic fraction into the excess fraction and etiologic redundant fraction. The excess fraction is the subset of the etiologic fraction that is non-redundant and is equal to the number of excess cases divided by the total number of diseased [1–3]. The difference between the etiologic fraction and the excess fraction is the etiologic redundant fraction, i.e. the number of etiologic redundant cases (i.e. people who got the disease from the exposure but would have gotten it from another SC by the end of the study period) divided by the total number of diseased. The non-causal fraction, the proportion of cases for whom the exposure was not a cause, is equal to the number of non-causal cases divided by the total number of diseased people at the end of the study. The relationship among etiologic, excess, etiologic redundant and non-causal fractions is also illustrated in Figure 3.
Below, we use disease response types to quantify these fractions, as well as the etiologic and excess causal effects and the discrepancy between them. We use the response types as a tool because they are simple and descriptive.
Further specification of response types
In their 1986 paper, Greenland and Robins [9] described four possible disease responses to a given exposure: doomed, susceptible-causative, susceptible-preventive and immune. Response types reflect the way a person will respond, in terms of disease outcome, to an exposure by the end of a defined observation period. Since these response types describe the disease status of exposed individuals at the end of the observation period, they assess the excess exposure effect. To assess the etiologic exposure effect, we have adapted these response types (see endnote 3). Below, we describe three possible disease responses to an exposure of interest; non-causal cases, etiologic cases and non-diseased. The response type of an exposed individual who becomes diseased over the observation period is determined by his/her constellation of components at the moment of disease onset. In contrast, the response type of an exposed individual who does not become diseased over the observation period is determined by his or her constellation of components at the end of the observation period.
An exposed individual who first acquires a SC that does not include the exposure of interest is a non-causal case. The exposure has no effect on disease occurrence among non-causal cases; these individuals are assigned response type 1. An exposed individual who first acquires the causal partners of the exposure is an etiologic case. For etiologic cases, the exposure determines whether and when the disease occurs; these individuals are assigned response type 2. An exposed individual who does not acquire the causal partners of the exposure and does not acquire another SC that does not include the exposure is non-diseased. The exposure has no effect on disease occurrence among the non-diseased; these individuals are assigned response type 4 (see endnote 4).
For example, referring to Figure 2: Independent Mechanism, an exposed person (i.e. someone who has already acquired E) who then acquires U2 and U3 is a non-causal (type 1) case. An exposed person who then acquires U1 is an etiologic (type 2) case. And an exposed person who does not acquire U1 or both U2 and U3 is non-diseased (type 4).
The proportions of individuals in the population who are non-causal cases, etiologic cases and non-diseased are labeled as p1, p2 and p4, respectively (see Figure 3). The proportion of people who develop the disease under the condition of exposure is equal to the sum of the proportions of non-causal and etiologic cases, or p1+p2. The proportion of people who will develop the disease under the etiologic counterfactual condition, i.e., the proportion of exposed individuals who would have been diseased at their moment of disease onset had they not been exposed, is equal to the proportion of non-causal cases, or p1. However, the proportion of people in the same population that will develop the disease under the excess counterfactual condition, i.e. the proportion of exposed that would have been diseased at the end of the study had they not been exposed, includes both: (1) the proportion of non-causal cases, and (2) the subset of the etiologic cases who are etiologic redundant cases. As we will later show, the proportion of etiologic redundant cases is directly related to the extent to which redundancy obscures causal identification.
To identify this subset of the etiologic fraction, we must further specify these response types by distinguishing between redundant and non-redundant cases. We will refer to a person who acquires only one type of SC of disease (i.e. SCs that include the exposure or SCs that do not include the exposure, but not both types) over the study period as "non-redundant". We will refer to a person who acquires both types of SC (where at least one SC includes the exposure and one does not) as "redundant". Among the non-causal cases, we define a person who only acquires a SC(s) that does not include the exposure as a type 1n, where '1n' refers to non-causal, non-redundant. Individuals who are type 1n develop the disease from a SC that does not include the exposure, and do not later acquire a SC that includes the exposure. The proportion of individuals who are non-causal, non-redundant over the study period is designated p1n. A non-causal case that acquires both types of SC(s) is labeled a type 1r, where '1r' refers to non-causal, redundant. Individuals who are type 1r develop the disease from a SC that does not include the exposure but later acquire another SC that does include the exposure. The proportion of individuals who are non-causal, redundant over the study period is designated p1r.
Likewise, among the etiologic cases, we define a person who only acquires a SC(s) that includes the exposure as a type 2n, where '2n' refers to etiologic, non-redundant. Individuals who are type 2n develop the disease from a SC that includes the exposure, and do not later acquire a SC that does not include the exposure (i.e. type 2n individuals are excess cases). The proportion of individuals who are etiologic, non-redundant over the study period is designated p2n. An etiologic case that acquires both types of SC(s) is labeled a type 2r, where '2r' refers to etiologic, redundant. Individuals who are type 2r develop the disease from a SC that includes the exposure but later acquire a SC that does not include the exposure. The proportion of individuals who are etiologic, redundant over the study period is designated p2r. Each of these specific response types, and their contribution to the proportion of exposed diseased individuals are labeled in Figure 3.
Relationship between relative etiologic effect and relative excess effect
Using these proportions, we can calculate the relative etiologic and excess causal contrasts. The numerator of both RRetiology and RRexcess is the proportion of the exposed who develop the disease by the end of the study period; (p1+p2) or (p1n+p1r+p2n+p2r). However, the denominators of RRetiology and RRexcess differ. The denominator of RRetiology includes those individuals who would have developed the disease at the moment they did even in the absence of the exposure; p1 or (p1n+p1r). The calculation of RRetiology in terms of these specific response types is shown in Equation 1.
The denominator of RRexcess includes all individual who would have developed the disease by the end of the study in the absence of the exposure. The denominator of RRexcess includes: (i) those individuals who, in the absence of the exposure, would have developed the disease at the moment they did; p1 or (p1n+p1r), and (ii) those individuals who would have developed the disease later over the study period; p2r. Thus, the denominator of RRexcess is (p1n+p1r+ p2r). The calculation of RRexcess in terms of these specific response types is shown below in Equation 2.
Thus, the only difference between RRetiology and RRexcess is p2r in the denominator of RRexcess. The relationship between these causal contrasts can be expressed as the ratio of RRexcess to RRetiology, which reduces to Equation 3.
As Equation 3 shows, RRexcess underestimates RRetiology to the extent that there are individuals who are etiologic redundant (i.e., type 2r) in the population. As p2r increases relative to p1, the discrepancy between RRetiology and RRexcess increases. As discussed earlier, p2r (and p1r, though it has no effect on excess versus etiology) will be larger as the prevalence of the components increases and/or when redundancy arises from dependent processes.
Here, we used the disease type notation to express the excess and etiologic causal effects in a population of people actually exposed. It is also possible to express redundancy in a general way for target populations of any size with varying exposure distribution (assuming exposure is randomly distributed) and all forms of disease frequency measures. Using the notation in Maldonado's and Greenland's paper "Estimating Causal Effects" (2002) [23], we provide an alternative expression of redundancy in Additional File 1.
In summary, the difference between RRexcess and RRetiology is the extent that there are redundant, etiologic cases in the population; the magnitude of this difference is driven by the structure of the SCC model and the prevalence of its components. Given that both RRexcess and RRetiology are both causal contrasts, there are two possible counterfactuals for the exposed. In addition to specifying whether the causal effect for the exposed, the unexposed or the entire population is of interest, epidemiologists should also specify whether the excess or etiologic causal effect is of interest. Because we are inherently interested in identifying causes, we conceptualize the excess effect as an underestimate of the etiologic effect. In the illustrative example below, we demonstrate the extent to which RRexcess can underestimate RRetiology.
Numeric example: A hypothetical study of 200 people
To make the remaining discussion less abstract, we use the simplistic hypothetical causal model for liver cancer shown in Figure 4. In this hypothetical causal model, there are six dichotomous causal components that make up two SCs of disease. Supposing these are the only causes of liver cancer, an individual can either get liver cancer from exposure to: (1) a metabolic polymorphism of the glutathione-S-transferase gene (the GSTm1 deletion), aflatoxin and U1, or (2) hepatitis C infection, alcohol and U2.
An exposed individual (i.e. a person with the GSTm1 deletion) who acquires hepatitis C, alcohol and U2 before acquiring aflatoxin and U1, is a non-causal (type 1) case. An exposed individual who acquires aflatoxin and U1 before acquiring hepatitis C, alcohol and U2 is an etiologic (type 2) case. An exposed individual who does not acquire both aflatoxin and U1 and does not acquire hepatitis C, alcohol and U2 is non-diseased (type 4).
Imagine a researcher began with 100 pairs of individuals who were identical except for their GSTm1 status; 100 individuals had the GSTm1 deletion and 100 did not. The individuals with the deletion were exposed, and the individuals with normal GSTm1 were their counterfactuals. Furthermore, (for simplicity) imagine that all 200 individuals had components U1 and U2 at the start of observation.
Since the researcher did not know when the individuals will develop liver cancer (i.e. the induction period is unknown), she decided to follow the study population for 12 years. Imagine that liver cancer is immediately apparent and the researcher had a perfect diagnostic test and so did not need to account for a latency period. By the end of the study period, five of 100 exposed and 3 of 100 unexposed individuals were diseased. The risk of disease among the exposed by the end of the 12-year period was 0.05 (5/100), the risk of disease among the unexposed by the end of the same time period was 0.03 (3/100), and observed risk ratio (RRobserved) was 1.7 (0.05/0.03).
Figure 5 displays the component acquisition and disease experience of the exposed individuals who developed liver cancer over the 12-year study period (labeled GD1 through GD5) and their unexposed counterparts (labeled GN1 through GN5). In addition to the 10 individuals shown in Figure 5, the study included 95 exposed people who remained non-diseased and their 95 unexposed counterparts who also remained non-diseased. For diseased individuals, the last component needed to first complete a SC is shown in bold font. In addition, each individual's disease status at the end of the study and specific response type are shown.
Assigning specific response types requires we examine the moment that each exposed individual developed the disease. If we examine the bolded factors in Figure 5, we can identify the moment of disease onset of each exposed individual (GD1 through GD5). Persons GD2 and GD4 were non-causal cases (types 1n and 1r, respectively), GD1 and GD5 were excess cases (type 2n) and GD3 was an etiologic redundant case (type 2r). Therefore, p1 was 0.02 (2/100), p2n was 0.02 (2/100) and p2r was 0.01 (1/100). RRexcess is 1.7 [(0.02+0.02+0.01)/(0.02+0.01)]. Since the unexposed group represents the counterfactuals for the exposed group (i.e. there is no confounding or bias), RRobserved is equal to RRexcess.
RRetiology is 2.5 [(0.02+0.02+0.01)/0.02]. Persons GD3 and GN3 are the only pair in which the disease status of the unexposed counterpart at year 12 is not equivalent to his/her disease status at the moment the exposed person became diseased. From the moment that GD3 and GN3 were exposed to alcohol, the SCs for GD3 were redundant and the counterpart, GN3, became diseased from SC2. At this moment, GN3 was no longer an appropriate comparison for GD3. As a result, RRetiology was underestimated by RRexcess.
Figure 5 also shows that there was also redundancy of SCs for GD4. After developing the disease from SC 2, GD4 also acquired aflatoxin, fulfilling SC 1. In this circumstance, the disease status of the counterpart (GN4) at the end of the study was equivalent to the disease status he or she had at the moment GD4 became diseased. Therefore, GN4 was an appropriate comparison for GD4, and the relationship between RRexcess and RRetiology was unaffected by this pair.
The researcher could correctly conclude that the GSTm1 deletion yielded a relative excess risk of disease of 1.7. However, she could not determine the full etiologic effect of the deletion. Using Equation 3, we can see that the researcher captured 67% [0.02/(0.02 + 0.01) = 0.67] of the relative etiologic effect, or conversely that she missed 33%.
In this example, one etiologic redundant case among the 100 exposed people (person GD3) resulted in an appreciable underestimation of the etiologic effect. Clearly, the underestimation would have been more severe had there been more etiologic redundant cases. For instance, if GD1 had also been an etiologic redundant case, the etiologic effect would remain the same, while the observed RR would have been 1.25 [(5/100)/(4/100) = 1.25]. In this case, the researcher would have missed 50% [1-(0.02/(0.02 + 0.02)) = 0.50] of the relative etiologic effect.
Even in the absence of confounding and bias, our studies underestimate the etiologic effect to the extent that there are etiologic redundant cases in the study population by the end of the study period. Because the presence of redundancy is related to when we compare exposed and unexposed people, it may seem intuitive that we may be able to avoid or correct this underestimation. In discussing the problem of redundancy with colleagues, we find that epidemiologists often believe that redundancy is a latency issue, or can be resolved using matching and/or rate-based measures. However, as we discuss below, this is not the case.