A counterfactual approach to bias and effect modification in terms of response types
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Abstract
Background
The counterfactual approach provides a clear and coherent framework to think about a variety of important concepts related to causation. Meanwhile, directed acyclic graphs have been used as causal diagrams in epidemiologic research to visually summarize hypothetical relations among variables of interest, providing a clear understanding of underlying causal structures of bias and effect modification. In this study, the authors aim to further clarify the concepts of bias (confounding bias and selection bias) and effect modification in the counterfactual framework.
Methods
The authors show how theoretical data frequencies can be described by using unobservable response types both in observational studies and in randomized controlled trials. By using the descriptions of data frequencies, the authors show epidemiologic measures in terms of response types, demonstrating significant distinctions between association measures and effect measures. These descriptions also demonstrate sufficient conditions to estimate effect measures in observational studies. To illustrate the ideas, the authors show how directed acyclic graphs can be extended by integrating response types and observed variables.
Results
This study shows a hitherto unrecognized sufficient condition to estimate effect measures in observational studies by adjusting for confounding bias. The present findings would provide a further understanding of the assumption of conditional exchangeability, clarifying the link between the assumptions for making causal inferences in observational studies and the counterfactual approach. The extension of directed acyclic graphs using response types maintains the integrity of the original directed acyclic graphs, which allows one to understand the underlying causal structure discussed in this study.
Conclusions
The present findings highlight that analytic adjustment for confounders in observational studies has consequences quite different from those of physical control in randomized controlled trials. In particular, the present findings would be of great use when demonstrating the inherent distinctions between observational studies and randomized controlled trials.
Keywords
Bias Causal inference Counterfactual Directed acyclic graphs Effect modification Exchangeability Randomization Response typesAbbreviations
- DAG
Directed acyclic graph
- RR
Risk ratio
Background
The counterfactual approach provides a clear and coherent framework to think about a variety of important concepts related to causation [1, 2]. In particular, the counterfactual approach to confounding has been widely accessible to epidemiologists since the publication of a classic methods paper by Greenland and Robins [3], and the concept of bias is now explained in the counterfactual framework [4, 5, 6, 7, 8, 9, 10, 11, 12]. (Note that an update of the classic methods paper was recently published [13]). Meanwhile, directed acyclic graphs (DAGs) have long been used as causal diagrams in epidemiologic research to visually summarize hypothetical relations among variables of interest [14, 15]. DAGs have been used extensively to determine the variables for which it is necessary to control for confounding bias to estimate causal effects [14, 15, 16, 17, 18, 19, 20]. Besides, Hernán et al. [21] showed that various types of selection bias share a common underlying causal structure, and referred to conditioning on common effects as selection bias. Furthermore, VanderWeele and Robins [22] provided a structural classification of effect modification by using DAGs. Indeed, the different approaches provide complementary perspectives, and can be employed together to provide a clearer understanding of causality [23].
In this study, we aim to further clarify the concepts of bias (confounding bias and selection bias) and effect modification in the counterfactual framework. To achieve this, we show how theoretical data frequencies can be described by using unobservable response types both in observational studies and in randomized controlled trials. These descriptions also demonstrate sufficient conditions to estimate effect measures in observational studies, which would provide a further understanding of the assumption of conditional exchangeability. To illustrate the ideas, DAGs are employed, and we show how one can extend the original DAGs by integrating response types and observed variables. We deal only with structural (systematic) relations among the underlying variables of interest, so that an issue of random variation does not arise. Throughout this article, we assume that the consistency condition is met [24, 25, 26, 27, 28].
Methods
Definitions and notation
A causal diagram and causal effects
Causal RRs in the subsets of C can be consistently estimated under the assumption of conditional exchangeability, or, equivalently, no unmeasured confounding given data on C (i.e., E∐D_{ e }|C for ∀e). Note that, when the causal effect of interest is the effect of E on D either in the total population or in the subsets of C, intervening on E is of concern, and one does not consider intervening on C. Indeed, as outlined by VanderWeele [31], intervening on C would only be of concern if the joint effect of E and C on D was of interest. Therefore, under the situation in which C is being considered as a potential confounder as well as direct effect modifier (Figure 1), intervening on C is not of interest.
When we show how theoretical data frequencies can be described by using unobservable response types in observational studies, however, it is of great use to elucidate the relations between C and E in the counterfactual framework. By so doing, we demonstrate sufficient conditions to estimate effect measures in observational studies, which would provide a further understanding of the assumption of conditional exchangeability.
Response types
Enumeration of 4 response types for exposure E and corresponding potential outcomes
Etype | Potential outcomes of E | |
---|---|---|
E^{T}( ω) | E_{ c }( ω) | |
E_{1}( ω) | E_{0}( ω) | |
1 | 1 | 1 |
2 | 1 | 0 |
3 ^{a} | 0 | 1 |
4 | 0 | 0 |
Enumeration of 16 response types for outcome D and corresponding potential outcomes
Dtype | Potential outcomes of D | |||
---|---|---|---|---|
D^{T}( ω) | D_{ ce }( ω) | |||
D_{11}( ω) | D_{01}( ω) | D_{10}( ω) | D_{00}( ω) | |
1 | 1 | 1 | 1 | 1 |
2 ^{b, c} | 1 | 1 | 1 | 0 |
3 ^{a, b, c} | 1 | 1 | 0 | 1 |
4 | 1 | 1 | 0 | 0 |
5 ^{a, b, c} | 1 | 0 | 1 | 1 |
6 | 1 | 0 | 1 | 0 |
7 ^{a, b, c} | 1 | 0 | 0 | 1 |
8 ^{b} | 1 | 0 | 0 | 0 |
9 ^{a, b, c} | 0 | 1 | 1 | 1 |
10 ^{a, b, c} | 0 | 1 | 1 | 0 |
11 ^{a} | 0 | 1 | 0 | 1 |
12 ^{a, b} | 0 | 1 | 0 | 0 |
13 ^{a} | 0 | 0 | 1 | 1 |
14 ^{a, b} | 0 | 0 | 1 | 0 |
15 ^{a, b} | 0 | 0 | 0 | 1 |
16 | 0 | 0 | 0 | 0 |
Enumeration of 16 response types for selection variable S and corresponding potential outcomes
Stype | Potential outcomes of S | |||
---|---|---|---|---|
S^{T}( ω) | S_{ ed }( ω) | |||
S_{11}( ω) | S_{01}( ω) | S_{10}( ω) | S_{00}( ω) | |
1 | 1 | 1 | 1 | 1 |
2 ^{b, c} | 1 | 1 | 1 | 0 |
3 ^{a, b, c} | 1 | 1 | 0 | 1 |
4 | 1 | 1 | 0 | 0 |
5 ^{a, b, c} | 1 | 0 | 1 | 1 |
6 | 1 | 0 | 1 | 0 |
7 ^{a, b, c} | 1 | 0 | 0 | 1 |
8 ^{b} | 1 | 0 | 0 | 0 |
9 ^{a, b, c} | 0 | 1 | 1 | 1 |
10 ^{a, b, c} | 0 | 1 | 1 | 0 |
11 ^{a} | 0 | 1 | 0 | 1 |
12 ^{a, b} | 0 | 1 | 0 | 0 |
13 ^{a} | 0 | 0 | 1 | 1 |
14 ^{a, b} | 0 | 0 | 1 | 0 |
15 ^{a, b} | 0 | 0 | 0 | 1 |
16 | 0 | 0 | 0 | 0 |
Finally, we integrate information about the potential outcomes discussed above to produce 2 types of compound potential outcomes, which are also called nested counterfactuals [2]. (Note that compound potential outcomes have been extensively used in the issues of mediation and direct/indirect effects [35, 36, 37, 38].) First, we combine the potential outcomes of E and the potential outcomes of D to define ${D}_{c{E}_{c\mathit{\text{'}}}}\left(\omega \right)$. In other words, the compound potential outcomes of D are defined by (i) confounder status (C(ω) = 1, C(ω) = 0) and (ii) potential exposure status following an intervention on confounder (E_{1}(ω), E_{0}(ω)). For each individual ω, there would thus be 4 possible compound potential outcomes ${D}_{1{E}_{1}}\left(\omega \right)$, ${D}_{1{E}_{0}}\left(\omega \right)$, ${D}_{0{E}_{1}}\left(\omega \right)$, and ${D}_{0{E}_{0}}\left(\omega \right)$. Second, we combine the potential outcomes of E, the potential outcomes of D, and the potential outcomes of S to define ${S}_{{E}_{c}{D}_{c\mathit{\text{'}}{E}_{c\mathit{\text{'}}\mathit{\text{'}}}}}\left(\omega \right)$. Note that the compound potential outcomes of S are defined by (i) potential exposure status following an intervention on confounder (E_{1}(ω), E_{0}(ω)) and (ii) the compound potential outcomes of D (${D}_{1{E}_{1}}\left(\omega \right)$, ${D}_{1{E}_{0}}\left(\omega \right)$, ${D}_{0{E}_{1}}\left(\omega \right)$, and ${D}_{0{E}_{0}}\left(\omega \right)$). Thus, for each individual ω, there would be 8 possible compound potential outcomes ${S}_{{E}_{1}{D}_{1{E}_{1}}}\left(\omega \right)$, ${S}_{{E}_{1}{D}_{1{E}_{0}}}\left(\omega \right)$, ${S}_{{E}_{1}{D}_{0{E}_{1}}}\left(\omega \right)$, ${S}_{{E}_{1}{D}_{0{E}_{0}}}\left(\omega \right)$, ${S}_{{E}_{0}{D}_{1{E}_{1}}}\left(\omega \right)$, ${S}_{{E}_{0}{D}_{1{E}_{0}}}\left(\omega \right)$, ${S}_{{E}_{0}{D}_{0{E}_{1}}}\left(\omega \right)$, and ${S}_{{E}_{0}{D}_{0{E}_{0}}}\left(\omega \right)$.
Enumeration of 48 EDS response types and corresponding potential outcomes
Etype | Dtype | Stype | Potential outcomes of E | Potential outcomes of D | Compound potential outcomes of D | Potential outcomes of S | Selection status | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
E^{T}( ω) | D^{T}( ω) | S^{T}( ω) | E_{ c }( ω) | D_{ ce }( ω) | ${\mathit{D}}_{\mathit{c}{\mathit{E}}_{\mathit{c}\mathit{\text{'}}}}\left(\mathit{\omega}\right)$ | S_{ ed }( ω) | ${\mathit{S}}_{{\mathit{E}}_{\mathit{c}}{\mathit{D}}_{\mathit{c}{\mathit{E}}_{\mathit{c}}}}\left(\mathit{\omega}\right)$ | |||||||||||
E _{1} | E _{0} | D _{11} | D _{01} | D _{10} | D _{00} | ${\mathit{D}}_{\mathbf{1}{\mathit{E}}_{\mathbf{1}}}$ | ${\mathit{D}}_{\mathbf{1}{\mathit{E}}_{\mathbf{0}}}$ | ${\mathit{D}}_{\mathbf{0}{\mathit{E}}_{\mathbf{1}}}$ | ${\mathit{D}}_{\mathbf{0}{\mathit{E}}_{\mathbf{0}}}$ | S _{11} | S _{01} | S _{10} | S _{00} | ${\mathit{S}}_{{\mathit{E}}_{\mathbf{1}}{\mathit{D}}_{{}_{\mathbf{1}}{\mathit{E}}_{{}_{\mathbf{1}}}}}$ | ${\mathit{S}}_{{\mathit{E}}_{\mathbf{0}}{\mathit{D}}_{{}_{\mathbf{0}}{\mathit{E}}_{{}_{\mathbf{0}}}}}$ | |||
1 | 1 | 1 | 1 | 1 | 1 | 1 | (1)^{a} | (1) | 1 | (1) | (1) | 1 | 1 | (1) | (1) | (1) | 1 | 1 |
1 | 1 | 4 | 1 | 1 | 1 | 1 | (1) | (1) | 1 | (1) | (1) | 1 | 1 | (1) | (0) | (0) | 1 | 1 |
1 | 1 | 6 | 1 | 1 | 1 | 1 | (1) | (1) | 1 | (1) | (1) | 1 | 1 | (0) | (1) | (0) | 1 | 1 |
1 | 1 | 16 | 1 | 1 | 1 | 1 | (1) | (1) | 1 | (1) | (1) | 1 | 0 | (0) | (0) | (0) | 0 | 0 |
1 | 4 | 1 | 1 | 1 | 1 | 1 | (0) | (0) | 1 | (1) | (1) | 1 | 1 | (1) | (1) | (1) | 1 | 1 |
1 | 4 | 4 | 1 | 1 | 1 | 1 | (0) | (0) | 1 | (1) | (1) | 1 | 1 | (1) | (0) | (0) | 1 | 1 |
1 | 4 | 6 | 1 | 1 | 1 | 1 | (0) | (0) | 1 | (1) | (1) | 1 | 1 | (0) | (1) | (0) | 1 | 1 |
1 | 4 | 16 | 1 | 1 | 1 | 1 | (0) | (0) | 1 | (1) | (1) | 1 | 0 | (0) | (0) | (0) | 0 | 0 |
1 | 6 | 1 | 1 | 1 | 1 | 0 | (1) | (0) | 1 | (1) | (0) | 0 | 1 | (1) | 1 | (1) | 1 | 1 |
1 | 6 | 4 | 1 | 1 | 1 | 0 | (1) | (0) | 1 | (1) | (0) | 0 | 1 | (1) | 0 | (0) | 1 | 0 |
1 | 6 | 6 | 1 | 1 | 1 | 0 | (1) | (0) | 1 | (1) | (0) | 0 | 1 | (0) | 1 | (0) | 1 | 1 |
1 | 6 | 16 | 1 | 1 | 1 | 0 | (1) | (0) | 1 | (1) | (0) | 0 | 0 | (0) | 0 | (0) | 0 | 0 |
1 | 16 | 1 | 1 | 1 | 0 | 0 | (0) | (0) | 0 | (0) | (0) | 0 | 1 | (1) | 1 | (1) | 1 | 1 |
1 | 16 | 4 | 1 | 1 | 0 | 0 | (0) | (0) | 0 | (0) | (0) | 0 | 1 | (1) | 0 | (0) | 0 | 0 |
1 | 16 | 6 | 1 | 1 | 0 | 0 | (0) | (0) | 0 | (0) | (0) | 0 | 1 | (0) | 1 | (0) | 1 | 1 |
1 | 16 | 16 | 1 | 1 | 0 | 0 | (0) | (0) | 0 | (0) | (0) | 0 | 0 | (0) | 0 | (0) | 0 | 0 |
2 | 1 | 1 | 1 | 0 | 1 | (1) | (1) | 1 | 1 | (1) | (1) | 1 | 1 | 1 | (1) | (1) | 1 | 1 |
2 | 1 | 4 | 1 | 0 | 1 | (1) | (1) | 1 | 1 | (1) | (1) | 1 | 1 | 1 | (0) | (0) | 1 | 1 |
2 | 1 | 6 | 1 | 0 | 1 | (1) | (1) | 1 | 1 | (1) | (1) | 1 | 1 | 0 | (1) | (0) | 1 | 0 |
2 | 1 | 16 | 1 | 0 | 1 | (1) | (1) | 1 | 1 | (1) | (1) | 1 | 0 | 0 | (0) | (0) | 0 | 0 |
2 | 4 | 1 | 1 | 0 | 1 | (1) | (0) | 0 | 1 | (0) | (1) | 0 | 1 | (1) | (1) | 1 | 1 | 1 |
2 | 4 | 4 | 1 | 0 | 1 | (1) | (0) | 0 | 1 | (0) | (1) | 0 | 1 | (1) | (0) | 0 | 1 | 0 |
2 | 4 | 6 | 1 | 0 | 1 | (1) | (0) | 0 | 1 | (0) | (1) | 0 | 1 | (0) | (1) | 0 | 1 | 0 |
2 | 4 | 16 | 1 | 0 | 1 | (1) | (0) | 0 | 1 | (0) | (1) | 0 | 0 | (0) | (0) | 0 | 0 | 0 |
2 | 6 | 1 | 1 | 0 | 1 | (0) | (1) | 0 | 1 | (1) | (0) | 0 | 1 | (1) | (1) | 1 | 1 | 1 |
2 | 6 | 4 | 1 | 0 | 1 | (0) | (1) | 0 | 1 | (1) | (0) | 0 | 1 | (1) | (0) | 0 | 1 | 0 |
2 | 6 | 6 | 1 | 0 | 1 | (0) | (1) | 0 | 1 | (1) | (0) | 0 | 1 | (0) | (1) | 0 | 1 | 0 |
2 | 6 | 16 | 1 | 0 | 1 | (0) | (1) | 0 | 1 | (1) | (0) | 0 | 0 | (0) | (0) | 0 | 0 | 0 |
2 | 16 | 1 | 1 | 0 | 0 | (0) | (0) | 0 | 0 | (0) | (0) | 0 | (1) | (1) | 1 | 1 | 1 | 1 |
2 | 16 | 4 | 1 | 0 | 0 | (0) | (0) | 0 | 0 | (0) | (0) | 0 | (1) | (1) | 0 | 0 | 0 | 0 |
2 | 16 | 6 | 1 | 0 | 0 | (0) | (0) | 0 | 0 | (0) | (0) | 0 | (1) | (0) | 1 | 0 | 1 | 0 |
2 | 16 | 16 | 1 | 0 | 0 | (0) | (0) | 0 | 0 | (0) | (0) | 0 | (0) | (0) | 0 | 0 | 0 | 0 |
4 | 1 | 1 | 0 | 0 | (1) | (1) | 1 | 1 | 1 | (1) | (1) | 1 | (1) | 1 | (1) | (1) | 1 | 1 |
4 | 1 | 4 | 0 | 0 | (1) | (1) | 1 | 1 | 1 | (1) | (1) | 1 | (1) | 1 | (0) | (0) | 1 | 1 |
4 | 1 | 6 | 0 | 0 | (1) | (1) | 1 | 1 | 1 | (1) | (1) | 1 | (1) | 0 | (1) | (0) | 0 | 0 |
4 | 1 | 16 | 0 | 0 | (1) | (1) | 1 | 1 | 1 | (1) | (1) | 1 | (0) | 0 | (0) | (0) | 0 | 0 |
4 | 4 | 1 | 0 | 0 | (1) | (1) | 0 | 0 | 0 | (0) | (0) | 0 | (1) | (1) | (1) | 1 | 1 | 1 |
4 | 4 | 4 | 0 | 0 | (1) | (1) | 0 | 0 | 0 | (0) | (0) | 0 | (1) | (1) | (0) | 0 | 0 | 0 |
4 | 4 | 6 | 0 | 0 | (1) | (1) | 0 | 0 | 0 | (0) | (0) | 0 | (1) | (0) | (1) | 0 | 0 | 0 |
4 | 4 | 16 | 0 | 0 | (1) | (1) | 0 | 0 | 0 | (0) | (0) | 0 | (0) | (0) | (0) | 0 | 0 | 0 |
4 | 6 | 1 | 0 | 0 | (1) | (0) | 1 | 0 | 1 | (1) | (0) | 0 | (1) | 1 | (1) | 1 | 1 | 1 |
4 | 6 | 4 | 0 | 0 | (1) | (0) | 1 | 0 | 1 | (1) | (0) | 0 | (1) | 1 | (0) | 0 | 1 | 0 |
4 | 6 | 6 | 0 | 0 | (1) | (0) | 1 | 0 | 1 | (1) | (0) | 0 | (1) | 0 | (1) | 0 | 0 | 0 |
4 | 6 | 16 | 0 | 0 | (1) | (0) | 1 | 0 | 1 | (1) | (0) | 0 | (0) | 0 | (0) | 0 | 0 | 0 |
4 | 16 | 1 | 0 | 0 | (0) | (0) | 0 | 0 | 0 | (0) | (0) | 0 | (1) | (1) | (1) | 1 | 1 | 1 |
4 | 16 | 4 | 0 | 0 | (0) | (0) | 0 | 0 | 0 | (0) | (0) | 0 | (1) | (1) | (0) | 0 | 0 | 0 |
4 | 16 | 6 | 0 | 0 | (0) | (0) | 0 | 0 | 0 | (0) | (0) | 0 | (1) | (0) | (1) | 0 | 0 | 0 |
4 | 16 | 16 | 0 | 0 | (0) | (0) | 0 | 0 | 0 | (0) | (0) | 0 | (0) | (0) | (0) | 0 | 0 | 0 |
Four hypothetical situations
Subsequently, Figure 2B shows a situation in which researchers can obtain the information about the total population, including those who dropped out. In this situation, a possibility of selection bias can be ruled out since researchers do not condition on S.
In observational studies, researchers usually aim to eliminate confounding bias by employing some statistical procedures, e.g., standardization and inverse-probability weighting method. In other words, they aim to analytically block or remove the path between C and E by making an adequate adjustment. (Note that outcome modeling techniques such as disease risk scores focus on the path between C and D[39].) By contrast, in randomized controlled trials, researchers manipulate the value of E by employing certain interventions; they physically prevent E from varying in response to variations in C. Thus, as shown in Figure 2C and D, C would no longer have effects on E, and the arrow from C to E is erased or removed [14]. This should be clearly distinguished from analytic control of C in observational studies.
In the following sections, we demonstrate significant differences between these 4 hypothetical situations, by describing theoretical data frequencies in terms of response types.
Results
Describing data from observational studies in terms of response types
As demonstrated above, under the situation described in Figure 1, individuals can be classified into one of the maximum of 1,024 EDS response types. Despite its sophistication and usefulness, however, the response type of each individual is unobservable. Indeed, this is called a fundamental problem of causal inference [40]. Nonetheless, we can show the conceptual link between unobservable response types and observed, or observable, data frequencies in the population. In this respect, the concept of compound potential outcomes is quite useful.
Notably, individuals of the same EDS response types can be potentially classified into 2 cells. For example, consider individual ω who is classified as E1D6S4 response type (see Table 4). This individual is, by definition, exposed to E = 1 irrespective of the value of C (i.e., E_{1}(ω) = E_{0}(ω) = 1). Further, individual ω is expected to experience outcome D if there had been interventions to set C to 1 (i.e., ${D}_{1{E}_{1}}\left(\omega \right)={D}_{11}\left(\omega \right)=1$), whereas this individual is expected not to experience outcome D if there had been interventions to set C to 0 (i.e., ${D}_{0{E}_{0}}\left(\omega \right)={D}_{01}\left(\omega \right)=0$). Finally, the information about this individual is, by definition, available to researchers had there been interventions to set C to 1 (i.e., ${S}_{{E}_{1}{D}_{1{E}_{1}}}\left(\omega \right)={S}_{11}\left(\omega \right)=1$), whereas this individual is lost to follow-up had there been interventions to set C to 0 (i.e., ${S}_{{E}_{0}{D}_{0{E}_{0}}}\left(\omega \right)={S}_{10}\left(\omega \right)=0$). Thus, in observational studies, individual ω of E1D6S4 response type can be classified into either one of the following 2 cells in Figure 3; one is E = 1, D = 1, C = 1, and S = 1 while the other is E = 1, D = 0, C = 0, and S = 0. Note that this depends on the probabilities that C is present or absent in individual ω (i.e., P_{C|E1D6S4} and ${P}_{\overline{C}|E1D6S4}$).
To summarize, Figure 3 shows theoretical data frequencies in an observational study (i.e., Figure 2A and B). The situation is, however, strikingly different when we conduct a randomized controlled trial, which will be demonstrated in the next section.
Describing data from randomized controlled trials in terms of response types
As noted above, researchers manipulate the value of E in randomized controlled trials. Since researchers physically prevent E from varying in response to variations in C, we do not need to consider E response types when describing theoretical data frequencies in ideal randomized controlled trials; rather we focus on D response types and S response types. In other words, observed exposure status and E response types become independent (i.e., E ∐ E^{T}) when researchers marginally intervene on E. Thus, theoretical data frequencies from randomized controlled trials can be described in terms of 256 (i.e., 16 × 16) possible DS response types, in contrast with 1,024 possible EDS response types.
Note that the numbers in the parentheses of left-hand sides of equations 3 and 4 are based on the subpopulation of C = 1 in observational studies (i.e., upper part of Figure 3), whereas the right-hand sides of these equations are based on the subpopulation of C = 1 in randomized controlled trials (i.e., upper part of Figure 4). In other words, these equations explain how individuals of subpopulation of C = 1 are re-distributed as a result of intervention on E.
Again, the numbers in the parentheses of left-hand sides of equations 5 and 6 are based on the subpopulation of C = 0 in observational studies (i.e., lower part of Figure 3), whereas the right-hand sides of these equations are based on the subpopulation of C = 0 in randomized controlled trials (i.e., lower part of Figure 4). In other words, these equations explain how individuals of subpopulation of C = 0 are re-distributed as a result of intervention on E. It should be noted that these re-distributions do not occur across the upper and lower parts of Figures 3 and 4 because C precedes E temporally and the value of C is, by definition, predetermined before intervention on E. These discussions also demonstrate that, in Figure 4, individuals of the same DS response types can be potentially classified into 4 cells, depending on the probability of being exposed or unexposed to C (i.e., P_{C|DjSk} or ${P}_{\overline{C}|\mathit{DjSk}}$) and the probability of being exposed or unexposed to E (i.e., P_{ E } or ${P}_{\overline{E}}$).
Note that, when the information about the total population is available, both marginal and conditional exchangeability assumptions are met in Figure 4; the distributions of DS response types are comparable between the exposed and unexposed groups. However, when the information about those who dropped out is not available, exchangeability assumptions do not hold, either conditionally or unconditionally. See (Additional file 1: Appendix 1) for a discussion of positivity – another fundamental assumption for causal inference [41, 42, 43].
Epidemiologic measures in terms of response types
The descriptions of data frequencies in Figures 3 and 4 have a crucial implication, demonstrating significant distinctions between association measures and effect measures [9]. In the following sections, we continue to focus our discussion on RRs, which can be extended to other measures. Note also that, although epidemiologic measures can be defined for a variety of target population (e.g., the exposed and the unexposed), the following discussion focuses on the situation in which target population is the total population. Furthermore, we also discuss epidemiologic measures in the subpopulation defined by C or S.
In observational studies (Figure 2A and B), researchers can readily calculate associational RRs by referring to the notations in Figure 3. In particular, when no information is available about those who dropped out (Figure 2A), one can calculate an associational RR_{S=1} by using the information about individuals classified into the inner rectangles in Figure 3. Then, as shown in (Additional file 2: Table S1), associational RR_{S=1} can be described in terms of a probability of being exposed or unexposed to C among the individuals of EiDjSk response type (i.e., P_{C|EiDjSk} or ${P}_{\overline{C}|\mathit{EiDjSk}}$) and a prevalence of the individuals of EiDjSk response type in the total population (i.e., P_{ EiDjSk }) (equation A1). Meanwhile, when researchers are capable of gathering information about those who dropped out (Figure 2B), the information about individuals of S response types 1 through 16 is available, which yields an associational RR (equation A4).
By contrast, when researchers obtain data from randomized controlled trials (Figure 2C and D), their frequencies can be described in a different way, as shown in Figure 4. In these cases, researchers can calculate causal RRs to infer causality between E and D. When no information is available about those who dropped out (Figure 2C), one can calculate a causal RR_{S=1} by using the information about individuals classified into the inner rectangles in Figure 4. Then, as shown in (Additional file 2: Table S2), causal RR_{S=1} can be described in terms of a probability of being exposed or unexposed to C among the individuals of DjSk response type (i.e., P_{C|DjSk} or ${P}_{\overline{C}|\mathit{DjSk}}$) and a prevalence of the individuals of DjSk response type in the total population (i.e., P_{ DjSk }) (equation A7). In ideal randomized controlled trials without loss to follow-up (Figure 2D), the information about individuals of S response types 1 through 16 is available, which yields a causal RR (equation A10). We should note that the causal RR shown in equation A10 is an alternative notation of the causal RR shown in equation 1 in terms of response types (see Additional file 1: Appendix 2).
Note that, even in ideal (either marginal or stratified) randomized controlled trials, one may observe a heterogeneity between stratum-specific causal RRs, which will be addressed in the section entitled “Modification of epidemiologic measures”.
Confounding bias
In this section, we aim to further clarify the concept of confounding bias in the counterfactual framework, by describing it in terms of response types.
We show a sufficient condition to estimate effect measures in observational studies by adjusting for confounding bias in terms of response types. In this case, we use effect measures in the total population in ideal randomized controlled trials (i.e., causal RR) as a gold standard. As noted above, confounding bias is induced by a common cause C of E and D. Thus, to show a sufficient condition to adjust for confounding bias, we need to compare association measures in the total population in observational studies (Figure 2B) and effect measures in the total population in randomized controlled trials (Figure 2D) In other words, a sufficient condition to adjust for confounding bias can be described as: adjusted associational RR = causal RR. Note that we here compare 2 distinct types of epidemiologic measures, which are obtained from distinct study designs.
The subtle differences between E^{T} ∐ D^{T}|C and E ∐ D^{T}|C are described graphically in the section entitled “Extended causal diagrams integrating response types”. It is worthwhile to mention that the condition E^{T} ∐ D^{T}|C is not guaranteed in randomized controlled trials.
The above discussion implies that analytic adjustment for C in observational studies has consequences quite different from those of physical control in randomized controlled trials. Even when adequate analytic control of C may be envisaged in observational studies, researchers cannot estimate effect measures without the assumption external to data. See Additional file 1: Appendix 5 for a discussion of recently-introduced assumptions to compensate for a lack of randomization.
Selection bias
In this section, we aim to further clarify the concept of selection bias in the counterfactual framework, by describing it in terms of response types.
We show sufficient conditions for non-selection bias in terms of response types. As explained above, selection bias is induced by conditioning on a common effect of E and D (Figure 2A and C). Thus, to show sufficient conditions for non-selection bias, we need to specify epidemiologic measures, i.e., association measures or effect measures. With regard to association measures, a sufficient condition for non-selection bias is described as associational RR_{S=1} = associational RR (see equations A1 and A4). Likewise, a sufficient condition for non-selection bias for effect measures is described as causal RR_{S=1} = causal RR (see equations A7 and A10). It is worthwhile to mention that, when discussing selection bias, one need to specify a stratum of S[21]. In most cases, researchers are interested in the presence and the degree of selection bias among the subjects who do not drop out. Thus, we here show sufficient conditions for non-selection bias in a stratum S = 1. As explained later by using extended causal diagrams, selection bias results in violation of E ∐ D^{T} even when exposure is randomly assigned.
Modification of epidemiologic measures
For decades, epidemiologists have used the term “effect modification” in a broad context, simply referring to a variation in the selected effect measure for the factor under study across levels of another factor [49]. In this respect, a recent paper clarified the distinction between interaction and effect modification within the counterfactual framework [31]. It is also well known that the presence, direction, and size of modification can be dependent on the choice of measure [50]. Since the term “effect modification” is ambiguous, it is now recommended to specify the measures more precisely, e.g., risk-difference modification [50]. The above discussion implies that researchers need to distinguish association-measure modification and effect-measure modification. For example, when the information about total population is available in a randomized controlled trial, causal-RR modification is defined to be present if stratum-specific causal RRs from each subpopulation varies across the strata of C, i.e., causal RR_{C=1} ≠ causal RR_{C=0} (see equations A11 and A12). When stratum-specific causal RRs are (approximately) homogeneous or uniform across strata, researchers usually pool the data to calculate a causal RR in the total population (i.e., causal RR). In a similar manner, one can define associational-RR modification (see equations A5 and A6). Only if it is appropriate to pool the data across the strata of C, one can validly interpret associational RRs in the total population.
Notably, the presence of association-measure modification does not necessarily imply the presence of effect-measure modification, and vice versa.
Extended causal diagrams integrating response types
In this section, we attempt to explain the concept of bias by extending causal diagrams, which integrate response types and observed variables. Although these causal diagrams, or extended DAGs, may appear less intuitive, they maintain the integrity of the original DAGs and would be of great use in graphically describing the findings discussed in this study. In particular, by integrating response types and observed variables, we can readily understand subtle differences between E^{T} ∐ D^{T}|C and E ∐ D^{T}|C, demonstrating sufficient conditions to estimate effect measures in observational studies.
Finally, it is worthwhile to mention that the perspectives of the extended DAGs are different from those of the twin network method, which has been developed to deal with counterfactual values in DAGs [2]. This graphical method uses two networks, one to represent the actual world and one to represent the hypothetical world. Thus, this method is used to represent the causal relations under intervention. The aim of our extended DAGs is to integrate response types and observed variables, which is thus applicable to observational studies as well as randomized controlled trials. As a consequence, we can use the extended DAGs to describe the sufficient conditions to infer causality in observational studies in terms of response types.
Discussion
We have clarified the concepts of bias and effect modification in the counterfactual framework, by describing theoretical data frequencies from observational studies and randomized controlled trials in terms of response types. Although these concepts have been extensively explained in the epidemiologic literature, most of the studies have discussed them separately. In this article, we have highlighted the relations between these concepts, by discussing them simultaneously. The present findings would somehow clarify the link between the assumptions for making causal inferences in observational studies and the counterfactual approach, demonstrating the inherent distinctions between observational studies and randomized controlled trials. The extension of DAGs using response types maintains the integrity of the original DAGs, which allows one to understand the underlying causal structure discussed in this study.
We have shown a hitherto unrecognized sufficient condition ${E}^{\mathrm{T}}\coprod {D}^{\mathrm{T}}\left|C\right.$ to estimate effect measures in observational studies by adjusting for confounding bias. This condition is stronger than the assumption of (full) conditional exchangeability, and it is not straightforward to discuss technical advantages of the hitherto unrecognized condition. Such consideration however would enable one to further understand the conceptual link between unobservable response types and observed, or observable, data frequencies in the population. This would also facilitate understanding of the underlying causal structures of bias and effect modification.
In this article, we use a simple hypothetical situation, including only 4 binary variables. Thus, it should be noted that the present study does not encompass more complicated situations, e.g., M-bias [51]. It is however worthwhile to mention that the condition ${E}^{\mathrm{T}}\coprod {D}^{\mathrm{T}}\left|C\right.$ is applicable even when an exposure and an outcome are polytomous variables, because our discussion based on the extended DAGs does not restrict the type of variables. When considering situations in which there are some confounders, the present finding would apply by defining and estimating a function of measured confounders that can be treated as a single confounder. It should be also noted that we focused only on direct effect modification, and thus, the present discussion does not necessarily apply to other types of effect modification, i.e., indirect effect modification, effect modification by proxy, and effect modification by a common cause [22]. Further, this study does not address the issue of information bias or measurement error. Recent studies have discussed how DAGs can be used to represent them [52, 53, 54, 55], which should be addressed further in future studies.
Conclusion
As shown in the present study, researchers should recognize inherent limitations of observational studies in estimating causal effects. It should be emphasized, however, that the recognition should come in the interpretation of the evidence when trying to draw conclusions, not in the statement of research goals or study design and conduct phases [56]. The data from observational studies yield measures of association and those who examine the data should strive to impose a meaning based on their expert knowledge on each occasion, which would improve causal interpretations.
Authors’ information
ES is Assistant Professor of Epidemiology at Okayama University. His primary research interest concerns improving causal interpretations of observational studies. TM was a Research Fellow of Epidemiology when this study was conducted. He is currently working as Assistant Professor in Center for Innovative Clinical Medicine at Okayama University Hospital. TT, as a Professor of Environmental Epidemiology, has evaluated a variety of health effects of environmental factors to advance the public’s health. EY, as a Professor of Statistics, is interested in contributing to the advancement of statistical theories necessary for causal inference.
Notes
Supplementary material
References
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