## Abstract

The minimal sufficient cause (MSC) model, also known as the sufficient component cause model, has been used to facilitate understanding of several key concepts in epidemiology. To improve the understanding of mediation, we introduce a causal model for mediation that is grounded in the MSC approach. First, we describe an unbiased model for mediation, to clarify the causal meaning of previously described indirect effects. Through the use of potential outcomes and response types, we express each indirect (and direct) effect in terms of component causes within the MSC model. Second, we use an MSC-based model to illustrate a common cause of the mediator and outcome, i.e. a confounder of the mediator–outcome relationship. By describing this potential source of bias within the MSC-based model, important complexities are noted that impact the magnitude of plausible confounding. In conclusion, an MSC-based approach leads to several important insights concerning the interpretation of indirect and direct effects, as well as the potential sources of bias in mediation analysis.

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## Acknowledgements

The author would like to acknowledge Sharon Schwartz for her insightful input, guidance, and mentorship. She would also like to thank Tyler VanderWeele, Maria Glymour, and Ezra Susser for their helpful comments on this manuscript.

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## Appendix: Direct effects (PDE and TDE) in terms of potential outcomes, response types, and the MSC model for mediation

### Appendix: Direct effects (PDE and TDE) in terms of potential outcomes, response types, and the MSC model for mediation

The pure direct effect (PDE) is the effect that the exposure would have if exposure did not cause the mediator. In terms of the current example, the PDE would be the effect of arsenic on death if arsenic did not cause skin lesions. Operationalized in terms of potential outcomes, the PDE is a comparison between (1) the risk of death if everyone were exposed to arsenic, but had the skin lesions they would have had were they unexposed \( \left[ {{\text{P}}\left( {{\text{Y}}_{{1{\text{M}}_{0} }} = 1} \right)} \right] \) and (2) the risk of death if everyone were unexposed to arsenic \( \left[ {{\text{P}}\left( {{\text{Y}}_{{0{\text{M}}_{0} }} = 1} \right)} \right]. \)

The difference between \( {\text{P}}\left( {{\text{Y}}_{{ 1 {\text{M}}_{0} }} = 1} \right) = {\text{P}}\left( {{\text{Y}}^{\text{T}}_{ 1} + {\text{Y}}^{\text{T}}_{ 2} + {\text{Y}}^{\text{T}}_{ 4} + {\text{M}}^{\text{T}}_{ 1} {\text{Y}}^{\text{T}}_{ 6} + {\text{M}}^{\text{T}}_{ 1} {\text{Y}}^{\text{T}}_{ 8} } \right) \) and \( {\text{P}}\left( {{\text{Y}}_{{0{\text{M}}_{0} }} = 1} \right) = {\text{P}}\left( {{\text{Y}}^{\text{T}}_{1} + {\text{M}}^{\text{T}}_{1} {\text{Y}}^{\text{T}}_{6} + {\text{M}}^{\text{T}}_{1} {\text{Y}}^{\text{T}}_{2} } \right) \) is: \( {\text{PDE}} = {\text{P}}\left( {{\text{Y}}^{\text{T}}_{ 4} + {\text{M}}^{\text{T}}_{ 1} {\text{Y}}^{\text{T}}_{ 8} + {\text{M}}^{\text{T}}_{ 2} {\text{Y}}^{\text{T}}_{ 2} + {\text{M}}^{\text{T}}_{ 4} {\text{Y}}^{\text{T}}_{ 2} } \right) \) (Table 5 and Appendix Table 1). Intuitively, the PDE is equal to the proportion of individuals for whom skin lesions directly causes death (Y
^{T}_{4}
); plus those who have skin lesions regardless of arsenic exposure (M
^{T}_{1}
) and for whom skin lesions and arsenic interact to cause death (Y
^{T}_{8}
); plus those who do not have skin lesions in the absence of arsenic exposure (M
^{T}_{2}
or M
^{T}_{4}
) and for whom either skin lesions or arsenic cause death (Y
^{T}_{2}
). In terms of component causes within the MSC model, the PDE consists of individuals who (1) do not have L or K, but have C or (2) do not have L or B, but have K and (C or F).

The total direct effect (TDE) is the effect that the exposure would have if lack of exposure did not prevent the mediator. Given the current example, the TDE would be the effect of arsenic on death if lack of arsenic did not prevent skin lesions; that is, if everyone (both the unexposed and the exposed) had the skin lesions they would have had were they exposed. In terms of potential outcomes, the TDE is a comparison between (1) the risk of death if everyone were exposed to arsenic \( \left[ {{\text{P}}\left( {{\text{Y}}_{{1{\text{M}}_{1} }} = 1} \right)} \right]. \) and (2) the risk of death if everyone were unexposed to arsenic, but had the skin lesions they would have were they exposed \( \left[ {{\text{P}}\left( {{\text{Y}}_{{0{\text{M}}_{1} }} = 1} \right)} \right]. \)

Based on Table 5, the TDE is the difference between \( {\text{P}}\left( {{\text{Y}}_{{ 1 {\text{M}}_{1} }} = 1} \right) = {\text{P}}\left( {{\text{Y}}^{\text{T}}_{ 1} + {\text{Y}}^{\text{T}}_{ 2} + {\text{Y}}^{\text{T}}_{ 4} + {\text{M}}^{\text{T}}_{ 1} {\text{Y}}^{\text{T}}_{ 6} + {\text{M}}^{\text{T}}_{ 1} {\text{Y}}^{\text{T}}_{ 8} + {\text{M}}^{\text{T}}_{ 2} {\text{Y}}^{\text{T}}_{ 6} + {\text{M}}^{\text{T}}_{ 2} {\text{Y}}^{\text{T}}_{ 8} } \right) \) and \( {\text{P}}\left( {{\text{Y}}_{{0{\text{M}}_{1} }} = 1} \right) = {\text{P}}\left( {{\text{Y}}^{\text{T}}_{ 1} + {\text{M}}^{\text{T}}_{ 1} {\text{Y}}^{\text{T}}_{ 6} + {\text{M}}^{\text{T}}_{ 1} {\text{Y}}^{\text{T}}_{ 2} + {\text{M}}^{\text{T}}_{ 2} {\text{Y}}^{\text{T}}_{ 6} + {\text{M}}^{\text{T}}_{ 2} {\text{Y}}^{\text{T}}_{ 2} } \right); \) it follows that the \( {\text{TDE}} = {\text{P}}\left( {{\text{Y}}^{\text{T}}_{4} + {\text{M}}^{\text{T}}_{1} {\text{Y}}^{\text{T}}_{8} + {\text{M}}^{\text{T}}_{2} {\text{Y}}^{\text{T}}_{8} + {\text{M}}^{\text{T}}_{4} {\text{Y}}^{\text{T}}_{2} } \right) \) (Table 5 and Appendix Table 1). Intuitively, the TDE is equal to the proportion of individuals for whom skin lesions directly causes death (Y
^{T}_{4}
); plus those who have skin lesions in the presence of arsenic exposure (M
^{T}_{1}
or M
^{T}_{2}
) and for whom skin lesions and arsenic interact to cause death (Y
^{T}_{8}
); plus those who do not have skin lesions, even in the presence of arsenic exposure (M
^{T}_{4}
) and for whom either skin lesions or arsenic cause death (Y
^{T}_{2}
). In terms of the MSC model, the TDE consists of individuals who (1) do not have L, K,or A, but have C or (2) do not have L or B, but have (K or A) and (C or F).

Similar to the indirect effects, the difference between the pure and total direct effect is due to the differential placement of parallelism and synergy. While the TDE includes mediated synergistic types (M
^{T}_{2}
Y
^{T}_{8}
), it excludes mediated parallel types (M
^{T}_{2}
Y
^{T}_{2}
). In contrast, the PDE includes mediated parallel types (M
^{T}_{2}
Y
^{T}_{2}
), while excluding mediated synergistic types (M
^{T}_{2}
Y
^{T}_{8}
).

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Hafeman, D.M. A sufficient cause based approach to the assessment of mediation.
*Eur J Epidemiol* **23**, 711–721 (2008). https://doi.org/10.1007/s10654-008-9286-7

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DOI: https://doi.org/10.1007/s10654-008-9286-7