Strength in causality: discerning causal mechanisms in the sufficient cause model

Abstract

The assessment of causality is fundamental to epidemiology and biomedical sciences. One well-known approach to distinguishing causal from noncausal explanations is the nine Bradford Hill viewpoints. A recent article in this journal revisited the viewpoints to incorporate developments in causal thinking, suggesting that the sufficient cause model is useful in elucidating the theoretical underpinning of the first of the nine viewpoints—strength of association. In this article, we discuss how to discern the causal mechanisms of interest in the sufficient cause model, which pays closer attention to the relationship between the sufficient cause model and the Bradford Hill viewpoints. To this end, we explicate the link between the sufficient cause model and the potential-outcome model, both of which have become the cornerstone of causal thinking in epidemiology and biomedicine. A clearer understanding of the link between the two models provides significant implications for interpretation of the observed risks in the subpopulations defined by exposure and confounder. We also show that the concept of potential completion times of sufficient causes is useful to fully discerning completed sufficient causes, which leads us to pay closer attention to the fourth of the nine Bradford Hill viewpoints—temporality. Decades after its introduction, the sufficient cause model may be vaguely understood and thus implicitly used under unreasonably strict assumptions. To strengthen our assessment in the face of multifactorial causality, it is significant to carefully scrutinize the observed associations in a complementary manner, using the sufficient cause model as well as its relevant causal models.

Appendix

In terms of the potential completion times of sufficient causes, Assumptions 1 and 2 can be algebraically rewritten as follows:

Assumption 1

$$ \begin{aligned} & {\text{Pr}}\left( {{\text{min}}\left\{ {d_{1} ,d_{2} ,d_{3} } \right\} \le h|\left( {A,C} \right) = \left( {1, 1} \right)} \right) = 0 \\ & {\text{Pr}}\left( {{\text{min}}\left\{ {d_{1} ,d_{2} ,d_{4} } \right\} \le h|\left( {A,C} \right) = \left( {0, 1} \right)} \right) = 0 \\ & {\text{Pr}}\left( {{\text{min}}\left\{ {d_{1} ,d_{3} ,d_{4} } \right\} \le h|\left( {A,C} \right) = \left( {1, 0} \right)} \right) = 0 \\ & {\text{Pr}}\left( {{\text{min}}\left\{ {d_{2} ,d_{3} ,d_{4} } \right\} \le h|\left( {A,C} \right) = \left( {0, 0} \right)} \right) = 0 \\ \end{aligned} $$

Assumption 2

$$ \begin{aligned} & {\text{Pr}}\left( {{\text{min}}\left\{ {d_{1} ,d_{2} ,d_{3} } \right\} \le h|\left( {A,C} \right) = \left( {1, 1} \right)} \right) = 0 \\ & {\text{Pr}}\left( {d_{1} \le h|\left( {A,C} \right) = \left( {0, 1} \right)} \right) = 0 \\ & {\text{Pr}}\left( {d_{1} \le h|\left( {A,C} \right) = \left( {1, 0} \right)} \right) = 0 \\ \end{aligned} $$

These clearly show that Assumption 2 is less strict than Assumption 1. As mentioned in the text, Assumption 3 is less strict than Assumption 2, which can be more readily seen by rewriting Assumption 3 as follows:

Assumption 3

$$ \begin{aligned} & {\text{Pr}}\left( {{\text{min}}\left\{ {d_{1} ,d_{2} ,d_{3} } \right\} \le {\text{min}}\left\{ {d_{4} ,h} \right\}|\left( {A,C} \right) = \left( {1, 1} \right)} \right) = 0 \\ & {\text{Pr}}\left( {d_{1} \le {\text{min}}\left\{ {d_{3} ,h} \right\}|\left( {A,C} \right) = \left( {0, 1} \right)} \right) = 0 \\ & {\text{Pr}}\left( {d_{1} \le {\text{min}}\left\{ {d_{2} ,h} \right\}|\left( {A,C} \right) = \left( {1, 0} \right)} \right) = 0 \\ \end{aligned} $$