To the Editor:
Causal diagrams1,2 have become widespread in epidemiologic research. Recently developed single-world intervention graphs explicitly connect the potential outcomes framework of causal inference with causal diagrams.3 Here, we provide a practical example demonstrating how single-world intervention graphs can supplement traditional causal diagrams.
A randomized controlled trial is conducted to evaluate whether a vaccine (A = 1 if vaccine, 0 if placebo) decreases the risk of disease (
;)
if disease, 0 otherwise). Individuals are enrolled at baseline, randomized to vaccine or placebo, followed 6 months, and monitored for disease. The vaccine is more likely to result in injection site pain (
if pain, 0 otherwise), and those with pain are more likely to drop out and have unobserved outcomes (
if dropped out, 0 otherwise). Participants with poor (unmeasured) health (
if poor health, 0 otherwise) are more likely to experience pain and get the disease. The scenario is summarized in Figure A.
There is selection bias if we condition on not dropping out (
) because the path
is opened. Stratifying on W does not block this path and may in fact induce more bias. Based on this causal diagram, it is not immediately clear how to identify the causal effect of the vaccine using the observed data (although see references 4, 5, or 6).
The single-world intervention graph in Figure B, however, clearly displays the independencies necessary to identify the effect of the vaccine from the observed data as follows (here, a variable
represents the value of
had the individual received vaccine level
):
The first equality holds by the law of total probability, the second by d-separation of
and
given
, the third by d-separation of
and
, the fourth by d-separation of
and
given
and
, and the last by causal consistency. All components of the final line of the equation, which is Robins’ g-formula,7 can be estimated from observed data. The key insight provided by the single-world intervention graph is that
is independent of
given
, but conditioning on
does not open any paths between
and
.
We conducted a simulation of 1,000,000 individuals for illustration (SAS code is available in the eAppendix; https://links.lww.com/EDE/B306). Individuals were randomly assigned vaccine with probability 0.5 and had probability 0.3 of being in poor health. The probability of injection site pain for healthy individuals was 0.2 if assigned placebo and 0.6 if assigned vaccine. Poor health increased the probability of pain by 0.3. The probability of dropping out was 0.1 for those without pain and 0.9 for those with pain. Finally, the probability of disease was 0.3 for healthy individuals assigned placebo, and it was increased by 0.5 by poor health and decreased by 0.2 by the vaccine.
The true effect of the vaccine on the disease was a 0.20 decrease in risk. The complete case analysis gave a 0.24 decrease in risk. Stratifying on injection site pain worsened the bias, giving a 0.26 decrease in risk. Finally, the g-formula with empirically estimated expectations and probabilities yielded the true decrease of 0.20.
An anonymous reviewer noted that the derivation above also holds with certain additional edges in the causal diagram, such as
or
. These would lead to, respectively, edges
or
in the single-world intervention graph. In the latter case,
is d-separated from
given
and
, thus
would remain independent of
conditional on
(Theorem 12 in Richardson and Robins3). The reviewer also noted that the derivation fails with unmeasured confounding between
and
or between
and
.
FIGURE: The causal diagram (A) corresponding to the vaccine trial. The single-world intervention template (B), the template used to construct single-world intervention graphs, corresponding to the vaccine trial is constructed by splitting the treatment node of the causal diagram, and replacing all descendants of the assigned treatment with their potential outcomes.
ACKNOWLEDGMENTS
The authors thank the anonymous reviewer of this letter for their helpful comments.
Alexander Breskin
Stephen R. Cole
Department of Epidemiology
University of North Carolina at Chapel Hill
Chapel Hill, NC
[email protected]
Michael G. Hudgens
Department of Biostatistics
University of North Carolina at Chapel Hill
Chapel Hill, NC
REFERENCES
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2. Greenland S, Pearl J, Robins JMCausal diagrams for epidemiologic research. Epidemiology. 1999;10:37–48.
3. Richardson TS, Robins JMSingle world intervention graphs (SWIGs): a unification of the counterfactual and graphical approaches to causality. Cent Stat Soc Sci Univ Washingt Ser Work Pap. 2013;128:2013.
4. Bareinboim E, Pearl JControlling selection bias in causal inference. Proc Learn Res. 2012;22:100–108.
5. Bareinboim E, Tian J, Pearl JRecovering from selection bias in causal and statistical inference. Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence. 2014;2410–2416.
6. Bareinboim E, Tian JRecovering causal effects from selection bias. Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence. 2015:3475–3481.
7. Robins JA new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect. Math Model. 1986;7:1393–1512.
