A Practical Example Demonstrating the Utility of Single-world Intervention Graphs

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Excerpt

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 (
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM1/v/2018-04-21T053452Z/r/image-tiff
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 (
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM2/v/2018-04-21T053452Z/r/image-tiff
if pain, 0 otherwise), and those with pain are more likely to drop out and have unobserved outcomes (
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM3/v/2018-04-21T053452Z/r/image-tiff
if dropped out, 0 otherwise). Participants with poor (unmeasured) health (
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM4/v/2018-04-21T053452Z/r/image-tiff
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 (
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM5/v/2018-04-21T053452Z/r/image-tiff
) because the path
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM6/v/2018-04-21T053452Z/r/image-tiff
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
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM7/v/2018-04-21T053452Z/r/image-tiff
represents the value of
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM8/v/2018-04-21T053452Z/r/image-tiff
had the individual received vaccine level
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM9/v/2018-04-21T053452Z/r/image-tiff
):
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM10/v/2018-04-21T053452Z/r/image-tiff
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM11/v/2018-04-21T053452Z/r/image-tiff
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM12/v/2018-04-21T053452Z/r/image-tiff
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM13/v/2018-04-21T053452Z/r/image-tiff
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM14/v/2018-04-21T053452Z/r/image-tiff
The first equality holds by the law of total probability, the second by d-separation of
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM15/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM16/v/2018-04-21T053452Z/r/image-tiff
given
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM17/v/2018-04-21T053452Z/r/image-tiff
, the third by d-separation of
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM18/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM19/v/2018-04-21T053452Z/r/image-tiff
, the fourth by d-separation of
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM20/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM21/v/2018-04-21T053452Z/r/image-tiff
given
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM22/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM23/v/2018-04-21T053452Z/r/image-tiff
, 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
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM24/v/2018-04-21T053452Z/r/image-tiff
is independent of
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM25/v/2018-04-21T053452Z/r/image-tiff
given
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM26/v/2018-04-21T053452Z/r/image-tiff
, but conditioning on
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM27/v/2018-04-21T053452Z/r/image-tiff
does not open any paths between
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM28/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM29/v/2018-04-21T053452Z/r/image-tiff
.
We conducted a simulation of 1,000,000 individuals for illustration (SAS code is available in the eAppendix; http://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
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM30/v/2018-04-21T053452Z/r/image-tiff
or
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM31/v/2018-04-21T053452Z/r/image-tiff
. These would lead to, respectively, edges
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM32/v/2018-04-21T053452Z/r/image-tiff
or
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM33/v/2018-04-21T053452Z/r/image-tiff
in the single-world intervention graph. In the latter case,
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM34/v/2018-04-21T053452Z/r/image-tiff
is d-separated from
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM35/v/2018-04-21T053452Z/r/image-tiff
given
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM36/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM37/v/2018-04-21T053452Z/r/image-tiff
, thus
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM38/v/2018-04-21T053452Z/r/image-tiff
would remain independent of
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM39/v/2018-04-21T053452Z/r/image-tiff
conditional on
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM40/v/2018-04-21T053452Z/r/image-tiff
(Theorem 12 in Richardson and Robins3). The reviewer also noted that the derivation fails with unmeasured confounding between
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM41/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM42/v/2018-04-21T053452Z/r/image-tiff
or between
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM43/v/2018-04-21T053452Z/r/image-tiff
and
JOURNAL/epide/04.02/00001648-201805000-00020/math_20MM44/v/2018-04-21T053452Z/r/image-tiff
.
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