To the Editor:
In the January 2014 issue of Epidemiology, Gatto et al1 provided a valuable organizational schema for epidemiologic causal effects. Using the potential outcomes framework, they illuminated the distinguishing characteristics of various causal effects and clarified their interpretation. Here, we highlight the significance of the fact that the notion of confounding can be defined with respect to both marginal distributions of potential outcomes and a specific effect measure.2,3
Let A denote a binary exposure of interest (1 = exposed, 0 = unexposed) and Y, a binary outcome (1 = outcome occurred, 0 = outcome did not occur). Then, we let
denote the potential outcomes for individual ω if, possibly contrary to fact, there had been interventions to set A to a. As a result, persons can be classified into 4 response types, ie, doomed (type 1), causal (type 2), preventive (type 3), and immune (type 4). Then, we let pi, qi, and ri, where i = 1–4, be proportions of response type i in the exposed group, the unexposed group, and the total population, respectively. Note that ri can be calculated as
represents the prevalence of A = a in the study population.
When a reference of exposure is A = 0, a causal risk difference in the total population is given by
, which is described as
using the response types (see the first row of Table 1 in the article by Gatto et al1). Then, as a weaker condition for no confounding, Gatto et al1 showed
This sufficient condition is implicitly derived from the notion of confounding in distribution.2 Note that this definition makes reference to the distribution of potential outcomes, and confounding is scale-independent in this case. Meanwhile, confounding has also been defined with respect to a specific effect measure. When we use risk difference as a measure of interest, a sufficient condition for no confounding in measure is given by
which is weaker than the “weaker” condition given by Gatto et al.1 Note that this definition makes reference to a particular measure, and confounding is scale-dependent in this case.2 When we are interested in a causal risk ratio in the total population with A = 0 as a reference of exposure, ie,
(see the first row of their eTable 1), a sufficient condition for no confounding in measure is given by
An analogous discussion applies when A = 1 is a reference of exposure (see the second rows of their Table 1 and eTable 1). More general discussion is available in the article by Suzuki et al.4
The above discussion clearly shows that sufficient conditions for no confounding vary according to the 2 notions of confounding (ie, in distribution and in measure), and we need to clarify which notion is used on each occasion. Notably, when the target population is the exposed or the unexposed, sufficient conditions for no confounding are identical in the 2 definitions of confounding;
is a sufficient condition when the target is the exposed, and
is a sufficient condition when the target is the unexposed (see the fifth and sixth rows of their Table 1, respectively). See the eAppendix (http://links.lww.com/EDE/A792) for sufficient conditions for no confounding when odds ratios are used.
Gatto et al1 also showed “natural population effects” by comparing a “factual” risk with a “hypothetical” risk. When a reference of exposure is A = 0, this measure is given by
(see the third row of their Table 1). Note that this effect is identical to the numerator of attributable fraction (population) or, more strictly speaking, attributable caseload (population).5 Gatto et al1 reported that conditions are unavailable for no confounding with regard to this measure. However, a sufficient condition for no confounding is given by
Note that this condition is scale-independent, and we can readily calculate the natural population risk difference by multiplying a crude risk difference (ie, (p1 + p2) − (q1 + q3)) by
, which is weaker than the condition,
, given by Gatto et al.1 An analogous discussion applies to a natural population risk difference with A = 1 as a reference of exposure (see the fourth row of their Table 1).
A clear schema is a useful method for teaching epidemiologic methods. We hope that our discussion further refines their organizational schema to achieve this goal.
Department of Epidemiology
Graduate School of Medicine, Dentistry and
Department of Information Science
Faculty of Informatics
Okayama University of Science
1. Gatto NM, Campbell UB, Schwartz S. An organizational schema for epidemiologic causal effects. Epidemiology. 2014;25:88–97
2. VanderWeele TJ. Confounding and effect modification: distribution and measure. Epidemiol Method. 2012;1:55–82
3. Greenland S, Robins JM, Pearl J. Confounding and collapsibility in causal inference. Stat Sci. 1999;14:29–46
4. Suzuki E, Mitsuhashi T, Tsuda T, Yamamoto E. A counterfactual approach to bias and effect modification in terms of response types. BMC Med Res Methodol. 2013;13:101
5. Suzuki E, Yamamoto E, Tsuda T. On the relations between excess fraction, attributable fraction, and etiologic fraction. Am J Epidemiol. 2012;175:567–575