A counterfactual approach to bias and effect modification in terms of response types

Background The counterfactual approach provides a clear and coherent framework to think about a variety of important concepts related to causation. Meanwhile, directed acyclic graphs have been used as causal diagrams in epidemiologic research to visually summarize hypothetical relations among variables of interest, providing a clear understanding of underlying causal structures of bias and effect modification. In this study, the authors aim to further clarify the concepts of bias (confounding bias and selection bias) and effect modification in the counterfactual framework. Methods The authors show how theoretical data frequencies can be described by using unobservable response types both in observational studies and in randomized controlled trials. By using the descriptions of data frequencies, the authors show epidemiologic measures in terms of response types, demonstrating significant distinctions between association measures and effect measures. These descriptions also demonstrate sufficient conditions to estimate effect measures in observational studies. To illustrate the ideas, the authors show how directed acyclic graphs can be extended by integrating response types and observed variables. Results This study shows a hitherto unrecognized sufficient condition to estimate effect measures in observational studies by adjusting for confounding bias. The present findings would provide a further understanding of the assumption of conditional exchangeability, clarifying the link between the assumptions for making causal inferences in observational studies and the counterfactual approach. The extension of directed acyclic graphs using response types maintains the integrity of the original directed acyclic graphs, which allows one to understand the underlying causal structure discussed in this study. Conclusions The present findings highlight that analytic adjustment for confounders in observational studies has consequences quite different from those of physical control in randomized controlled trials. In particular, the present findings would be of great use when demonstrating the inherent distinctions between observational studies and randomized controlled trials.


Appendix 1: positivity condition
In addition to exchangeability, recent studies have emphasized the significance of positivity condition, sometimes referred to as the experimental treatment assignment assumption, to infer causation [1][2][3]. Positivity means that we must ensure that there is a nonzero probability of being assigned to each of the treatment levels at every combination of the values of the observed confounder(s) in the population under study. By using the notation in the present study, positivity is described as: if [ ] 0 P C c  then [ | ] 0 P E e C c    for  e. In (either marginally or stratified) randomized controlled trials, positivity is taken for granted. In observational studies, however, positivity is not guaranteed.
When the information about those who did not drop out is available in observational studies ( Figure  2A), positivity assumption can be described in terms of EDS response types as follows (see Figure  3): If [ 1]

 
Then, by referring to which is equivalent to the causal RR in equation 1. In other words, the RR in equation A10 is an alternative notation of causal RR in terms of response types.

Appendix 3: sufficient conditions to estimate effect measures in observational studies by adjusting for confounding bias
We show that the weighted average of stratum-specific associational RRs in equation 7 is equivalent to the causal RR in equation 1 if E D e |C for  e holds. First, to simplify the explanation, we show the RR in equation 7 by using the conventional notation of probability as Then, if E D e |C for  e holds, this can be rewritten by referring to Tables 1 and 2 as   T  T  T  T   1,2,3,4,5,6,7,8 1,2  1,2,3,4,9,10,11,12 1,3  T  T   1,2  1,3 TT 1,2,5,6,9,10,13,14 3,4 T T T 1,2,5,6,9,10,13,14 1,3,5,9,11,13,15 e e e e e e e P C P E which is equivalent to the causal RR in equation 1. The first equation is derived from      Finally, we show that the weighted average of stratum-specific associational RRs in equation 7 is equivalent to the causal RR in equation 1 if E T D T |C holds. By using the notation in the present study, we can analogously prove this referring to Table 2

Appendix 4: relations between the sufficient conditions in Appendix 3
Here, we show a proof of the following inclusion relation: Next, note that the following equivalence relation holds:

Appendix 5: assumptions of monotone treatment response and monotone treatment selection
As has been well noted [4], randomization is so highly valued because it is expected to produce exchangeability, that is, the potential outcomes of D and the observed exposure E are independent. In observational studies, however, (conditional) exchangeability is not guaranteed, and researchers are required to use their expert knowledge to enhance its plausibility. On a related issue, in the field of econometrics, assumptions of monotone treatment response (MTR) [5] and monotone treatment selection (MTS) [6] were recently introduced to compensate for the lack of randomization. Although the detail of these assumptions is beyond the scope of this paper, it is worth mentioning that the MTR assumption is equivalent to the assumption of positive monotonic effect, as discussed in this study. Meanwhile, the MTS assumption can be described by using the notation in the present study as follows: Note that, although the MTR assumption is at an individual level, the MTS assumption is at a population level. Also, we should note that the MTR assumption may be relevant in randomized controlled trials as well as observational studies. By contrast, the MTS assumption is primarily relevant in observational studies, except for the situation in which adherence to treatment is not perfect in randomized controlled trials. (Notably, the MTS assumption in observational studies is related to the presence of the 3 marginally open paths between E and D T in Figure 8, i.e., E←E T ←U1→D T , E←E T ←U2→D T , and E←C←U1→D T . Even when we condition on C, only the third path can be blocked, and E and D T remain connected via the first 2 paths. Thus, structurally, the MTS assumption does not always refer to the issue of "selection." In Figure 6, there are no open paths between E and D T , which demonstrates that the MTS assumption is irrelevant in randomized controlled trials if adherence to treatment is perfect. Further, when using stratified randomization of E by C as shown in Figure 7, the only open path between E and D T , i.e., E←C←U1→D T , can be blocked by adjusting for C.) When the information about those who did not drop out is available in observational studies (Figure 2A), the MTS assumption can be described in terms of EDS response types as follows (see Figure 3): The full enumeration of EDS response types in this study would provide new assumptions at an individual level. For example, by extending positive monotonic assumption (or, the MTR assumption) to compound potential outcomes, one may be interested in the following assumption: