What assumption(s) are necessary for an association to equal causation?

Essay 3

In this essay, I argue that for an association to equal causation, we need to assume that the causal effects are identifiable due to the absence of or corrections for any threats to exchangeability and errors in the measurement process. I will first state the conditions used by Greenland and Robins 1986 (GR from here on) as they map to the necessary assumptions underlying the essay’s primary claim. I will use the potential outcomes framework (POF) and the causal diagrams (modern epidemiological thinking) introduced in Hernan & Robins 2020 (HR here on) to defend assumptions. I will also demonstrate how the assumptions fall in the traditional epidemiological thinking articulated by Rothman et al. 2008 Chapter 9 (Rothman here on).

Three clarifying points at the outset. First, causation in the current essay refers to the effects of causes or the causal effects (Y^a=1 - Y^a=0) as defined in the essays from the past weeks. In other words, there is a treatment (A) whose effect we are interested in¹ and we have chosen an outcome (Y) to measure. Second, I believe (though I cannot warrant this) that DAGs (causal and non-causal diagrams - for measurement errors) provide a way to conceptualize a shared taxonomy of different kinds of bias. For instance, visualizing backdoor pathways for different instances of confounding and selection bias, as shown in the Chapters 7 and 8 figures of HR, is useful to better understand the similarities and differences in the causal structure of the biases. Also, visualizing how factors can differently influence the measurement error in A and Y can help understand the types of measurement bias (figures 9.1-9.7 of HR). This essay will rely on structural definitions and DAGs as an aide. Third and arguably the most important clarifying point, building on the last couple of weeks, we cannot observe or measure individual or group-level causal effects. Hernan 2018 claims further that it is a limitation shared by observational and experimental studies (e.g., RCTs). However, RCTs might require fewer assumptions for an associational effect to be inferred as a causal effect. Any study provides (effect) measures capturing associations that can be used, under the right assumptions, to lay causal claims. Hence, what are the assumptions for E[Y=1|A=1] to be equal to E[Y^a=1=1].

GR notes that identifiable effects are those where the different contrasts of the treatment can accurately predict different distributions of the effects. The impossibility of doing that in/for an individual study unit has been previously discussed as the fundamental problem in causal inference in the context of individual-level effects by Holland and others. Solving the non-identifiability problem would require another study unit (or group of units if one is interested in average effects) that: a) does not have the same treatment and b) is equivalent to (or comparable to) the original study unit for the outcome (i.e., equality of the incidence proportions of the groups in the absence of treatment). To clarify, I am assuming the exchangeability of study units: if the treatments of the two units had been exchanged, the same distribution of the outcomes (response types) would have resulted. In other words, identifiability² of effect is rendered due to the assumption about the exchangeability of the study units³. Within POF, the exchangeability assumption can be defined in terms of (marginal or conditional) independence between potential outcomes (but not the observed outcomes) and the treatment states. Y^a⊥A is the notation for marginal exchangeability. In instances where such exchangeability assumption breaks, the effects would not be correctly identified. Hence, the association would not be causation. The absence of or adequate corrections for threats to exchangeability ensures that we can identify causal effects, assuming no measurement errors.

The most commonly recognized threat to exchangeability is confounding. HR present a generic (going beyond epidemiology) and structural (not dependent on statistical or other study data-related considerations) definition of confounding as: when treatment and an outcome share a common cause (measured confounding denoted by L and unmeasured denoted by U) that opens a backdoor pathway between A and Y. Regardless of the magnitude and direction of confounding, in such instances, the estimated effect is due to mixing of two effects along the pathways A→Y and A←L/U→Y. Hence, we cannot equate Y^a with Y|A. As noted above, association is causation when the effect measure for Y=1|A=1 is the same as that for Y^a=1=1. The presence of confounding L or U opens a backdoor pathway between A and Y that breaks the assumption of marginal exchangeability thereby making the causal effect of A on Y non-identifiable. If we condition on L, the backdoor pathway can be blocked. This would not achieve marginal exchangeability but it can help achieve partial or conditional exchangeability denoted by Y^a⊥A|L. This means that the treated and the untreated are not marginally exchangeable but are conditionally exchangeable given L. Conditioning on the common cause of the treatment and outcome can help close the backdoor pathway to achieve conditional exchangeability that can render identification of conditional effects. A potential confounder is thus any measured or unmeasured variable that causes confounding as it is defined above.

This structural definition of confounding is different from confounding as (just) a bias, commonly held by traditional epidemiologists or the statistical definition rejected by both modern and traditional epidemiologists. Thinking of the existence of a common cause of treatment and the outcome (the structural definition) points to the reality of the study population (i.e., the causal world consisting of the causal pies and causal field) and does not depend on the study methods. Contrastingly, the definition of confounding as a form of bias relies on the confounder to be: a) predictive or “extraneous risk factor” of the outcome (Y) in the untreated (A=0) group, b) associated with the treatment (A) in the source population from which the study population is derived, and c) not lie on the causal pathway A→Y (Rothman pgs 4-5 and GR pg 417). Traditional epidemiologists consider confounders that satisfy the above necessary criteria a threat to the study's internal validity. The threat can be practically dealt with at the study design (randomization) or analysis (standardization, inverse probability weighting, etc.) phases noting the dependence of confounding on study methods. The statistical definition is concerned with the covariate balance across groups and judging the presence or absence of confounding based on whether the crude effect measure estimate is different from or the same as the stratified estimates to propose whether the confounder is collapsible. Such a definition depends not just on the source population and method of data collection but also on the effect measure of choice (GR pg 418). Apart from their rejection of the statistical definition, modern and traditional epidemiologists agree on the broad notion that confounding is the mixing or confusion of effects such that the effect that should not be ascribed to the treatment is ascribed to it (HR pg 91 and Rothman pg 2).

The second threat to exchangeability is selection bias. The structural definition is: bias(es) that arise from conditioning on a common effect (C) of two variables, one of which is either the treatment or a cause of treatment, and the other is either the outcome or a cause of the outcome. Conditioning on C opens a backdoor pathway that breaks the exchangeability assumption rendering the effect of A on Y non-identifiable. This definition covers multiple versions of a causal structure including those representing non-response bias, healthy worker/volunteer bias, self-selection, Berksonian bias, differential loss to follow-up, and even censoring (where only those who are uncensored (C=0) can be thought of as being selected in the study).

Traditional epidemiological thinking considers selection bias as the post-treatment procedures that make the study population non-representative of the source population (Rothman pgs 6 and 8). Such thinking then considers self-selection and healthy worker bias as somewhat different from the Berksonian bias. The latter is considered a ‘special case’ of selection bias that depends on the outcome and treatment (Rothman pg 6).

Selection bias is different from confounding. At the definition level, causal structures for confounding (common cause of treatment and outcome: e.g., A←L→Y) and selection bias (common effect of treatment, outcome, or their causes: e.g., A→C←Y). Traditional epidemiological thinking agrees with these differences although it does not present the articulation in generic and rigorous terms. Practically, randomization can solve confounding since the treatment is exogenous, ensuring independence between potential outcomes and the assigned treatment. However, randomization cannot solve non-exchangeability issues due to selection bias when the selection occurs after the treatment assignment; as also hinted by Rothman (pg 8). On a separate note, GR present a case where the average causal effect is identified but the conditional effect is not due to confounding within small strata (see GR pgs 415-416). Under the structural considerations, HR define this as selection bias in the conditional effect since it is an instance of conditioning on the common effect (L) of two marginally independent variables, one of which is associated with A and the other with Y (see HR pg 91 and figure 7.4). Hence, the structural definitions and DAGs provide more clear descriptions of confounding and selection bias that help delineate the differences between them. At times, the structure of bias may not matter so long as the factor (confounding or selection) needs to be adjusted to achieve comparability between the untreated and treated groups. Contrastingly, traditional epidemiological thinking is where there is a sharp line between confounding and selection bias.

Going beyond threats to exchangeability, we turn to: “errors in the measurement process”. While measurement error does not threaten exchangeability, it threatens the construct validity of A or/and Y by introducing measurement (or information) bias. Structurally, such bias can be understood based on the independence and non-differentiability of measurement errors associated with A (where A^* is the incorrectly measured construct and V_A is the measurement error) and Y. If V_A and V_Yare not dependent on some common V, then they are considered independent. If V_A (V_Y) does not differ by Y (A), then measurement errors in A (Y) are considered non-differential by outcome (treatment). Hence, there are four kinds of errors: independent nondifferential (HR figure 9.2), dependent nondifferential (HR figure 9.3), independent differential (HR figures 9.4 and 9.5), and dependent differential (HR figures 9.4 and 9.5). Examples of such bias include recall bias and reverse causation. The depiction of measurement bias differs from that of confounding and selection bias in that it uses some non-causal arrows in DAGs (A→A^*). However, DAGs still provide a common taxonomy for all. Traditional epidemiological thinking holds a similar view of different types of information biases, with a greater emphasis on the direction of bias, without evoking a clear structure. Regardless, in modern and traditional thinking, the absence of measurement errors or their appropriate correction removes measurement bias that ensures construct validity of outcome and treatment needed for equating an association to causation.

1. For clarity of notation, I will use A instead of X for treatment in this essay.

2. Due to lack of space, I am not explicitly diving into the differences between assumptions for partial or full identifiability as explained in GR.

3. These are other assumptions about positivity and consistency necessary for identifiability of effects. For the sake of this essay, we do not focus on them and investigate only exchangeability.