La Constitución Inca en el Valle del Cusco

Although identification of individual-level causal effects is generally agreed to be impossible within the counterfactual framework, identification of average causal effects is possible and forms the basis of a great deal of epidemiological causal inference (6). Average causal effects may be identified by creating exchangeable groups of individuals and comparing their average outcomes. This is often achieved through randomisation (29, 30).

2.4.1 Average causal effects for time-fixed exposures

To demonstrate the principle of using randomisation to identify the average causal effect for a time-fixed exposure, we consider a specific example involving the effect of chemotherapy versus radiotherapy on two-year survival amongst breast cancer patients. We illustrate how both unconditionally and conditionally exchangeable groups of individuals may be created by randomisation.

2.4.1.1 Exchangeability

2.4.1.1.1 Unconditional exchangeability

Epidemiologists have long considered the randomised controlled trial (RCT) to be the ‘gold standard’ for demonstrating causality because, if implemented correctly, it guarantees unconditional exchangeability (31). An RCT in our example context might involve randomly assigning each patients to receive either chemotherapy and radiotherapy, and then comparing the average outcome for each treatment group.

In this situation, the group who received chemotherapy is unconditionally exchangeable with the group who received radiotherapy. This is because randomisation ensures that the outcome is equally likely in both groups prior to the intervention, and so a simple comparison of the average outcome for each group after the intervention is sufficient to identify an average causal effect (32). In other words, those who received chemotherapy, had they instead received radiotherapy, would have experienced the same average outcomes as those who actually did receive radiotherapy (6), i.e. they are unconditionally exchangeable.3

2.4.1.1.2 Conditional exchangeability

We could alternately consider a conditionally randomised controlled trial (CRCT), in which each patient is randomly assigned to receive either chemotherapy or radiotherapy based on

3 _{If there exists differential loss to follow-up, then exchangeability may not be ensured by this process}

(33, 34). However, this is an additional complexity which we do not cover here, since our purpose is simply to illustrate the conceptual rationale behind such designs.

their initial cancer stage. For example, individuals in stage IV are randomised to receive chemotherapy with a higher probability than radiotherapy.

Here, a simple comparison of the average outcome for each treatment group cannot be assumed sufficient, as any difference in two-year survival might be attributable to the fact that the chemotherapy group has, on average, a worse prognosis at the beginning of the study. Nevertheless, we are still able to identify an average causal effect by comparing the average two-year survival between those who received chemotherapy and those who received radiotherapy among individuals who had the same initial cancer stage. Thus, within each subgroup of cancer stage, those who received chemotherapy, had they instead received radiotherapy, would have experienced the same average outcomes as those who actually did receive radiotherapy (6). The two treatment groups are conditionally exchangeable, i.e. they are exchangeable conditional on initial cancer stage.

2.4.2 Average causal effects for time-varying exposures

To demonstrate the principle of using randomisation to identify an average causal effect for a time-varying exposure, we return to the example from Section 2.3.1.2 involving the use of antibiotics to clear a chest infection, in which a dose of antibiotics may be prescribed at the point of initial diagnosis or at a subsequent follow-up visit.

We illustrate in this context how unconditionally and conditionally exchangeable groups of individuals may be manufactured by sequential randomisation.

2.4.2.1 Exchangeability

2.4.2.1.1 (Sequential) unconditional exchangeability

An RCT in our example context might involve randomly assigning each patient to receive each dose of antibiotics. This is referred to as ‘sequential randomisation’ (26) because patients are randomised at each time point. In this way, we create four treatment groups – those who received two doses, no doses, only the first dose, or only the second dose.

The proportion of people whose infections subsequently cleared in each of the treatment groups may then be directly compared. The process of sequential randomisation ensures that the outcome is equally likely in all groups prior to treatment both at the point of diagnosis and at the point of follow-up. Thus, a simple comparison of the average outcome for each group after the final intervention is sufficient to identify an average causal effect. For example, those who received both doses of antibiotics, had they instead received one of the other dosing regimes, would have experienced the same average outcomes as those who actually did receive those other dosing regimes (6), i.e. they are unconditionally exchangeable. 2.4.2.1.2 (Sequential) conditional exchangeability

By contrast, a CRCT in our example context might instead involve randomly assigning each patient to receive each dose of antibiotics based on the severity of their infection at the time.

For example individual who are initially judged to have more severe infections may be randomised to receive the first dose of antibiotics with a higher probability than those with less severe infections. Similarly, individuals with more severe infections at the follow-up visit may be randomised to receive the second dose with a higher probability.

Because each treatment group (i.e. those who received two doses, no doses, only the first dose, or only the second dose) is likely to have a different average outcome prognosis as a result of the way in which individuals were randomised, they cannot be directly compared. Moreover, we cannot even identify an average causal effect by comparing the proportion cleared chest infections among individuals who had the same infection severity at both time points, because infection severity at the second time point is itself affected by whether or not an individual received the first dose of antibiotics (i.e. infection severity is not randomised). However, within subgroups defined by initial infection severity, receipt of the first dose of antibiotics, and follow-up infection severity, those who received the second dose of antibiotics, had they instead not received the second dose of antibiotics, would have experienced the same distribution of outcomes as those who actually did not receive the second dose. The average outcome for each of the treatment groups may then be compared within levels of baseline and follow-up infection severity because they are (sequentially) conditionally exchangeable, i.e. they are exchangeable at each time point conditional on current infection severity.

We will return to this concept in Section 2.5.4, where we present a clearer graphical depiction of this issue (§2.5.4.1) and the challenges associated with identifying casual effects in

sequentially randomised contexts (§2.5.4.2).