Underlying assumptions

However, given that outcome data are available for only a small sample rather than all of the population, it is necessary to accept certain assumptions before the ACE, or any other causal parameter, can be estimated.

If treatment assignment is randomised and adhered to perfectly, randomisation provides independence between outcome and allocation of treatment; thus, under the assumption that the sample is generalisable to the target population, the outcome data 𝐸(𝑌𝑗) = 𝐸(𝑌|𝑍 = 𝑗) can be used to estimate this average quantity ACE by 𝐸(𝑌|𝑍 = 1) − 𝐸(𝑌|𝑍 = 0) = 𝐸(𝑌₁) − 𝐸(𝑌₀) , a result first demonstrated by Neyman (85). However any selection process which violates the independence between allocation of treatment and outcome (such as that typically introduced by deviation

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from randomised treatment or by observational rather than experimental assignment of treatment) invalidates this estimation (84).

As such, the underlying treatment assignment mechanism is of fundamental importance. The causal methods to be discussed presently rely on randomisation to provide balance on all baseline variables (including all potential random variables and thus counterfactual outcomes) between groups. Methods for observational studies rely on the assumption that the randomisation balance can be simulated by conditioning on all relevant variables which confound the a priori independence between treatment allocation and outcome, invoking the so-called strong ignorability assumption (where “strong” implies an unconfounded and therefore causal interpretation), also known as the assumption of no unmeasured confounders (NUC), such that the relationship between the potential outcomes and received treatment can be assumed to be independent, given adjustment for all such confounders. In the case of dynamic treatment regimens, the sequential ignorability assumption would be required, adjusting for relevant (time-varying) covariates at each time point, to simulate sequential randomisation of treatments at each time 𝑘 (107, 108). This assumption can be interpreted as, conditional on all pertinent baseline and time-varying confounders, those who do and do not change treatment at time 𝑡 have the same probability of outcome, such that the decision on whether to change treatment is independent of underlying untreated outcome given these variables.

Formally, randomisation ensures that the probability that an individual with potential outcome𝑌0 and 𝑌1 is assigned a certain treatment is a constant that does not depend on their potential outcomes 𝑌0 and 𝑌1, such that 𝑃(𝑍|𝑌0, 𝑌1) = 𝑃(𝑍) ∀ 𝑌0, 𝑌1, whereas the NUC assumption states that the probability of assignment is independent of all missing (𝑌_𝑚𝑖𝑠) and observed (𝑌_𝑜𝑏𝑠) outcomes, and can be assumed to be random once

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conditioned on observed covariates 𝑋, i.e. 𝑃(𝑍|𝑋, 𝑌𝑜𝑏𝑠, 𝑌𝑚𝑖𝑠) = 𝑃(𝑍|𝑋) ∀ 𝑌𝑚𝑖𝑠, 𝑌𝑜𝑏𝑠 (109).

Another basic assumption commonly invoked for causal analysis, necessary to allow statistical inference based on the assumption that individuals’ observations are statistically independent, is the stable-unit-treatment-value assumption (SUTVA), which assumes that a full set of potential outcomes exist for each person (even though only one is observed) which are independent of outcomes and treatment status of all other subjects (110). Note that the SUTVA differs from the usual assumption of identical independent distributions of outcomes (i.e. that the outcome of an individual is unaffected by outcomes of other participants) required for standard statistical analysis, as it relates to stability of treatment received as well as outcome status. This assumption is necessary for the notation 𝑌𝑖𝑘 to be sufficient in denoting the outcome of unit 𝑖 with treatment 𝑘; otherwise, notation would need to include information on treatment received by all other units and the basic assumption of statistical independence between subjects (necessary for all statistical inference) would also fail (111). The SUTVA means that the observed outcome for unit i will equal one of the potential outcomes for that unit, no matter how the treatment was received (63).

This assumption also incorporates the notion of treatment stability, i.e. that each level or form of treatment is given to different units in an identical manner (so that there are no hidden versions of treatments), or equivalently that even if the treatment delivery varies slightly from individual to individual, the effect (i.e. potential outcomes) will be consistent. This assumption is required in order that the following equality will always hold: 𝑌𝑖 = 𝑌1𝑖 ∗ 𝐴𝑖 + 𝑌0𝑖∗ (1 − 𝐴𝑖) (112).

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The SUTVA may be contravened in studies of infectious disease or when behavioural interventions are affected by interaction between individuals (85) or in cluster randomised behavioural or educational trials where participants may observe or influence each other’s outcomes. Treatment stability may not hold in cases where natural variation occurs in the treatment process, for example when the rapport between therapist and individual affects the delivery of treatment for psychological disorders, or when clinicians’ expertise influences the quality of surgical or medical treatment, or when practitioners’ experience with one patient influences their subsequent treatment of another patient. In practice, some treatment variation is likely to occur, but such heterogeneity is assumed to remain within a reasonable range (113). The exclusion restriction (ER) is a commonly employed assumption which implies no direct effect of treatment assignment, such that the offer of treatment itself has no influence on outcome, but instead any effect of randomisation on outcome is entirely due to its effect on treatment receipt. An alternative interpretation is that a patient who receives treatment will experience the same outcome whether or not they were originally randomised to receive it (114). In a continuous compliance setting, this assumption may be considered a “zero dose-response” assumption, such that without treatment, assignment will have no effect (115). The exclusion restriction is so called because the assignment mechanism is assumed to influence outcome “exclusively” through its effect on actual treatment received (112).

The exclusion restriction is likely to hold in double-blind trials (116) where the psychological effect of treatment assignment is (theoretically) negligible, but when participants are aware of the treatment they have been allocated to, there is potential for a strong placebo effect (or expectation of treatment) and thus this assumption may not hold. Unblinded behavioural intervention studies may be particularly prone to

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invalidation of this assumption, given that the impact of being assigned a certain behavioural intervention may affect an individual’s outcome for reasons other than the intervention itself (for example, due to changes in a patient’s mood or motivation levels, or because of an underlying rapport with therapist). Furthermore, the exclusion restriction will not hold if additional treatments or extra medical attention tend to be given in a certain treatment arm.

Before moving on to consider various approaches to causal methods which make these assumptions, it is worthy to note that, in making explicit the nature of relationships between variables and the underlying implicit assumptions regarding these relationships required for causal inferences, cDAGs implicitly demonstrate a number of key features of the causal techniques to be discussed.

5.2.5. cDAGs demonstrate underlying features and assumptions of