3. Heterogeneity in causal effects
An important issue is the assessment of differences in causal effects between individuals. Clearly, a necessary condition for a factor Z to be a modifier of the effect of X on Y is that Z precedes the outcome Y. If such a potential effect-modifier Z is associated with X, the parameter that describes the modification of the effect of X on Y is not identified without making further assumptions. Effect-modifiers are typically assessed with interaction terms in regression models.
Choice of the effect measure
Whether and, if yes, to what extent the degree of an effect differs according to the values of Z depends, however, on the choice of the model and the associated index of effect magnitude. As mentioned above, some effect measures (e.g. the odds ratio) usually serve only to quantify the magnitude of a causal effect supposed to be constant between the individuals.
Moreover, the risk difference is the only measure for which effect heterogeneity is logically linked with causal co-action in terms of counterfactual effects. To explain this, it is necessary to define the causal synergy of two binary factors, Xi and Zi (coded as 0 or 1), on a binary outcome Yi in an individual i (at fixed time).
Clearly, if Xi and Zi do not act together in causing the event Yi = 1, then
(a) if Yi = 1 is caused by Xi only,
Yi = 1 if (Xi = 1 and Zi = 0) or
(Xi = 1 and Zi = 1)
and Yi = 0 in all other cases. Thus, Yi = 1 occurs in all cases where Xi = 1 and in no other cases.
(b) if Yi = 1 is caused by Zi only,
Yi = 1 if (Xi = 0 and Zi = 1) or
(Xi = 1 and Zi = 1)
and Yi = 0 in all other cases. Thus, Yi = 1 occurs in all cases where Zi = 1 and in no other cases.
Therefore, causal synergy means that 1) Yi = 1 if either one or both factors are present and 2) Yi = 0 if neither factor is present. Now, one is often interested in superadditive risk differences, where the joint effect of X = 1 and Z = 1 is higher than the sum of the effects of (X = 1 and Z = 0) and (X = 0 and Z = 1) as compared to the risk for Y = 1 under (X = 0 and Z = 0), that is,
P(Y = 1 | X = 1, Z = 1) > P(Y = 1 | X = 1, Z = 0) + P(Y = 1 | X = 0, Z = 1) - P(Y = 1 | X = 0, Z = 0).
If superadditivity is present, one can show that there must be causal synergy between X
, at least for some individuals [[2
], chap. 18; [26
]]. This relation does not apply in the opposite direction: If there is causal synergy among some individuals there may be no superadditivity. Thus, one can demonstrate rather a causal interaction than its non-existence. Note that other logical relations do not exist and the risk difference is the only measure for which such a logical link exists [[2
]; chap. 18; [26
]]. Also, other measures like correlations, standardised mean differences or the fraction of explained variability do not serve to quantify the degree of causal effects because they mix up the herefore solely relevant mean difference with parameters of exposure and outcome variability [28
Another crucial point for the choice of effect index is whether the interaction terms in regression models corresponds with so-called mechanism-based
(e.g. biological) interactions [29
]. For instance, if the dose of intake of a particular drug is known to influence the release of a certain hormone linearly, then the interaction term of another factor with drug intake in a linear model corresponds to the presence of a biological interaction.
Deterministic versus probabilistic causality
A fundamental question relating to heterogeneity in causal effects is the distinction between deterministic and probabilistic causality [[2
], chap. 1; [30
], chap. 1]. The functional-deterministic understanding of causality is based on the Laplacian conception of natural phenomena, which are assumed to follow universally valid natural laws. Here, in the absence of measurement error and other biases, the observable heterogeneity in Y
– given X
and the other observed covariates – would be attributed solely to unobserved factors. If we knew the causal mechanism completely (how complicated it may be) and the values of all the causal factors, the outcome Y
would be exactly determined. Note that I have implicitly used this assumption in the previous discussions.
Within the probabilistic understanding of causality, individual variation exists within the outcome Y
, which can not be explained by unconsidered factors. This variation might be called real randomness
and can be found in quantum physics [14
]. It is possible to incorporate real randomness into counterfactual models because one can specify a probability distribution for a potential outcome of a fixed individual at a fixed time [[7
] and references therein]. In real situations, however, the distinction between deterministic and probabilistic causality does not play a major role in systems that are complex enough for substantial residual heterogeneity in the modelled effect to be expected. Here, the effect is practically probabilistic. Such a situation is rather the rule than the exception in medical and behavioural sciences.
On the other hand, after incorporating major effect-modifiers into a model, the effect of X
should be sufficiently homogeneous to allow for uniform interventions in the subpopulations defined by the values of the effect-modifiers. As a consequence of the existence of effect-modifiers, a variation in their distribution across different populations implies that one would expect to estimate different effects if the modifiers were not considered in a model. Thus, differences in estimates of effects do not imply that different causal mechanisms act; instead, they might be solely due to different distributions of hidden effect-modifiers [[2
], chap. 18; [16
]]. Interactions with intrinsic variables; that is, individuals' immutable properties like sex, race and birth date are often regarded as an indication of a narrow scope of a model [31
]. On the other hand and as mentioned above, nonmanipulable properties are hardly subject to counterfactual arguments.