El estudio de radio

Still using Figure 3-3, assume that X, L, and M are all binary and that Y is continuous. There are four effects which can be of interest in causal mediation analysis: the controlled direct effect (CDE); the

41 natural direct effect (NDE); the natural indirect effect (NIE); and the total causal effect (TCE) (Daniel et al., 2011, VanderWeele, 2015). Each is calculated as the marginal mean difference between two potential outcomes. Thus, it is useful to return to counterfactual notation. The CDE is the most intuitive. Between the two potential outcomes it changes the value of the exposure while holding the mediator at some control value ‘m’ (e.g. 0 for a binary variable). It is defined as:

CDE = 𝐸[𝑌 (1, 𝑚)] − 𝐸[𝑌 (0, 𝑚)]

Like the potential outcomes discussed above, the CDE can be interpreted as a contrast between the potential outcomes of two counterfactual scenarios. In the first, all participants are exposed, while in the second none are exposed. In both the mediator is held at m. Effectively then the CDE statistically adjusts for the mediator while counterfactually intervening on the exposure (e.g. by using simulation techniques given conditional exchangeability). However because the CDE “is a function of m” the marginal mean difference may vary under different values of m (Daniel et al., 2011). A consequence of this is that the difference between the total effect and the CDE does not then correspond to a controlled indirect effect. Note that a difference between the total effect and the CDE may indicate that there is some indeterminate degree of mediation, although this is far from a robust approach to identifying indirect effects. For this reason, the CDE is generally not used in this thesis.

A nested counterfactual, however, can treat the mediator in such a way as to always estimate the same marginal mean difference regardless of the control value used and thus quantify both a direct and an indirect effect. Nested counterfactuals take the form: E[Y(X, M(x))]. The difference is that the mediator is no longer held at an a priori control value for all participants, but instead takes on a different value for each, dependent on the (possibly counterfactual) value of the exposure.

Specifically, a nested counterfactual is used to calculate the potential outcome Y when X is set to 0 or 1, and when M is set to the value that it would have taken for each participant when X is set to 0 or 1. Thus the counterfactual value or the mediator is ‘nested’ within a counterfactual value of the exposure such that the value that the mediator takes is one that “evolves naturally” within the counterfactual scenario - thus the natural effect (Daniel et al., 2011).

The natural direct effect is defined as:

NDE = 𝐸[𝑌(1, 𝑀(0))] − 𝐸[𝑌(0, 𝑀(0))]

It compares two potential outcomes. The first, E[Y(1, M(0))], is the potential outcome in which everyone is exposed but the mediator is set to the value it would have taken under no exposure. The second, E[Y(0, M(0))], is the potential outcome in which no one is exposed and the mediator is set to

42 the value it would have taken under no exposure. Thus, it is very similar in form to the CDE in that the mediator is held at a value while the exposure is intervened upon. The difference is that the value the mediator takes derives from counterfactual values of the exposure. As a consequence, it is thus allowed to vary between participants rather than being held at the same value for the full sample. Consider the prior data example where the sample was 100 adolescents, the outcome was the number of units consumed per week and the exposure was a binary variable indicating whether the

adolescent’s mother drinks more than 14 units of alcohol per week. Assume that the mediator was a binary variable indicating whether the adolescent had negative expectations around alcohol

consumption (M=0) or positive expectations around alcohol consumption (M=1). Calculating the potential outcome E[Y(1, M(0))] would involve counterfactually exposing the sample to all mothers consuming more than 14 units per week while letting adolescent alcohol expectations take on the value that it would have taken if all of the sample was counterfactually ‘controlled’ such that none of the mothers consumed more than 14 units of alcohol per week. This would then be contrasted with the potential outcome E[Y(0, M(0))] in which no one’s mother drank more than 14 units per week and everyone’s mediator took on the value corresponding to the same exposure status.

The corresponding NIE is then defined as:

NIE = 𝐸[𝑌(1, 𝑀(1))] − 𝐸[𝑌(1, 𝑀(0))]

It contrasts a scenario in which everyone is exposed and the mediator is set to the value that it would have taken under exposure E[Y(1, M(1))], to one in which everyone is exposed and the mediator is set to the value that it would have taken under no exposure E[Y(1, M(0))]. Thus the NIE holds the exposure value at one level while varying the mediator.

The TCE is perhaps more intuitive. It is defined as;

TCE = 𝐸[𝑌(1, 𝑀(1))] – 𝐸[𝑌(0, 𝑀(0))]

In the first scenario E[Y(1, M(1))], all are exposed and the mediator takes the value that it would have taken under exposure. In the second scenario E[Y(0, M(0))], none are exposed and the mediator takes the value that it would have under no exposure. Thus, the estimate produced is simply the difference between the exposed and the unexposed. In effect it is equivalent to the difference between Y(1) and Y(0) and as such is analogous to the ACE from above.

43 This thesis uses the Stata package gformula to calculate the TCE, NDE, and NIE (Daniel et al., 2011). Gformula is an implementation of the mediational g-computation procedure. The same assumptions are required for valid causal mediation as for calculating the ACE, with some extensions. Where conditional exchangeability can be assumed when calculating an ACE if there is no

uncontrolled confounding of the X→Y relationship, mediation analysis additionally requires that there is no uncontrolled confounding of the X→M or M→Y relationships, including M→Y confounding caused by L (i.e. there are no EIMOCs) (Wang and Arah, 2015). Given a set of confounders C, a set of EIMOCs L, and no uncontrolled confounding, mediational g-computation works by using Monte Carlo simulation to simulate each post-exposure variable in sequence

(according to a DAG, for example) to produce a potential outcome. The TCE, NIE and NDE are then estimated by comparing these potential outcomes. For example, assume there were two variables in L as per the below DAG in Figure 3-4. Assume also that C, X, L1, L2, and M are binary while Y is continuous.

Figure 3-4: Mediation model with two variables in L

The process of ‘setting’ variables to specific values in order to estimate potential outcomes from nested counterfactuals is achieved by first modelling the relationships in the observed data as per the DAG and then using the parameters from these models to direct Monte Carlo simulation under the interventions of interest. For example, for potential outcome 𝐸[𝑌(1, 𝑀(0))], in which everyone is exposed but the mediator takes on the value it would have taken if no one was exposed, L1* is simulated on its conditional distribution given C and X when X is replaced with 0. L2 is then simulated on the conditional distribution given X, C, and L1*. M* is then simulated given X, C, L1* and L2*. Finally, the potential outcome Y* is simulated given X, C, L1*, L2*, and M*. The process

44 for estimating the other potential outcome necessary for the NIE, 𝐸[𝑌(1, 𝑀(1))], is identical except that X is replaced with 1, not 0. The NIE is then the difference between the two simulated potential outcomes (Daniel et al., 2011, VanderWeele, 2015, Thoemmes and Ong, 2015, Wang and Arah, 2015). Mediational g-computation thus only ever conditions on past variables. This is what Daniel et al argue differentiates it from approaches which cannot handle EIMOCs, such as the Baron & Kenny approach (Daniel et al., 2011). The other strength of meditational computation is that exposure- mediator interactions can also be simulated. However, this was not an important focus for this thesis as very few exposure-mediator interactions were detected (covered in detail in Chapter 9). Similar to the IPW estimators described above gformula uses bootstrapping to calculate confidence intervals for the TCE, NDE, and NIE.

Gformula has been used in multiple settings, including quantifying epigenetic mediators of the effect of smoking on lung cancer incidence (Fasanelli et al., 2015), estimating the direct effect of intrauterine smoking on offspring mental health (Menezes et al., 2013), investigating whether “biogeographical ancestry” has an indirect effect on prevalence of type 2 diabetes via ethnicity (Piccolo et al., 2014), investigating whether number of live births mediates the effect of birthweight on later diagnoses of endometriosis (Gao et al., 2019), and more. It is used in Chapter 9 of this thesis to explore mediators of the effect of parental influences on adolescent alcohol harm, including other parental influences (e.g. parenting and the parent-child relationship), peer effects, and more.