Table 1

Definitions of the four-way decomposition as applied to the current study

DefinitionExplanationApplication*

Total effect (TE)

Total effect
Total effect of X (changing x* to x) on Y
What is the risk of alcohol misuse in adulthood among individuals whose parents misused alcohol relative to individuals whose parents did not misuse alcohol?
Controlled direct effect (CDE)Due to neither mediation nor interaction
Effect of X (changing x* to x) on Y, intervening to fix M to m
What would be the risk of alcohol misuse in adulthood among individuals whose parents misused alcohol, relative to individuals whose parents did not misuse alcohol, if everyone had a high level of school performance (ie, above-average school marks)?

Reference interaction

(INTref)

Due to interaction only
An additive interaction that operates only if M is present when X is x
What is the combined risk of alcohol misuse in adulthood among individuals whose parents misused alcohol and who had a low level of school performance (ie, at or below-average school marks), if having alcohol misusing parents is not necessary for a low level of school performance?

Mediated interaction

(INTmed)

Due to mediated interaction
An additive interaction that operates only if X (changing x* to x) has an effect on M
What is the combined risk of alcohol misuse in adulthood among individuals whose parents misused alcohol and who had a low level of school performance (ie, at or below average school marks), if having alcohol misusing parents is necessary for a low level of school performance?

Pure indirect effect (PIE)

Due to mediation only
Effect of the mediator (changing m* to m) on Y when X is x, multiplied by the effect of X (changing x* to x) on M
What is the risk of alcohol misuse in adulthood among individuals who had a lower level of school performance, if having alcohol misusing parents is necessary for a low level of school performance (ie, at or below-average school marks)?
• Y = Outcome (Alcohol misuse in adulthood).

• X = Exposure (Parental alcohol misuse); x* = No; x = Yes.

• M = Mediator (School performance); m* = At or below-average school marks; m = Above-average school marks.

• *Here, we build on the definitions proposed by Bean et al.23