Table 1 Formal definition and interpretation of the four-way decomposition of the total effect.

Controlled Direct Effect (CDE)

(Y₁₀ − Y₀₀)

The effect of poverty and family adversity on young people’s mental health when emotional support is set to be high (i.e. the effect due to neither to mediation nor interaction)

Reference Interaction (INT_ref)

(Y₁₁ − Y₁₀ − Y₀₁ + Y₀₀)(M₀)

The effect of poverty and family adversity on young people’s mental health due to the interaction between poverty and family adversity and emotional support that operates when emotional support would be high in the absence of poverty and adversity. (i.e. the effect only due to interaction)

Mediated Interaction (INT_med)

Y₁₁ − Y₁₀ − Y₀₁ + Y₀₀)(M₁ − M₀)

The effect of poverty and family adversity on young people’s mental health due to the interaction between poverty and family adversity and emotional support and the fact that poverty and family adversity are causing low emotional support. (i.e. the effect due to both mediation and interaction)

Pure Indirect Effect (PIE)

(Y₀₁ − Y₀₀)(M₁ − M₀) = (Y_0M1 − Y_0M0)

The effect of poverty and family adversity on young people’s mental health through emotional support not accounting for the possible interaction between poverty and family adversity and emotional support. (i.e. the effect only due to mediation)