Moderation and mediation are terms for measuring conditional effects. They are frequently discussed in the context of SEMs. However, both can be estimated in standard regression settings.

Let \(X\) be a predictor, \(Y\) be a response and we are studying whether \(Z\) moderates the relationship between \(X\) and \(Y\). In other words, we want to know whether \(Z\) modifies the strength (or direction) of the relationship between \(X\) and \(Y\). For example, we could study the relationship between socio-economic status (\(X\)) and frequency of breast exams (\(Y\)). Among younger women, breast exams are rare because current recommendations are for women age 40 and over to get exams. However, in older women, there is a relationship because those with higher SES tend to be more concerned with health. Age (\(Z\)) moderates the relationship between SES and frequency of breast exams.

Moderation is represented in a regression model with nothing more than an interaction term. There is a causal aspect to it, in that there needs to be the assumption that the predictor (\(X\)) has a causal relationship on the response (\(Y\)). If \(X\) is randomized, this assumption is satisfied. If the \(X\) is not randomized, this assumption must be based on theory and domain specific knowledge.

Fit the interaction model,

\[ Y = \beta_0 + \beta_1X + \beta_2Z + \beta_3XZ + \epsilon. \]

Therefore, \(\beta_3\) measures the moderation effect.

Again, let \(X\) be a predictor and \(Y\) be a response. We are studying whether \(Z\) mediates the relationship between \(X\) and \(Y\). Mediating variables are more complicated that moderators. A variable (\(Z\)) is said to be a mediator if it partially or fully explains the relationship between the predictor (\(X\)) and the response (\(Y\)). The easiest way to explain this is visually.

The unmediated and mediated model follow.