A mediator is a variable that plays an intermediate role in a causal chain. Ie, the 1st variable in the chain causes the mediator, which in turn causes the 3rd variable. Partial mediation is when the 1st variable causes the 3rd both through the mediator & directly.
Questions involving chains of influence spanning more than two variables in length will likely involve mediation. This will include many structural equation models, path analyses, and time series. In time series, mediation is often assumed, but may be only partial (as in p-order autoregression).
An intro from the Wikipedia page on mediation:
In statistics, a mediation model is one that seeks to identify and explicate the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third explanatory variable, known as a mediator variable. Rather than hypothesizing a direct causal relationship between the independent variable and the dependent variable, a mediational model hypothesizes that the independent variable influences the mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables. In other words, mediating relationships occur when a third variable plays an important role in governing the relationship between the other two variables.
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