The two main ways of reasoning about cause and effect in statistics are causal graphs and counterfactuals.
With causal graphs, you write down variables and draw arrows representing direct effects of one variable on another, and then work with a set of axioms that summarise what it means for one variable to affect another.
With counterfactuals, you talk about the effect of a variable in terms of the difference between the actual outcome with the variable set one way and the ‘potential outcome’ if it had been set another way. This is a fairly natural way to think about cause and effect – saying you have a hangover because you got drunk is saying that you have a hangover, and if you had not got drunk you would not have had a hangover. Reasoning about potential outcomes requires assumptions about conditional independence between variables.
Thomas Richardson (of UW) and James Robins (of Harvard) have a new working paper showing that graph and potential outcome approaches to causal inference are equivalent, and that the equivalence can be constructed in a much technically simpler and metaphysically more straightforward way than was previously available.