So what is?
This is, ultimately the domain of the statistical subfield of causal inference. When can we say that A causes B, rather than merely asserting that there exists some manner of associational relationship?
To get beyond mere correlation, the fundamental principle of causation lies in understanding the counterfactual. The intellectual model upon which most modern work rests is called the Rubin Causal Model (so named for Donald Rubin, the father of causal inference). The seminal work of note is “Bayesian Inference for Causal Effects: The Role of Randomization” published by Rubin in 1978. This framework is often referred to as the “Potential Outcomes” framework. Embedded in this is the idea of the counterfactual.
Suppose we have some binary treatment (Let it be denoted as $latex D_i$ for the $latex i$th unit or individual under study) we wish to administer. We want to understand the causal effect of this treatment on some outcome variable (call it $latex Y_i$) defined over the units in the study. We can denote this individual causal effect as the following: [latex]Y_i^1 - Y_i^0[/latex] where superscripts indicate whether the subject received treatment or not. You will notice something that is very difficult, however. The causal effect for an individual is the difference in the potential outcomes when they receive treatment ([latex] Y_i^1[/latex]) and when they do not receive treatment ([latex] Y_i^0[/latex]). The Fundamental Problem of Causal Inference is that we can never observe both.
To get around this, we instead want to think about average causal effects. I will discuss this in greater depth in my next entry. But merely by thinking about things in terms of counterfactuals (or potential outcomes), we can already reason about the observation that correlation is not causation. Consider the following correlation:
What is the counterfactual? If we only modified our level of lemon importation could we plausibly expect this to affect traffic fatalities? No. This is a trivial example, but there is more yet to come.