Activities

• Intervention and prediction
• Experiment and causality

Learning objectives

Additional sections

• Get formatted versions: Word : PDF
• Instructor orientation
• Role in statistical practice
• Classroom discussion
• Tips for an active classroom
• Student pre-requisites
• Pitfalls

Orientation for instructors

There is an epigram that is familiar to all statistics instructors:

Correlation is not causation.

Put on a more explicit logical footing, this is equivalent to:

Correlation does not imply causation.

But as familiar as this epigram is, it’s wrong. We legitimately can say this:

No causation implies no correlation.

The “no correlation” part of this is pretty easy. We have many statistical techniques for working with two variables X and Y – t-tests, regression, etc – that can establish correlation. So by “no correlation” we mean that any of these tests have failed to demonstrate a correlation.

But “no causation” is more subtle. Knowing what this means requires a bit of formalism. The formalism we will use is that of diagrams: directed acyclic graphs.

Here are several:

We can define “no causation” in terms of these graphs. Here’s a useful approximation: The procedure is to imagine putting ants into one of the nodes of the graph. And these imaginary ants can (and will!) move only along the arrows in the direction of the arrowhead. The ants can’t walk against the flow of the arrow.

By “no causation” between X and Y we mean that there is no node from which ants can get to both X and Y.

This definition works well enough so long as the variables X, Y, and C are left to their own devices, being caused according to the arrows in the diagram. But if we intervene, say holding some node at a constant value, then we might close down an otherwise legitimate along-the-arrows pathway involving X and Y, or we might even open up a pathway, like that through C in diagram (n), that ordinary wouldn’t let ants get to X and Y from C. Such a situation is called a “collider.”

It’s starting to get complicated. I think this is part of the motivation for the traditional canon of introductory statistics, namely:

1. Correlation is not causation.
2. Experimental intervention is the only way to establish causation.
3. In an experiment we impose variability on X and look for correlated variability in Y.

This is a nice story, but it’s incomplete. For example, not all experiments involve imposing variability on X. An experimental intervention could be holding some other variable constant. And there are three ways to hold a variable constant:

1. Physically pin the variable to a constant value, e.g. by regulating the temperature in an experimental chamber.
2. Select a subset of cases from the data for which the variable happens to be at a constant value.
3. In a model, include the variable to be held constant among the explanatory variables. Consequently, when using the model to generate simulated data, we can intervene to hold that variable constant. Effectively, we build a model of the world and do experiments on that model.

1. and (b) are the most compelling, because we can’t really know if the model in (c) actually represents the real-world system.

What we miss out entirely in the canonical intro stats curriculum is the kind of experiment that involves holding variables constant. Such experiments can actually be used to reject hypotheses about causal networks. If we observe a correlation when the hypothetical network (augmented, perhaps, by holding variables constant) says we shouldn’t, we can reject the hypothesis. Or if we see no correlation when the network says we should see one, we can reject the hypothesis. This process of elimination, involving experiments where variables are held constant, can sometimes resolve a dispute between two rival hypotheses.

Going back to the statement that “no causation implies no correlation,” we can take the contrapositive to arrive at this, more meaningful statement:

Correlation implies causation.

It’s just that the correlation doesn’t by itself distringuish between rival hypotheses, telling us, for example, whether X causes Y or the other way around, or whether there is a common cause shaping both X and Y. But by doing hold-a-variable-constant experiments, we can distinguish between different possible arrangements of X, Y, and C.