Causal Terror — Bayes Thinking Lab

When are two things associated, and what should I take into account in an analysis? Here you learn it step by step.

What is this about?

In data, two things are often associated, for example exercise and heart health. Why? Exercise might protect the heart. Active people might also be younger. Or the way the sample was selected plays a role. To sort this out, you draw the causes you assume as a graph with arrows (DAG). An arrow A → B means: A influences B.

In three stages, the tool takes you from the question “When are two variables associated?” to the question of which variables you should take into account (adjust for) in an analysis. You need no prior knowledge of causal analysis.

The key terms

• Edge: a single arrow between two variables, for example A → B.
• Path: a sequence of one or more edges connecting two variables. A → B ← C is a path made of two edges. When walking along a path, the direction of the arrows does not matter. A path is causal only if all arrows point in the same direction (A → B → C).
• Information flows: If a path is open, one variable tells you something about the other. We then see an association.
• Adjusting (also controlling for or partialling out): You compare only people who are equal on one variable, for example only people of the same age.
• Confounder / fork (A ← B → C): a common cause of two variables. It creates a spurious association. Adjusting closes this unwanted path.
• Mediator / chain (A → B → C): an intermediate station through which a cause acts. Adjusting closes this causal path. Whether that is what you want depends on the effect you are after: for the total effect the path belongs to it, for the direct effect it does not.
• Collider (A → B ← C): a common effect of two variables. The path is closed at first. Adjusting opens this previously closed, unwanted path and creates an association that does not exist in the world.
• d-separated and d-connected: All paths between two variables are blocked (d-separated), or at least one is open (d-connected).
• Intervention (also called manipulation, do-operator): You deliberately make someone do something instead of just observing.

The core message

Adjusting for a confounder → an unwanted path is closed.
Adjusting for a mediator → a causal path is closed.
Adjusting for a collider → a previously closed, unwanted path is opened.
Which of these consequences you want depends on which effect you want to estimate. So the first question is always: Which effect am I after? Then you ask: What happens to the paths between X and Y when I adjust for Z?

The three rules of do-calculus

If you want to compute the effect of an intervention from observational data, you can apply three rules. You can read them as questions to your graph:
① Ignore observations: Which additional observed variables provide any relevant information at all?
② Replace intervention with observation: When may an interventional effect be inferred from observed associations?
③ Ignore intervention: Which variables are affected by the intervention at all?
In stage 3 the tool applies these rules to your adjustment. The derivation is in the expandable section “Why is this solution formally valid?”, after you have solved the task intuitively.

All numbers come from simulations with linear relationships. The arrows are assumptions that you can see in the graph and change.

How to proceed

① Information flow: You get to know the three patterns mediator, confounder and collider. The tool shows when an association arises between two variables and what adjusting does to it.
② Reading paths: You examine larger graphs and check for each path whether it is open or blocked. The tool shows what that means for the effect.
③ Application: Using exercise and heart health as an example, you determine the total effect (A) and the direct effect (B) in two tasks. The indirect effect follows from these. You choose yourself what to adjust for. After “Check my solution” the tool shows the consequence.

What to watch out for

• Whether an adjustment helps depends on which paths it opens or closes and on whether the path belongs to the effect you are after.
• An adjustment can also do harm even though it sounds plausible. Variables that change no path have no influence on the effect.
• Opposing paths can cancel each other out. An open path therefore does not always show an association.

And then?

How to implement the chosen adjustment in R is shown by the Golem Builder.

Stages – please choose a stage: ↓

Stage 1

Why are two things associated? An association arises when information can flow between them — along arrows, no matter which direction they point. That is important to understand: in data we initially see only associations, not the causal direction. The direction comes only from our theoretical model. For the data, it does not matter which way we believe(!) an arrow points. There are three basic patterns, and they differ in what happens when you adjust for a variable. Each pattern comes with a research question. Choose a pattern and compare A and C without and with adjustment for B.

Adjusting means: you compare only people who are equal on B (a group with the same B value). This is also called “controlling for B” or “partialling out B”. In the scatter plot you see this as colored groups.

Edge and path: An edge is a single arrow between two variables. A path consists of one or more edges. In A → B → C there are two edges, but only one path from A to C.

Pattern & information flow

Arrow strength 0.70

Data: A against C

Path A – C—

Association r — without adjustment—

Association r — with adjustment for B—

Remember: what adjusting does to the path

Pattern

Without adjustment

With adjustment

Consequence

Mediator (Chain)

Path open

Path blocked

Closes a causal path. The part running through B is no longer part of the estimate of the total effect. This only makes sense if you want to exclude exactly this part (direct effect).

Confounder (Fork)

Path open: spurious association

Path blocked

Closes an unwanted path: the spurious association disappears.

Collider

Path closed

Path opened

Opens a previously closed, unwanted path: an association arises that does not exist in the world.

Stage 2

Real research questions involve more than three variables. Then there are often several paths between two variables X (cause) and Y (effect). Choose an example graph. X and Y are given there; you examine the paths between them. Decide what you adjust for and read the paths: the tool checks at every station whether the information gets through.

A path consists of one or more edges (single arrows). It is open if it is blocked at no station. You block a mediator and a confounder by adjusting for them. You open a collider that way — even if you only adjust for one of its consequences. If at least one path is open, X and Y are d-connected (an association is expected). If all are blocked, they are d-separated (no association).

Tip to get started: load the examples “Confounder”, “Mediator” and “Collider” one after another and adjust for the middle variable. Dashed nodes are unobserved: you cannot adjust for them.

Graph

Example graph

In this example X (cause) and Y (effect) are given. You examine what adjusting does to the paths between them.

Tool

open causal pathopen non-causal pathadjusted✕ blocked: station where a path closes◉ opened: collider opened by adjustment

Selected node: none

Edit mode: Drag an arrow from one node to another, select an arrow and change its strength, or move nodes. “Reset changes” restores the original graph.

Paths from X to Y

What does this mean for the effect?

Simulated data (visual cross-check)

Correlation r(X, Y) — without adjustment—

r(X, Y | Z) — with adjustment—

r is a correlation, not an effect. The effect estimates (regression coefficients) are shown below under “Estimating the effect”.

Estimating the effect: comparing methods

Stage 3

In psychology we often observe associations, but what we are really interested in are causal effects: what would happen if we deliberately intervened on X?

Here you apply what you learned in stages 1 and 2. The example: how does exercise (X) affect heart health (Y)? How to proceed: the task defines which effect you want to estimate. Then you click the variables in the graph that you want to adjust for, and decide for yourself. “Check my solution” brings up the evaluation; after that it updates with every change. If you get stuck, “Help: show paths” colors the paths in the graph. “Show solution” becomes available after your first check. The formal derivation with the three rules of do-calculus is at the very bottom and explains why a solution is valid.

Reminder from stage 1: Adjusting for a confounder closes an unwanted path. Adjusting for a mediator closes a causal path. Adjusting for a collider opens a previously closed, unwanted path. Whether that helps depends on the effect you are after.

Example: exercise and heart health

We want to know what causal effect regular exercise has on heart health.

It is not enough to simply compare active and less active people. People differ, for example, in age and income, and both can be related to exercise behavior as well as to heart health.

At the same time, exercise can affect heart health indirectly through better fitness. Further variables such as gym proximity, resting heart rate or hospital admission are also part of the assumed causal system. The DAG below shows these assumptions. Select the variables that fit the effect you are after.

Now it is your turn: Which variables do you have to adjust for in order to identify
A) the total effect or
B) the direct effect
of exercise on heart health?

Task and graph

Click variables = adjust. Adjusted variables are filled. You can also toggle them here:

ExerciseHeart healthadjusted

positive influencenegative influenceNumber and thickness: strength

Which causal assumptions does this example contain?

Each arrow is an assumption about how the world works. The number states by how much the target changes on average when the cause increases by one unit. Gym proximity, hospital admission (people who enter the sample via a hospital) and resting heart rate are variables in their own right.

Live evaluation

What does this mean for the estimate?

What the total effect is made of

Why is this solution formally valid? Derivation with the three rules

The live evaluation above shows intuitively which paths are open. Here you check the same selection formally with the three rules of do-calculus for the current task. At the end it says whether the effect you are after can be computed from observational data (identified) or not.