Rubin causal model

Key takeaways

Rubin causal model belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rubin causal model to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rubin causal model from memory before moving on to harder problems.

Reference excerpt

The Rubin causal model (RCM), also known as the Neyman–Rubin causal model, is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was coined by Paul W. Holland. The potential outcomes framework was first proposed by Jerzy Neyman in his 1923 Master's thesis, though he discussed it only in the context of completely randomized experiments. Rubin extended it into a general framework for thinking about causation in both observational and experimental studies.

Introduction The Rubin causal model is based on the idea of potential outcomes. For example, a person would have a particular income at age 40 if they had attended college, whereas they would have a different income at age 40 if they had not attended college. To measure the causal effect of going to college for this person, we need to compare the outcome for the same individual in both alternative futures. Since it is impossible to see both potential outcomes at once, one of the potential outcomes is always missing. This dilemma is the "fundamental problem of causal inference." Because of the fundamental problem of causal inference, unit-level causal effects cannot be directly observed. However, randomized experiments allow for the estimation of population-level causal effects. A randomized experiment assigns people randomly to treatments: college or no college. Because of this random assignment, the groups are (on average) equivalent, and the difference in income at age 40 can be attributed to the college assignment since that was the only difference between the groups. An estimate of the average causal effect (also referred to as the average treatment effect or ATE) can then be obtained by computing the difference in means between the treated (college-attending) and control (not-college-attending) samples. In many circumstances, however, randomized experiments are not possible due to ethical or practical concerns. In such scenarios there is a non-random assignment mechanism. This is the case for the example of college attendance: people are not randomly assigned to attend college. Rather, people may choose to attend college based on their financial situation, parents' education, and so on. Many statistical methods have been developed for causal inference, such as propensity score matching. These methods attempt to correct for the assignment mechanism by finding control units similar to treatment units.

An extended example Rubin defines a causal effect:

Intuitively, the causal effect of one treatment, E, over another, C, for a particular unit and an interval of time from t 1 {\displaystyle t_{1}} to t 2 {\displaystyle t_{2}} is the difference between what would have happened at time t 2 {\displaystyle t_{2}} if the unit had been exposed to E initiated at t 1 {\displaystyle t_{1}} and what would have happened at t 2 {\displaystyle t_{2}} if the unit had been exposed to C initiated at t 1 {\displaystyle t_{1}} : 'If an hour ago I had taken two aspirins instead of just a glass of water, my headache would now be gone,' or 'because an hour ago I took two aspirins instead of just a glass of water, my headache is now gone.' Our definition of the causal effect of the E versus C treatment will reflect this intuitive meaning."

According to the RCM, the causal effect of your taking or not taking aspirin one hour ago is the difference between how your head would have felt in case 1 (taking the aspirin) and case 2 (not taking the aspirin). If your headache would remain without aspirin but disappear if you took aspirin, then the causal effect of taking aspirin is headache relief. In most circumstances, we are interested in comparing two futures, one generally termed "treatment" and the other "control". These labels are somewhat arbitrary.

Potential outcomes Suppose that Joe is participating in an FDA test for a new hypertension drug. An all-knowing observer would know the outcomes for Joe under both treatment (the new drug) and control (either no treatment or the current standard treatment). The causal effect, or treatment effect, is the difference between these two potential outcomes.

Y t ( u ) {\displaystyle Y_{t}(u)} is Joe's blood pressure if he takes the new pill. In general, this notation expresses the potential outcome which results from a treatment, t, on a unit, u. Similarly, Y c ( u ) {\displaystyle Y_{c}(u)} is the effect of a different treatment, c or control, on a unit, u. In this case, Y c ( u ) {\displaystyle Y_{c}(u)} is Joe's blood pressure if he doesn't take the pill. Y t ( u ) − Y c ( u ) {\displaystyle Y_{t}(u)-Y_{c}(u)} is the causal effect of taking the new drug. From this table we only know the causal effect on Joe. Everyone else in the study might have an increase in blood pressure if they take the pill. However, regardless of what the causal effect is for the other subjects, the causal effect for Joe is lower blood pressure, relative to what his blood pressure would have been if he had not taken the pill. Consider a larger sample of patients:

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Worked examples

Example 1 — a first encounter with Rubin causal model

Start with the simplest possible case. Write down what Rubin causal model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rubin causal model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rubin causal model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rubin causal model

In research: Rubin causal model appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rubin causal model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rubin causal model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Causal inference, Econometric models, Experiments, so understanding it makes those chapters shorter.
In everyday life: Look for Rubin causal model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Rubin causal model in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rubin causal model means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rubin causal model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rubin causal model in simple terms?

The Rubin causal model (RCM), also known as the Neyman–Rubin causal model, is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was coined by Paul W.

Why does Rubin causal model matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rubin causal model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rubin causal model.