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Statistics Definition

Rubin causal model

Builds on Random variableExpected valueIndependence of random variables

Definition

Consider units i=1,…,ni = 1, \dots, n and a binary treatment. In the Rubin causal model the data are described as follows.

  1. Each unit ii has two potential outcomes Yi(1)Y_i(1) and Yi(0)Y_i(0): the outcome under treatment and under control.
  2. Each unit receives a treatment Wi∈{0,1}W_i \in \set{0, 1}, and the observed outcome is Yi=WiYi(1)+(1−Wi)Yi(0)Y_i = W_i Y_i(1) + (1 - W_i) Y_i(0).
  3. The potential outcomes of unit ii do not depend on the treatments of the other units. There is only one version of each treatment (the stable unit treatment value assumption, SUTVA).

The unit causal effect is Yi(1)−Yi(0)Y_i(1) - Y_i(0). When the units are drawn from a population, the average treatment effect is τ=E[Y(1)−Y(0)]\tau = E[Y(1) - Y(0)], an expected value over the population.

Remarks

Since only one of Yi(1)Y_i(1) and Yi(0)Y_i(0) is observed, unit effects are never observed; this is called the fundamental problem of causal inference. The assignment is unconfounded (ignorable) given covariates XX if WW is independent of (Y(0),Y(1))(Y(0), Y(1)) conditionally on XX.

Example: in a completely randomised experiment, randomization gives WW independent of (Y(0),Y(1))(Y(0), Y(1)). Then E[Y∣W=1]=E[Y(1)∣W=1]=E[Y(1)]E[Y \mid W = 1] = E[Y(1) \mid W = 1] = E[Y(1)], and likewise for W=0W = 0, so τ=E[Y∣W=1]−E[Y∣W=0]\tau = E[Y \mid W = 1] - E[Y \mid W = 0].

Non-example: in a vaccine trial in which unvaccinated people can be protected by their vaccinated neighbours, the outcome of a unit depends on the treatments of others. Clause 3 fails, and the two potential outcomes per unit are not enough to describe the situation.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Expected value, concept, no evidence levelExpected valueconceptIndependence of random variables, concept, no evidence levelIndependence of rando…conceptRandom variable, concept, no evidence levelRandom variableconceptRubin causal model, definition, no evidence levelRubin causal modeldefinition
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  1. Expected value concept
  2. Independence of random variables concept
  3. Random variable concept
  4. Rubin causal model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rubin_causal_model
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