Exam Pattern: Potential Outcomes Table Analysis

This note abstracts the normal midterm causal-inference question built around a table of potential outcomes, treatment assignment, and a pre-treatment covariate.

Source Question

  • STA305 Midterm W2026, Question 6

Setup

A table gives, for each unit:

  • $Y_i(0)$
  • $Y_i(1)$
  • the observed treatment $T_i$
  • a pre-treatment covariate $x_i$

The question then asks for several causal-inference tasks in sequence.

Workflow

Step 1: Interpret the potential outcomes

  • $Y_i(0)$ is the outcome unit $i$ would have under control
  • $Y_i(1)$ is the outcome unit $i$ would have under treatment

Step 2: Compute an individual causal effect

For a named unit,

$$ \tau_i = Y_i(1) - Y_i(0). $$

Step 3: State SUTVA

SUTVA combines two ideas:

  • no interference across units
  • no hidden versions of treatment

Step 4: Compute the average treatment effect

The population average treatment effect is

$$ ATE = \frac{1}{N}\sum_{i=1}^{N}\left(Y_i(1) - Y_i(0)\right). $$

In the source problem, the six individual effects sum to $52$, so

$$ ATE = \frac{52}{6} \approx 8.667. $$

Step 5: State unconfoundedness given $x$

Unconfoundedness means

$$ T \perp (Y(0), Y(1)) \mid x. $$

Step 6: Check the table within strata of $x$

The question is not whether treated and untreated units have identical potential outcomes. The question is whether, after conditioning on $x$, treatment assignment still appears to depend on the potential outcomes.

Exam Checklist

  • Distinguish potential outcomes from observed outcomes
  • Use $Y_i(1) - Y_i(0)$ for an individual effect
  • Average over all listed units for the ATE
  • State unconfoundedness with conditional independence notation
  • Evaluate assignment separately within covariate strata