Causal Catalog

Assumptions

Consistency

Definition	The observed outcome \(Y\) for a unit is the same as the potential outcome \(Y(T=t)\) for that unit under the treatment that was actually observed \(T=t\).
Mathematical definition	\(Y = Y(T)\) For binary treatment, that means: \(Y = Y(1) \text{ if } T=1\) \(Y = Y(0) \text{ if } T=0\)
Intuition/Examples	Expired aspirin (Lecture 2, slide 31) High or low energy dog (Neal reading, p14)

Exchangeability

Definition	The distribution of \(Y(0)\) and \(Y(1)\) for the \(T=1\) and \(T=0\) groups are the same. Also known as ignorability.
Mathematical definition	\(Y(0), Y(1) \perp T\)
Intuition/Examples	Caffeine and exam performance brainstorm

Conditional Exchangeability

Definition	The distribution of \(Y(0)\) and \(Y(1)\) for the \(T=1\) and \(T=0\) groups are the same, conditional on some variables \(X\). Also known as unconfoundedness. We determine which variables \(X\) to condition on based on the backdoor criterion.
Mathematical definition	\(Y(0), Y(1) \perp T \mid X\)
Intuition/Examples	Activity 7 on how to determine which variables satisfy the backdoor criterion

Positivity

Definition	Every unit has a non-zero probability of receiving the treatment, and every unit has a non-zero probability of not receiving the treatment. We can also view it as: every subgroup \(x\) in our sample has to have some units that received the treatment, and some units that did not receive the treatment. Also known as overlap.
Mathematical definition	For all values of covariates \(x \in X\):          \(0 < P(T = 1 \mid X = x) < 1\)
Intuition/Examples	See Lecture 8, beginning at slide 17 for intuition and PollEverywhere example, as well as Activity 8 for seeing bins of covariates in Yeager et al. (2019).

Study Designs

Randomized Experiments

Assumptions needed	Consistency No interference
Assumptions ensured	Exchangeability
Causal quantities identified	Average treatment effect (ATE)

Pros/cons

Advantages	Disadvantages
no biases, gets rid of confounding variables easier to ensure the validity of the causal claim ensures exchangeability	ethical concerns: do participants have informed consent? hard and expensive to execute smaller sample sizes due to difficulty in running results might not be representative of the whole population

Observational Studies

Assumptions needed	Consistency No interference Conditional exchangeability / unconfoundedness Positivity
Assumptions ensured	None 😞
Causal quantities identified	Average treatment effect (ATE) Average treatment effect on the treated (ATT) Average treatment effect on the untreated (ATU)

Pros/cons

Advantages	Disadvantages
more ethical than randomized experiments cheaper and easier to execute representative of the population since it models real-world scenarios can use pre-existing data	exchangeability is not guaranteed doesn't necessarily cover all of the groups of individuals that we want to measure the effect of the treatment on dealing with confounders raises the issue of positivity don't have full control over the data and treatment assignment