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Causal inference course note - Week 5
This is my note for the “A Crash Course in Causality: Inferring Causal Effects from Observational Data” course by Jason A. Roy on Coursera.
- Week 1. Welcome and Introduction to Causal Effects
- Week 2. Confounding and Directed Acyclic Graphs (DAGs)
- Week 3. Matching and Propensity Scores
- Week 4. Inverse Probability of Treatment Weighting (IPTW)
- Week 5. Instrumental Variables Methods
Table of Contents
- Week 5. Instrumental Variables Methods
Week 5. Instrumental Variables Methods
Introduction to instrumental variables
Confounding
Classic confounding situation: $X$ affects treatment $A$ and affects outcome Y: $X \rightarrow A \rightarrow Y \leftarrow X$. If $X$ is observed, we can analyze data using: matching, PS matching, IPTW. Even if there are risk factors, $V$, such that $V \rightarrow Y$, it’s valid to simply control for $X$.
Unmeasured confounding
Suppose there are unmeasured/unobserved variables, $U$, that affect $A$ and $Y$: $A \dashleftarrow U \dashrightarrow Y$. Then we have unmeasured confounding.
- Ignorability assumption violated.
- Biased estimates of causal effects.
- Cannot control for confounders and average over the distribution, if we do not observe them all.
Instrumental variables
Instrumental variables (IV) is an alternative causal inference method that does not rely on the ignorability assumption. Here $Z$ is an IV:
\[Z \rightarrow A \rightarrow Y \leftarrow X \rightarrow A\]It affects treatment, but does not (directly) affect the outcome. Think of $Z$ as encouragement.
Example
- $A$: somking during pregnancy (yes/no)
- $Y$: birthweight
- $X$: parity, mother’s age, weight, etc.
Concern: Could be unmeasured confounders.
Challenge: not ethical to randomly assign smoking to pregnant women.
Encouragement design
- $A$: somking during pregnancy (yes/no)
- $Y$: birthweight
- $X$: parity, mother’s age, weight, etc.
- $Z$: randomize to either receive encouragement to stop smoking ($Z=1$) or receive usual care ($Z=0$).
An intention-to-treat analysis would focus on the causal effect of encouragement:
\[E(Y^{Z=1}) - E(Y^{Z=0})\]This is a valid causal effect and would likely be of some interest.
What can we say about the causal effect of smoking itself? This is the focus of IV methods.
Instrumental variables
Sometimes the IV is randomly assigned as part of the study. Other times, the IV is believed to be randomized in nature (natural experiment):
- Mendelian randomization.
- Quarter of birth.
- Geographic distance to specialty care provider.
We will look at both situations in other lectures.
Randomized trials with noncompliance
Setup
Randomize trial:
- $Z$: randomization to treatment (1 if randomized to treatment, 0 otherwise).
- $A$: treatment received (1 if receive treatment, 0 otherwise).
- $Y$: outcome.
- Note: Typically, not everyone assigned to treatment will actually receive the treatment (non-compliance).
DAG
- Non-compliance makes a randomized trial like an observational study.
- There could be confounding based on treatment received.
- Common causes of treatment received and the outcome.
- It might be reasonable to assume that treatment assignment does not directly affect Y.
Potential treatment
- Observed data: $(Z, A, Y)$.
- For a given subject, they were assigned treatment $Z$ and received treatment $A$.
- Their treatment received might have been different had they been assigned treatment $1 - Z$.
- Each subject as two potential values of treatment:
- $A^{Z=1} = A^1$: Value of treatment if randomized to $Z=1$.
- $A^{Z=0} = A^0$: Value of treatment if randomized to $Z=0$.
Causal effect of assignment on receipt
We can think of the average causal effect of treatment assignment on treatment received as:
\[E(A^1 - A^0)\]- This is proportion treated if everyone had been assigned to received treatment, minus the proportion treated if no one had been assigned to receive the treatment.
- If perfect compliance, this would be equal to 1.
- This is generally estimable from the observed data, as, by randomization and consistency:
- $E(A^1) = E(A \vert Z = 1)$
- $E(A^0) = E(A \vert Z = 0)$
Causal effect on assignment on outcome
We can think of the average causal effect of treatment assignment on the outcome as:
\[E(Y^{Z=1} - Y^{Z=0})\]- This is average values of the outcome if everyone had been assigned to received treatment, minus the average outcome if no one had been assigned to receive the treatment.
- Intention-to-treat effect.
- If perfect compliance, this would be equal to the causal effect of treatment.
- This is generally estimable from the observed data, as, by randomization and consistency:
- $E(Y^{Z=1}) = E(Y \vert Z=1)$
- $E(Y^{Z=0}) = E(Y \vert Z=0)$
Causal effect of treatment
What about the causal effect of treatment received on the outcome?
- $Z$ can be thought of as (strong) encouragement to receive the treatment.
- It’s an IV. Does this help us estimate causal effects of treatment?
- This will be explored in another video.
- It’s an IV. Does this help us estimate causal effects of treatment?
Compliance classes
Potential values of treatment
We can classify people based on potential treatment.
| $A^0$ | $A^1$ | Label |
|---|---|---|
| 0 | 0 | Never-takers |
| 0 | 1 | Compliers |
| 1 | 0 | Defiers |
| 1 | 1 | Always-takers |
Angrist, Imbens, & Rubin, JASA, 1996.
Subpopulations
We can think of these as subpopulations of people:
- Never-takers: Do not take treatment, regardless of randomization (encouragement does not work). We would not learn anything about the effect of treatment in this subpopulation, as there is no variation in treatment received.
- Compliers: Take treatment when encouraged to, and do not otherwise. In this group, treatment received is randomized.
- Defiers: Do the opposite of what they are encouraged to do. In this group treatment received is also randomized, but in the opposite way.
- Always-takers: Always take treatment. In this group there is no variation in treatment received. No information about causal effect.
Causal effects
A motivation for using IV methods in general is concern about possible unmeasured confounding.
- If there is unmeasured confounding, cannot marginalize over all confounders (via matching, IPTW, etc).
IV methods do not focus on the average causal effect for the population.
- They focus on a local average treatment effect.
Local average treatment effect
In IV methodcs, the target of inference is:
\[E(Y^{Z=1} \vert A^0 = 0, A^1 = 1) - E(Y^{Z=0} \vert A^0 = 0, A^1 = 1)\] \[= E(Y^{Z=1} - Y^{Z=0} \vert \text{compliers})\] \[= E(Y^{A=1} - Y^{A=0} \vert \text{compliers})\]This is causal because it contrasts counterfactuals in a common population. Known as complier average causal effect (CACE).
- This is a causal effect in a subpopulation.
- A “local” causal effect.
- No inference about defiers, always-takers, or never-takers.
Observed data
For each person we observe an $A$ and a $Z$, not ($A^0$, $A^1$).
| $Z$ | $A$ | $A^0$ | $A^1$ | Class |
|---|---|---|---|---|
| 0 | 0 | 0 | ? | Never-takers or compliers |
| 0 | 1 | 1 | ? | Always-takers or defiers |
| 1 | 0 | ? | 0 | Never-takers or defiers |
| 1 | 1 | ? | 1 | Always-takers or compliers |
Without additional assumptions, we cannot classify each subject into one of these categories. We can narrow it down to two options, however.
Identifiability
Compliance classes above are also known as principal strata. These are latent (not directly observable).
How can we estimate the complier average causal effect? What assumptions are needed? These are topics for other lectures.
Assumptions
Assumptions about IVs
A variable is an instrumental variable (IV) if:
- It is associated with the treatment.
- It affects the outcome only through its effect on treatment (known as the exclusion restriction).
If $Z$ is an IV, $Z$ must not directly affect $Y$ (exclusion restriction).
When there is unmeasured confounding,
\[Z \rightarrow A \rightarrow Y\] \[A \leftarrow X \rightarrow Y\] \[A \dashleftarrow U \dashrightarrow Y\]$Z$ must not affect unmeasured confounders $U$. $Z$ cannot directly, or indirectly through its effect on $U$, affect the outcome $Y$.
Realistic?
IF $Z$ is random treatment assignment, are the IV assumptions met?
- It should affect treatment received. (We can check this.)
- If we think of it as a coin flip, then it should not affect the outcome or unmeasured confounders.
- However, if subjects are not blinded, knowledge of what they were assigned to could affect them.
- If clinicians are not blinded to assignment, it could affect them.
- Need to examine this assumption carefully for any given study.
Causal effects
Assuming we have a valid IV, we are interested in using it to help us estimate the complier average causal effect:
\[E(Y^{A=1} - Y^{A=0} \vert \text{compliers})\]The causal effect of treatment among subjects who only take treatment if they are randomized to $Z=1$.
Identification challenge
Challenge: We are interested in compliers, but we do not know who the compliers are.
Monotonicity assumption
The monotonicity assumption is there are no defiers.
- No one consistently does the opposite of what they are told.
- It is called monotonicity because the assumption is that the probability of treatment should increase with more encouragement.
With monotonicity,
| $Z$ | $A$ | $A^0$ | $A^1$ | Class |
|---|---|---|---|---|
| 0 | 0 | 0 | ? | Never-takers or compliers |
| 0 | 1 | 1 | Always-takers |
|
| 1 | 0 | 0 | Never-takers |
|
| 1 | 1 | ? | 1 | Always-takers or compliers |
Causal effect identification and estimation
[TODO]