# Introduction to Instrumental Variables

### Unmeasured confounding

• Suppose there are unobserved variables $$U$$ that affect both $$A$$ and $$Y$$, then $$U$$ is an unmeasured confounding

• This violates ignorability assumption

• Since we cannot control for the unobserved confounders $$U$$ and average over its distribution, if using matching or IPTW methods, the estimates of causal effects is biased

• Solution: instrumental variables

### Instrumental variables

• Instrumental variables (IV): an alternative causal inference method that does not rely on the ignorability assumption

• $$Z$$ is an IV

• It affects treatment $$A$$, but does not directly affect the outcome $$Y$$
• We can think of $$Z$$ as encouragement (of treatement)

### Example of an encouragement design

• $$A$$: smoking during pregnancy (yes/no)
• $$Y$$: birth weight
• $$X$$: mother’s age, weight, etc

• Concern: there could be unmeasured confounders
• Challenge: it is not ethical to randomly assign smoking
• $$Z$$: randomized to either received encouragement to stop smoking ($$Z=1$$) or receive usual care ($$Z=0$$)

• Causal effect of encouragement, also called intent-to-treat (ITT) effect, may be of some interest $E\left(Y^{Z=1}\right)-E\left(Y^{Z=0}\right)$
• Focus of IV methods is still causal effect of the treatment $E\left(Y^{A=1}\right)-E\left(Y^{A=0}\right)$

### IV is randomized

• Like the previous smoking example, sometimes IV is randomly assigned as part of the study

• Other times IV is believed to be randomized in nature (natural experiment). For example,

• Mendelian randomization (?)
• Quarter of birth
• Geographic distance to specialty care provider

## Randomized trials with noncompliance

### Randomized trials with noncompliance

• Setup
• $$Z$$: randomization to treatment (1 treatment, 0 control)
• $$A$$: treatment received, binary (1 treatment, 0 control)
• $$Y$$: outcome
• Due to noncompliance, not everyone assigned treatment will actually receive the treatment, and vice verse ($$A \neq Z$$)
• There can be confounding $$X$$, like common causes affecting both treatment received $$A$$ and the outcome $$Y$$
• It may be reasonable to assume that $$Z$$ does not directly affect $$Y$$

### Causal effect of assignment on receipt

• Observed data: $$(Z, A, Y)$$

• Each subject has two potential values of treatment

• $$A^{Z=1} = A^1$$: value of treatment if randomized to treatment
• $$A^{Z=0} = A^0$$: value of treatment if randomized to control
• Average causal effect of treatment assignment on treatment received $E\left(A^1 - A^0\right)$
• If perfect compliance, this would be $$1$$
• By randomization and consistency, this is estimable from the observed data $E\left(A^1\right) = E(A \mid Z=1), \quad E\left(A^0\right) = E(A \mid Z=0)$

### Causal effect of assignment on outcome

• Average causal effect of treatment assignment on the outcome $E\left(Y^{Z=1} - Y^{Z=0}\right)$

• This is intention-to-treat effect
• If perfect compliance, this would be equal to the causal effect of treatment received
• By randomization and consistency, this is estimable from the observed data $E\left(Y^{Z=1}\right) = E(Y \mid Z=1), \quad E\left(Y^{Z=0}\right) = E(Y \mid Z=0)$

## Compliance classes

### Subpopulations based on potential treatment

$$A^0$$ $$A^1$$ Label
0 0 Never-takers
0 1 Compliers
1 0 Defiers
0 0 Always-takers
• For never-takers and always-takers,
• Encouragement does not work
• Due to no variation in treatment received, we cannot learn anything about the effect of treatment in these two subpopulations
• For compliers, treatment received is randomized
• For defiers, treatment received is also randomized, but in the opposite way

### Local average treatment effect

• We will focus on a local average treatment effect, i.e., the complier average causal effect (CACE)

\begin{align*} & E\left(Y^{Z=1} \mid A^0=0, A^1=1 \right) - E\left(Y^{Z=0} \mid A^0=0, A^1=1 \right)\\ = & E\left(Y^{Z=1} - Y^{Z=0} \mid \text{compliers} \right)\\ = & E\left(Y^{a=1} - Y^{a=0} \mid \text{compliers} \right) \end{align*}

• “Local”: this is a causal effect in a subpopulation
• No inference about defiers, always-takers, or never-takers

## Instrumental variable assumptions

### IV assumption 1: exclusion restriction

1. $$Z$$ is associated with the treatment $$A$$

1. $$Z$$ affects the outcome only through its effect on treatment

• $$Z$$ cannot directly, or indirectly though its effect on $$U$$, affect $$Y$$

### Is the exclusion restriction assumption realistic?

• If $$Z$$ is a random treatment assignment, then the exclusion restriction assumption is met

• It should affect treatment received
• It should not affect the outcome or unmeasured confounders
• However, it the subjects or clinicians are not blinded, knowledge of what they are assigned to could affect $$Y$$ or $$U$$

• We need to examine the exclusion restriction assumption carefully for any given study

### IV assumption 2: monotonicity

• Monotonicity assumption: there are no defiers

• No one consistently does the opposite of what they are told
• 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 1 Always-takers or defiers
1 0 0 0 Never-takers or defiers
1 1 ? 1 Always-takers or compliers

# Estimate Causal Effects with Instrumental Variables

### Estimate CACE: 1. rewrite the ITT effect

• Due to randomization, we can identify the ITT effect $E\left( Y^{z=1} - Y^{z=0} \right) = E(Y\mid Z=1) - E(Y\mid Z=0)$

• Expand the first term in the above ITT effect \begin{align*} E(Y\mid Z=1) = & E(Y\mid Z=1, \text{always takers})P(\text{always takers}\mid Z=1)\\ & + E(Y\mid Z=1, \text{never takers})P(\text{never takers}\mid Z=1)\\ & + E(Y\mid Z=1, \text{compliers})P(\text{compliers}\mid Z=1) \end{align*}

• Note 1: among always takers and never takes, $$Z$$ does nothing
• $$E(Y\mid Z=1, \text{always takers}) = E(Y\mid \text{always takers}), \quad \text{etc.}$$
• Note 2: by randomization,
• $$P(\text{always takers}\mid Z=1) = P(\text{always takers}), \quad \text{etc.}$$

### Estimate CACE: 1. rewrite the ITT effect, cont.

• Therefore, the first term in the ITT effect is \begin{align*} E(Y\mid Z=1)=& E(Y\mid\text{always takers})P(\text{always takers})\\ & + E(Y\mid \text{never takers})P(\text{never takers})\\ & + E(Y\mid Z=1, \text{compliers})P(\text{compliers}) \end{align*}

• Similarly, the second term is \begin{align*} E(Y\mid Z=0)=& E(Y\mid\text{always takers})P(\text{always takers})\\ & + E(Y\mid \text{never takers})P(\text{never takers})\\ & + E(Y\mid Z=0, \text{compliers})P(\text{compliers}) \end{align*}

• Their difference is \begin{align*} & E(Y\mid Z=1) - E(Y\mid Z=0)\\ = & \left[E(Y\mid Z=1, \text{compliers})- E(Y\mid Z=0, \text{compliers})\right]P(\text{compliers}) \end{align*}

### Estimate CACE: 2. compute proportion of compliers

• Thus, the relationship between CACE and ITT effect is $\text{CACE} = \frac{E(Y\mid Z=1) - E(Y\mid Z=0)}{P(\text{compliers})}$

• To compute $$P(\text{compliers})$$, note that

• $$E(A\mid Z=1)$$: proportion of always takers plus compliers
• $$E(A\mid Z=0)$$: proportion of always takers
• Thus the difference is $P(\text{compliers}) = E(A\mid Z=1) - E(A\mid Z=0)$

### Estimate CACE: final formula

$\text{CACE} = \frac{E(Y\mid Z=1) - E(Y\mid Z=0)} {E(A\mid Z=1) - E(A\mid Z=0)}$

• Numerator: ITT, causal effect of treatment assignment on the outcome

• Denominator: causal effect of treatment assignment on the treatment received
• Denominator is between 0 and 1. Thus, CACE $$\geq$$ ITT
• ITT is underestimate of CACE, because some people assigned to treatment did not take it
• If perfect compliance, CACE $$=$$ ITT

## IVs in observational studies

### IVs in observational studies

• IVs can also be used in observational (non-randomized) studies

• $$Z$$: instrument
• $$A$$: treatment
• $$Y$$: outcome
• $$X$$: covariates
• $$Z$$ can be thought of as encouragement
• If binary, just encouragement yes or no
• If continuous, a ‘dose’ of encouragement
• $$Z$$ can be thought of as randomizers in natural experiments

• The key challenge: think of a variable that affects $$Y$$ only through $$A$$
• Only the assumption $$Z$$ affecting $$A$$ can be checked with data
• The validity of the exclusion restriction assumption rely on subject matter knowledge

### Natural experiment example 1: calendar time as IV

• Rationale: sometimes treatment preferences change over a short period of time

• $$A$$: drug A vs drug B

• $$Z$$: early time period (drug A is encouraged) vs late time period (drug B is encouraged)

• $$Y$$: BMI

### Natural experiment example 2: distance as IV

• Rationale: shorter distance to NICU is an encouragement

• $$A$$: delivery at high level NICU vs regular hospital

• $$Z$$: differential travel time from nearest high level NICU to nearest regular hospital

• $$Y$$: mortality

### More examples of natural experiments

• Mendelian randomization: some genetic variant is associate with some behavior (e.g., alcohol use) but is assumed to not be associated with outcome of interest

• Provider preference: use treatment prescribed to previous patients as an IV for current patient

• Quarter of birth: to study causal effect of years in school on income

## Two stage least squares

### Ordinary least squares (OLS) fails if there is confounding

• In OLS, one important assumption is that the covariate $$A$$ is independent with residuals $$\epsilon$$

$Y_i = \beta_0 + A_i \beta_1 + \epsilon_i$

• However, if there is confounding, $$A$$ and $$\epsilon$$ are correlated. So OLS fails.

• Two stage least squares can estimate causal effect in the instrumental variables (IV) setting

### Two stage least squares (2SLS)

• Stage 1: regress $$A$$ on $$Z$$ $A_i = \alpha_0 + Z_i \alpha_1 + e_i$
• By randomization, $$Z$$ and $$e$$ are independent
• Obtain the predicted value of $$A$$ given $$Z$$ for each subject $\hat{A}_i = \hat{\alpha}_0 + Z_i \hat{\alpha}_1$
• $$\hat{A}$$ is projection of $$A$$ onto the space spanned by $$Z$$
• Stage 2: regress $$Y$$ on $$\hat{A}$$ $Y_i = \beta_0 + \hat{A}_i \beta_1 + \epsilon_i$
• By exclusion restriction, $$Z$$ is independent of $$Y$$ given $$A$$

### Interpretation of $$\beta_1$$ in 2SLS: the causal effect

• Consider the case where both $$Z$$ and $$A$$ are binary $\beta_1 = E\left(Y \mid \hat{A}=1 \right) - E\left(Y \mid \hat{A}=0 \right)$

• There are two values of $$\hat{A}$$ in the 2nd stage model, $$\hat{\alpha}_0$$ and $$\hat{\alpha}_0 + \hat{\alpha}_1$$

• When we go from $$Z=0$$ to $$Z=1$$, what we observe is going from $$\hat{\alpha}_0$$ to $$\hat{\alpha}_0 + \hat{\alpha}_1$$
• We observe a mean difference of $$\hat{E}(Y\mid Z=1) - \hat{E}(Y\mid Z=0)$$ with a $$\hat{\alpha}_1$$ unit change in $$\hat{A}$$
• Thus, we should observe a mean difference of $$\frac{\hat{E}(Y\mid Z=1) - \hat{E}(Y\mid Z=0)}{\hat{\alpha}_1}$$ with $$1$$ unit change in $$\hat{A}$$

• The 2SLS estimator is a consistent estimator of the CACE $\beta_1 = \text{CACE} = \frac{\hat{E}(Y\mid Z=1) - \hat{E}(Y\mid Z=0)}{\hat{E}(A\mid Z=1) - \hat{E}(A\mid Z=0)}$

### More general 2SLS

• 2SLS can be used

• with covariates $$X$$, and
• for non-binary data (e.g, a continuous instrument)
• Stage 1: regression $$A$$ on $$Z$$ and covariates $$X$$

• and obtain the fitted values $$\hat{A}$$
• Stage 2: regress $$Y$$ on $$\hat{A}$$ and $$X$$

• Coefficient of $$\hat{A}$$ is the causal effect

## Sensitivity analysis and weak instruments

### Sensitivity analysis

• Sensitivity analysis method studies when each of the IV assumption (partly) fails

• Exclusion restriction: if $$Z$$ does affect $$Y$$ by an amount $$p$$, would my conclusion change? Vary $$p$$
• Monotonically: if the proportion of defiers was $$\pi$$, would my conclusion change?

### Strength of IVs

• Depend on how well an IV predicts treatment received, we can class it as a strong instrument or a weak instrument

• For a weak instrument, encouragement barely increases the probability of treatment

• Measure the strength of an instrument: estimate the proportion of compliers $E(A \mid Z=1) - E(A \mid Z=0)$

• Alternatively, we can just use the observed proportions of treated subjects for $$Z=1$$ and for $$Z=0$$

### Problems of weak instruments

• Suppose only 1% of the population are compliers

• Then only 1% of the samples have useful information about the treatment effect

• This leads to large variance estimates, i.e., estimate of causal effect is unstable
• The confidence intervals can be too wide to be useful

### References

• Coursera class: “A Crash Course on Causality: Inferring Causal Effects from Observational Data”, by Jason A. Roy (University of Pennsylvania)