Instrumental Variable Analysis in Clinical Research

Introduction

Clinical research often asks a deceptively simple question: does receiving a particular treatment actually cause a better outcome?

In a randomized clinical trial, randomization creates the foundation for a causal comparison. In observational research, however, patients who receive different treatments may differ systematically before treatment begins.

For example, patients who receive an aggressive therapy may have more severe disease than patients who receive standard care. If mortality is higher among the treated patients, a naive comparison could incorrectly attribute the difference to the treatment rather than to the underlying disease severity.

Adjustment for measured confounders can help, but it cannot automatically solve unmeasured confounding. Instrumental variable analysis provides one strategy for addressing this problem when a credible external source of treatment variation is available.

The Basic Causal Problem

Let:

• \(A\) = treatment received
• \(Y\) = clinical outcome
• \(X\) = measured baseline covariates
• \(U\) = unmeasured factors that influence treatment and outcome

The causal estimand might be the average treatment effect:

where \(Y(1)\) is the potential outcome if a patient receives treatment and \(Y(0)\) is the potential outcome if the same patient does not receive treatment.

The fundamental problem is that we never observe both potential outcomes for the same patient.

If treatment assignment is confounded, the observed difference

does not necessarily equal the causal treatment effect.

Why Confounding Is a Problem in Clinical Research

Suppose physicians preferentially prescribe a new therapy to patients with particularly severe disease.

If the new-treatment group experiences worse outcomes, there are at least two possible explanations:

1. The treatment causes worse outcomes.
2. The treatment group began with higher underlying risk.

This is the classic problem of confounding by indication.

What Is an Instrumental Variable?

Let \(Z\) denote the proposed instrumental variable.

The instrument does not need to be a treatment itself. Instead, it provides variation in treatment assignment.

Conceptually:

The desired causal structure can be represented as:

while problematic pathways include:

or:

because those pathways would undermine the interpretation of the instrument.

The Four Core IV Assumptions

A credible instrumental variable analysis requires strong assumptions. The exact formalization depends on the causal model and estimand, but four concepts are especially important.

Assumption 1: Relevance

The instrument must actually change treatment assignment.

Formally, the instrument must provide information about \(A\):

In a simple binary setting, relevance can be expressed as:

If the instrument barely changes treatment probability, it contains little useful information for identifying the treatment effect.

Assumption 2: Independence

The instrument should not be associated with unmeasured factors that independently affect the outcome.

In a simplified formulation:

after accounting for any covariates required by the causal design.

This assumption is often plausible in randomized encouragement designs, where the instrument itself is randomized.

It is much harder to defend when the instrument is derived from observational features such as physician preference, geographic distance, calendar time, or hospital characteristics.

Assumption 3: Exclusion Restriction

The exclusion restriction states that the instrument affects the outcome only through treatment.

Conceptually:

and not:

For example, suppose hospital prescribing preference is used as an instrument. The exclusion restriction would require that the physician or hospital preference affect the patient's outcome only because it changes which treatment the patient receives.

That can be difficult to defend if hospitals differ in staffing, expertise, supportive care, referral patterns, or other factors that independently influence outcomes.

Assumption 4: Monotonicity

Monotonicity is particularly important when the IV estimate is interpreted as a local average treatment effect.

For a binary instrument and binary treatment, define potential treatment decisions:

• \(A(1)\) = treatment received if \(Z=1\)
• \(A(0)\) = treatment received if \(Z=0\)

Monotonicity requires:

for every individual under the conventional "no defiers" formulation.

This allows:

Monotonicity rules out the last group.

What Does IV Actually Estimate?

One of the most important points in instrumental variable analysis is that the IV estimate does not necessarily equal the treatment effect for every patient.

Under the standard assumptions, the common IV estimand for a binary instrument identifies the local average treatment effect, or LATE:

The conditioning set \(A(1)>A(0)\) identifies the patients whose treatment status is changed by the instrument: the compliers.

Why LATE Matters Clinically

Suppose the instrument is whether a patient was randomly encouraged to use a new therapy.

Some patients may take the therapy regardless of encouragement. Others may refuse regardless of encouragement. The patients whose treatment behavior actually changes because of the encouragement are the compliers.

The IV estimate primarily describes the treatment effect for that subgroup.

The Simple Wald Estimator

For a binary instrument and binary treatment, the basic IV estimator can be written as the ratio of two differences.

The numerator is the effect of the instrument on the outcome.

The denominator is the effect of the instrument on treatment receipt.

Therefore, the estimator asks: how much does the outcome change per unit of treatment induced by the instrument?

A Simple Numerical Example

Suppose an encouragement intervention changes treatment uptake from 40% to 70%.

The instrument effect on treatment is:

The instrument effect on the outcome is:

Therefore:

The IV estimate is therefore a 0.30 increase in the outcome probability per unit increase in treatment probability induced by the instrument.

Two-Stage Least Squares

When treatment and outcome are continuous or the model includes covariates, instrumental variable estimation is commonly implemented using two-stage least squares, or 2SLS.

The first stage predicts treatment from the instrument and covariates.

The fitted treatment value is:

The second stage relates the outcome to the predicted treatment:

The coefficient associated with the endogenous treatment is the IV treatment-effect estimate under the model and identification assumptions.

Why Ordinary Regression Can Fail

Suppose the true outcome model is:

where \(U\) affects both treatment selection and outcome.

If \(A\) is correlated with \(U\), then:

under the reduced-form representation relevant to the regression.

The ordinary least squares estimator can therefore be biased for the causal coefficient.

An appropriate instrument can isolate variation in \(A\) that is attributable to \(Z\), rather than to the problematic component of treatment selection.

A Clinical Example: Physician Preference

Consider an observational study comparing an invasive treatment with standard medical management.

Patients are not randomized. Physicians differ in their tendency to prescribe the invasive treatment.

Suppose \(Z\) represents physician treatment preference measured before the patient's treatment decision.

The first-stage relationship might be:

and the second-stage model:

The critical question is not merely whether physician preference predicts treatment. It is whether the entire causal interpretation required for the IV estimate is credible.

Why Physician Preference May Be a Problematic Instrument

Physician preference can affect much more than treatment choice.

For example, physicians with different treatment preferences may also differ in:

• clinical experience;
• patient selection;
• use of supportive care;
• referral behavior;
• monitoring intensity;
• hospital characteristics; and
• access to specialized resources.

If those factors influence the outcome, the exclusion restriction may fail.

A Randomized Encouragement Design

A particularly attractive IV setting is a randomized encouragement design.

Suppose patients are randomized to receive an intervention encouraging them to use a treatment, but patients are free to accept or refuse the treatment.

Randomized encouragement creates variation in treatment uptake without requiring perfect adherence.

This can be useful when an intention-to-treat analysis answers the effect of being offered or encouraged to receive treatment, while investigators also want to estimate the effect of actually receiving treatment.

IV Versus Intention-to-Treat Analysis

These estimands are not interchangeable.

The Reduced Form

The reduced-form effect is the association between the instrument and the outcome:

The first-stage effect is the association between the instrument and treatment:

The Wald IV estimator is therefore:

This decomposition is useful because it makes the logic of IV transparent.

First-Stage Strength

The first-stage relationship deserves careful attention.

In a linear IV model, investigators often examine the first-stage F statistic as a diagnostic for weak instruments.

The important principle is: an instrument that barely changes treatment cannot provide much information about the causal treatment effect.

Weak Instruments

Weak instruments are especially dangerous because the IV estimator is a ratio:

If the denominator is close to zero, small sampling fluctuations can produce large changes in the estimated treatment effect.

Consequences can include:

• very large standard errors;
• unstable point estimates;
• confidence intervals with poor coverage;
• finite-sample bias toward ordinary regression estimates; and
• misleading conventional hypothesis tests.

Exclusion Restriction Is Usually Unverifiable

One of the most important limitations of IV analysis is that the exclusion restriction generally cannot be tested directly from observed data.

You can show that:

but observing this relationship does not establish that:

through some pathway other than treatment.

Consequently, IV analysis requires substantive scientific reasoning in addition to statistical computation.

Negative Control and Balance Checks

Although no diagnostic proves the IV assumptions, investigators can examine patterns that would make an instrument less credible.

For example, when baseline variables should not be affected by the instrument, large systematic differences may raise concerns about independence.

Multiple Instruments

Clinical datasets may contain several proposed instruments.

With multiple instruments, the first stage can be written:

Multiple instruments can improve treatment prediction, but they introduce additional assumptions.

Each instrument must have a defensible causal interpretation. Adding more instruments is not automatically beneficial if some violate exclusion or independence.

Overidentification Tests

When there are more excluded instruments than endogenous treatment variables, overidentification tests can sometimes be used to assess whether the instruments are collectively consistent with certain model restrictions.

However, a non-significant overidentification test does not prove that every instrument is valid.

Continuous Treatments and Instruments

IV methods are not limited to binary treatment variables.

For a continuous treatment, the first stage can still be written:

and the second-stage structural model can be:

The interpretation depends on the treatment scale, model, instrument, and identifying assumptions.

Binary Outcomes Require Extra Care

When the outcome is binary, ordinary 2SLS produces a linear probability-type structural interpretation that may not be appropriate for every scientific question.

Nonlinear IV models can be considered, but the interpretation and assumptions become more complicated.

For clinical research, investigators should clearly specify whether they want:

• a risk difference;
• a treatment effect on another continuous scale;
• an odds ratio;
• a risk ratio; or
• another causal estimand.

The choice of model should follow the scientific estimand rather than being driven solely by the availability of a familiar regression procedure.

Worked Clinical Example: Treatment Adherence

Suppose a randomized clinical study encourages patients to begin a therapy. However, not everyone follows the recommendation.

Let:

• \(Z=1\): patient receives treatment encouragement;
• \(Z=0\): patient does not receive encouragement;
• \(A=1\): patient actually receives treatment;
• \(A=0\): patient does not receive treatment;
• \(Y=1\): clinical response.

Suppose the observed data are:

Treatment uptake is:

and:

The first-stage effect is therefore:

The response probabilities are:

and:

The reduced-form effect is:

The Wald IV estimate is therefore:

Under the IV assumptions, the estimated treatment effect is therefore a 30 percentage-point increase in response among the patients whose treatment receipt was changed by the randomized encouragement.

Why the IV Estimate Can Be Larger Than the ITT Effect

The randomized encouragement changes treatment receipt by only 30 percentage points, while the outcome changes by 9 percentage points.

The ITT effect is therefore:

The IV estimate divides the ITT effect by the treatment uptake difference:

This is sometimes called the treatment-on-the-treated effect among compliers under the standard IV assumptions.

R Implementation: Wald Estimator

For a simple binary instrument and binary treatment, the Wald estimator can be computed directly in R.

p_treat_z0 <- 200 / 500
p_treat_z1 <- 350 / 500

p_outcome_z0 <- 50 / 500
p_outcome_z1 <- 95 / 500

first_stage <-
  p_treat_z1 - p_treat_z0

reduced_form <-
  p_outcome_z1 - p_outcome_z0

iv_estimate <-
  reduced_form / first_stage

first_stage
reduced_form
iv_estimate

The calculation returns:

first_stage
# 0.30

reduced_form
# 0.09

iv_estimate
# 0.30

R Implementation: Two-Stage Least Squares

For a continuous outcome and continuous or binary treatment, the ivreg framework is commonly used for 2SLS.

library(ivreg)

fit_iv <- ivreg(
  outcome ~ treatment + age + baseline_risk |
    instrument + age + baseline_risk,
  data = dat
)

summary(fit_iv)

The expression to the left of the vertical bar specifies the outcome model.

The expression to the right specifies the instruments and exogenous covariates.

Inspecting the First Stage in R

The first-stage relationship should be examined explicitly.

first_stage <- lm(
  treatment ~ instrument + age + baseline_risk,
  data = dat
)

summary(first_stage)

Questions to examine include:

• Does the instrument materially predict treatment?
• How large is the first-stage effect?
• Is the instrument weak?
• Are there unexpected baseline patterns?
• Does the first-stage relationship make clinical sense?

Model Diagnostics in R

fit_iv <- ivreg(
  outcome ~ treatment + age + baseline_risk |
    instrument + age + baseline_risk,
  data = dat
)

summary(
  fit_iv,
  diagnostics = TRUE
)

The diagnostics can help identify issues involving the first stage, weak instruments, and overidentification when multiple instruments are present.

These diagnostics should be treated as evidence about the model rather than as automatic proof that the IV assumptions hold.

Covariate Adjustment in IV Analysis

Covariates can be included when they improve precision or are required by the causal design.

For example:

fit_iv <- ivreg(
  outcome ~ treatment + age + sex + baseline_score |
    instrument + age + sex + baseline_score,
  data = dat
)

The same exogenous covariates generally appear in both the structural equation and the instrument set.

Importantly, adding covariates does not eliminate the need to defend the instrument's exclusion and independence assumptions.

Instrument Selection in Clinical Research

Potential instruments encountered in clinical research include:

Common Mistakes

1. Using an instrument simply because it predicts treatment. Relevance is necessary but not sufficient. An instrument can strongly predict treatment while violating exclusion.
2. Assuming IV automatically removes all confounding. IV removes bias only under the assumptions supporting identification.
3. Ignoring weak instruments. A weak first stage can make IV estimates highly unstable.
4. Interpreting LATE as the universal treatment effect. The IV estimate is generally local to the patients whose treatment decisions are affected by the instrument.
5. Assuming exclusion can be tested directly. The exclusion restriction is generally a substantive causal assumption.
6. Using ordinary standard errors after manually performing two stages. Proper IV variance estimation is required.
7. Adding every available variable as an instrument. More instruments can increase precision, but invalid instruments can introduce bias.
8. Ignoring the clinical pathway from instrument to outcome. A statistically convenient instrument may be clinically implausible.
9. Reporting only the IV coefficient. A complete analysis should describe the instrument, first stage, assumptions, estimand, diagnostics, and sensitivity analyses.

IV Analysis Versus Propensity Score Methods

These approaches are not necessarily competitors.

Covariate adjustment can be incorporated into an IV analysis, and researchers may use multiple complementary approaches to evaluate whether conclusions are robust.

Sensitivity Analysis

Because some IV assumptions cannot be verified directly, sensitivity analysis is particularly important.

Possible analyses include:

• using alternative plausible instruments;
• testing different covariate specifications;
• examining first-stage strength;
• checking baseline balance;
• investigating possible direct pathways from instrument to outcome;
• performing weak-IV robust inference;
• examining results under alternative definitions of treatment; and
• comparing IV estimates with randomized or observational estimates when available.

What to Report in a Clinical IV Analysis

A clear statistical analysis plan or manuscript should describe:

• the causal question;
• the treatment definition;
• the outcome definition;
• the proposed instrument;
• the scientific rationale for the instrument;
• the relevance assumption;
• the independence assumption;
• the exclusion restriction;
• the monotonicity assumption when LATE is invoked;
• the target estimand;
• the first-stage model;
• the second-stage model;
• the covariates;
• the weak-instrument diagnostics;
• the variance estimation method;
• the treatment of missing data;
• sensitivity analyses; and
• the limitations of the IV assumptions.

A Practical IV Analysis Workflow

How to Interpret an IV Estimate

Suppose an analysis reports:

A poor interpretation would be:

A more appropriate interpretation is:

The exact wording should be adapted to the treatment scale, outcome, model, and identified estimand.

When IV Analysis Is Especially Useful

Instrumental variable methods may be particularly valuable when:

• treatment is strongly confounded;
• important confounders are unmeasured;
• a credible source of treatment variation exists;
• randomized assignment affects adherence but not treatment receipt perfectly;
• treatment switching or nonadherence complicates a randomized comparison;
• policy or implementation changes generate plausibly exogenous treatment variation; or
• the research question specifically concerns the effect of treatment induced by a particular source of variation.

When IV May Not Be Appropriate

IV analysis should not be used merely because an observational dataset contains a variable that predicts treatment.

It may be inappropriate when:

• the proposed instrument has a weak first-stage association;
• the instrument plausibly affects the outcome directly;
• the instrument is associated with unmeasured outcome determinants;
• the treatment effect being estimated is not clinically meaningful for the instrument-defined subgroup;
• the assumptions are substantially less credible than alternative study designs; or
• the instrument was selected after examining outcome associations without a credible causal rationale.

IV Analysis in the Context of Randomized Trials

Instrumental variables can also arise naturally within randomized clinical trials.

For example, randomized treatment assignment \(Z\) can serve as an instrument for actual treatment received \(A\) when adherence is imperfect.

The randomized assignment is independent of baseline prognostic factors by design, while actual treatment receipt may not be.

This provides a particularly transparent IV structure:

The resulting IV estimate can address a different question from the intention-to- treat estimate.

Important Distinction: Nonadherence Is Not Automatically a Reason to Use IV

Nonadherence alone does not determine the correct analysis.

The scientific question should determine the estimand.

If the trial asks whether assigning patients to a treatment strategy improves outcomes, ITT is often the primary estimand.

If investigators specifically want the causal effect of receiving treatment among patients whose treatment receipt is influenced by randomized assignment, an IV analysis may be informative.

The Most Important Concept

Instrumental variable analysis is not simply a statistical trick for making confounding disappear.

It is a causal identification strategy that exchanges one difficult problem (unmeasured confounding) for a different set of strong assumptions about an external source of treatment variation.

A valid IV analysis therefore requires both:

• statistical strength — the instrument must meaningfully predict treatment; and
• scientific credibility — the causal assumptions must be defensible.

References

Angrist, J.D., Imbens, G.W., & Rubin, D.B. (1996). Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91, 444–455.

Angrist, J.D., & Pischke, J.-S. (2009). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press.

Imbens, G.W., & Angrist, J.D. (1994). Identification and estimation of local average treatment effects. Econometrica, 62, 467–475.

Stock, J.H., & Yogo, M. (2005). Testing for weak instruments in linear IV regression. In Identification and Inference for Econometric Models. Cambridge University Press.

Hernán, M.A., & Robins, J.M. (2020). Causal Inference: What If. Chapman & Hall/CRC.

Wooldridge, J.M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press.