Instrumental Variable

Instrumental Variable: Concept Architecture

Orientation and learning roadmap

This page moves from the confounding problem to the assumptions that make a proposed instrument credible, then uses an encouragement example to show what its variation can identify. It closes with diagnostic limits, health economic uses, and distinctions that prevent an instrument from being mistaken for an estimation method or a general cure for bias.

Why an instrument is proposed

In routine care, people receiving a treatment can differ from untreated people in prognosis, disease severity, or access to care, even after measured covariates are adjusted for. A proposed instrument supplies a source of variation in treatment uptake that may be less entangled with these determinants of the outcome, but its causal status depends on a defensible design and assumptions.

Let $Z$ denote the proposed instrument, $D$ the treatment received, $Y$ the outcome, and $U$ unmeasured common causes of treatment and outcome. The intended causal route is $Z \rightarrow D \rightarrow Y$, while $U$ may influence both $D$ and $Y$; a direct route from $Z$ to $Y$ would defeat the usual exclusion claim. Random assignment to treatment encouragement in a trial can supply a particularly clear candidate $Z$ when some patients do not take up the assigned treatment.

Conditions for a credible instrument

Instrument validity is a claim about a particular intervention, population, outcome, and analysis, not an intrinsic property of a variable's name. The first condition can be examined in observed data, whereas the independence and exclusion conditions require substantive causal reasoning and cannot generally be established by a significance test.

For a binary instrument and treatment, relevance can be displayed as $P(D=1\mid Z=1) \ne P(D=1\mid Z=0)$. This observed inequality is necessary for the simple Wald contrast, but it cannot by itself establish independence or exclusion. Covariate balance and negative-control outcomes may expose problems with a proposed instrument, yet passing such checks does not prove the untestable parts of validity.

Treatment effects and the population they describe

Even a credible instrument does not automatically identify the average treatment effect for every eligible patient. With a binary encouragement and binary treatment, the familiar local average treatment effect interpretation additionally requires consistent versions of treatment and outcome, no relevant interference, a nonzero first stage, and monotonicity: nobody would take treatment only when not encouraged.

Under those conditions, the identifiable effect concerns compliers, whose treatment receipt changes with the instrument. Always-takers and never-takers contribute to the observed groups but do not change treatment because of $Z$; their individual treatment effects are not identified by the simple instrument contrast. Effect heterogeneity means this complier effect need not equal an effect among all patients, all treatment users, or the people targeted by a later policy.

Worked example: trial encouragement and event risk

Consider a randomised offer of extra appointment support intended to raise uptake of a defined treatment, and suppose the offer has no effect on the adverse event except through receiving that treatment. These are illustrative numbers, and the interpretation also assumes that encouragement assignment is independent of potential outcomes, monotonicity holds, and outcome measurement and follow-up are comparable between groups.

The first-stage difference is $0.60-0.20=0.40$ and the outcome difference by encouragement is $0.30-0.38=-0.08$. The simple Wald contrast divides the latter by the former:

$$\frac{E(Y\mid Z=1)-E(Y\mid Z=0)}{E(D\mid Z=1)-E(D\mid Z=0)} =\frac{-0.08}{0.40}=-0.20.$$

On the risk-difference scale, this represents an estimated reduction of 20 percentage points among compliers under the stated assumptions. If the extra appointment support also changes event risk through monitoring or co-interventions, the exclusion restriction fails and the ratio cannot be given that interpretation. The assignment to encouragement is the proposed instrument; the treatment received is the exposure whose effect is sought.

Appraisal and reporting

Authors should explain why the proposed instrument arose, when it was assigned, how it affects uptake, and what direct or confounded pathways to the outcome might remain. Reporting should quantify the first-stage association and its uncertainty, examine plausible assumption violations, and state the treatment effect's scale and target population.

An apparently strong first stage does not make exclusion or independence true, while a weak instrument can produce severe imprecision and misleading conventional approximations. Multiple candidate instruments and overidentification tests cannot certify that every exclusion restriction is valid. Sensitivity analysis should connect plausible direct effects or residual confounding to the direction and size of the inferred treatment effect.

Health economic interpretation and boundaries

An instrument can help address treatment selection when estimating clinical effects or utilisation changes for an economic evaluation, provided its assumptions match the decision problem. Costs, quality-adjusted life-years (QALYs), time horizon, and the population affected by a policy still require their own measurement and modelling; a complier-specific clinical effect should not be silently treated as a population-wide effect.

An instrument differs from a confounder, which helps cause both treatment and outcome, and from a mediator, which lies on a causal route from treatment to outcome. Instrumental variable analysis is the subsequent family of estimation procedures that uses the instrument; selecting a plausible $Z$ is the identification step addressed here. “Natural experiment” describes a design context and is not itself a guarantee that a particular variable satisfies the instrument conditions.

Further learning and source basis

The foundational potential-outcomes account explains when an instrument identifies a complier effect, while methodological work in epidemiology clarifies the limits of extrapolating that effect to broader populations. The linked papers provide the primary technical basis for the assumptions and interpretation used here.

• Angrist, Imbens, and Rubin (1996), “Identification of Causal Effects Using Instrumental Variables”, Journal of the American Statistical Association, 91:444–455.
• Swanson, Miller, Robins, and Hernán (2015), “Think globally, act globally: An epidemiologist’s perspective on instrumental variable estimation”, Statistical Science, 30:79–96.
• Hernán and Robins (2006), “Instruments for causal inference: an epidemiologist’s dream?”, Epidemiology, 17:360–372.