Instrumental Variable Analysis

Instrumental Variable Analysis: Concept Architecture

Orientation and learning roadmap

This page follows an analysis from the causal question through instrument assessment, estimation, inference, and interpretation for a health economic decision. It shows how the simplest binary example relates to two-stage least squares, then explains what weak instruments and outcome scale change in practice.

Specify the question before estimating

An analysis begins by naming the treatment or exposure $D$, outcome $Y$, proposed instrument $Z$, eligible population, timing, and causal contrast of interest. This order matters because the effect identified by instrument-induced treatment changes may apply only to patients whose treatment choice responds to $Z$, rather than everyone whose care a policy might change.

In a randomised encouragement design, $Z$ may be the assignment to encouragement and $D$ the treatment actually received after assignment. The instrument itself is the candidate source of variation; instrumental variable (IV) analysis is the estimation and interpretation workflow using that source. Authors should state whether the economic question concerns a risk difference, change in cost, change in quality-adjusted life-years (QALYs), or another explicitly defined outcome.

Identification assumptions and design checks

An association between the instrument and treatment is necessary but does not settle whether the analysis is causal. The causal interpretation requires relevance, independence of the instrument from potential outcomes and treatment responses (possibly conditional on justified baseline variables), and exclusion of routes from instrument to outcome except through the defined treatment.

Balance checks and negative controls may reveal a violation but cannot establish all these requirements. Treatment and outcome definitions also need consistency and appropriate handling of interference, missingness, and selection into the analytic sample. If effect heterogeneity is important, report the local target population and the additional assumptions needed for any broader extrapolation.

Binary example: two contrasts and one ratio

Suppose a trial randomises $Z=1$ to an offer of appointment support and $Z=0$ to usual contact, with $D=1$ denoting receipt of a defined treatment and $Y=1$ an adverse event. This invented example assumes the offer changes outcomes only through treatment receipt, assignment is independent of the relevant potential outcomes, monotonicity holds, and the outcome is observed comparably across groups.

The intention-to-treat outcome contrast by assignment is $0.30-0.38=-0.08$, and the first-stage treatment contrast is $0.60-0.20=0.40$. Dividing the outcome contrast by the uptake contrast yields the binary Wald estimate on an additive risk scale:

$$\widehat{\tau}_{\mathrm{Wald}} =\frac{\overline{Y}_{Z=1}-\overline{Y}_{Z=0}} {\overline{D}_{Z=1}-\overline{D}_{Z=0}} =\frac{-0.08}{0.40}=-0.20.$$

Under the stated identification assumptions, this is a 20-percentage-point reduction in adverse-event risk for compliers, the people whose receipt changes with the offer. The $-0.08$ assignment effect is a different contrast and must not be labelled the treatment effect among all patients. The table contains group proportions, so a confidence interval cannot be calculated from it without sample sizes or more detailed uncertainty information.

Two-stage least squares and covariates

Two-stage least squares (2SLS or TSLS) extends the ratio approach to suitable linear models with one or more instruments and prespecified covariates. In the first stage, treatment is projected on the instrument and covariates; in the second, the outcome is projected on fitted treatment and those same exogenous covariates, with variance estimated by a proper IV procedure rather than a naive second-stage regression.

For a continuous outcome in a simple linear specification, the structural coefficient $\beta$ represents an additive outcome change per unit of treatment under the model and identifying conditions. With a binary $D$ and a single binary $Z$ in an unadjusted, just-identified model, 2SLS gives the same point estimate as the Wald ratio. Conditional adjustment requires assumptions about instrument validity given the covariates; selecting post-instrument variables can introduce bias.

Weak instruments, inference, and model choice

An instrument with only a small effect on treatment leaves little usable variation with which to estimate a treatment effect. Analysts should show the size and uncertainty of the first stage and consider weak-instrument-robust confidence sets or tests when strength is in doubt; a conventional first-stage $F$ statistic is a diagnostic in its specified setting, not a universal validity threshold.

Standard errors must account for the two-stage estimation and the actual sampling structure, including clustering when assignment or treatment variation is clustered. For binary outcomes, a linear IV model targets an additive risk-difference type contrast and may produce fitted values outside $[0,1]$ without invalidating that contrast as a point estimand. A risk ratio, odds ratio, or nonlinear model needs its own structural assumptions and estimator; substituting fitted probabilities into an ordinary logistic regression is not generally equivalent to a valid IV analysis.

Economic evaluation and interpretation

IV analysis can contribute causal estimates of health outcomes, resource use, or costs when uptake is confounded, but each outcome may have a different time horizon and missing-data mechanism. An economic model using an IV-derived clinical effect should document whether the complier population, comparator, and treatment definition match the population and interventions in the decision problem.

For example, an estimated risk reduction among patients whose treatment responds to an appointment offer cannot automatically be applied to patients who would always or never take the treatment. Cost-effectiveness analysis still requires incremental costs, relevant outcomes such as QALYs, uncertainty, and an explicit decision perspective. Neither a precise 2SLS coefficient nor an overidentification test proves that exclusion and independence hold.

Common reporting errors and further learning

Clear reporting separates the assignment effect, first-stage treatment change, IV estimate, and target population, then labels all assumptions required for a causal interpretation. It should also explain whether observations were excluded after assignment, whether the proposed instrument could change other care, and how results respond to plausible violations.

• Angrist, Imbens, and Rubin (1996), “Identification of Causal Effects Using Instrumental Variables”, Journal of the American Statistical Association, 91:444–455. This establishes the potential-outcomes interpretation for compliers.
• Staiger and Stock (1997), “Instrumental Variables Regression with Weak Instruments”, Econometrica, 65:557–586. This develops weak-instrument asymptotic theory.
• Rassen and colleagues (2009), “Instrumental variable analysis for estimation of treatment effects with dichotomous outcomes”, American Journal of Epidemiology, 169:273–284. This examines outcome-scale choices in health research.
• Hernán and Robins (2006), “Instruments for causal inference: an epidemiologist’s dream?”, Epidemiology, 17:360–372. This discusses additional assumptions and structural interpretations.