Sensitivity Analysis for Instrument-Outcome Dependence

Why Examine Instrument–Outcome Dependence?

An instrumental-variable analysis can be sensitive to violations of instrument independence. Even when an instrument changes treatment uptake and has no direct effect on the outcome, assignment groups may differ in their potential outcomes for reasons unrelated to the treatment. The observed outcome contrast then combines a treatment-induced contrast with this pre-existing imbalance. A sensitivity analysis asks how large that imbalance would need to be to alter the IV conclusion.

Here the focus is a transparent model in which the treatment is binary and the treatment effect is constant. This restriction makes it possible to express the bias with one outcome-scale parameter. It is an assumption of the analysis, not a general property of instrumental-variable estimators. In particular, the adjusted ratio below should not be treated as a general correction for arbitrary dependence or treatment-effect heterogeneity.

Model and Sensitivity Parameter

The parameter \(\delta\) is measured in the same units as the outcome. A positive value means that the group with \(Z=1\) has a higher mean untreated potential outcome than the group with \(Z=0\). It represents dependence between assignment and a potential outcome; it is not a causal effect of \(Z\). Because the treatment effect is constant, \(Y(1)=Y(0)+\tau\), so the assignment-group difference in mean \(Y(1)\) is also \(\delta\).

The independence assumption for \(D(0)\) and \(D(1)\) is deliberately stated separately from the allowed dependence involving \(Y(0)\). It ensures that the observed first stage identifies \(\pi\), even though the outcome groups may be imbalanced. If assignment is also associated with treatment-uptake potentials, the observed first stage can itself be distorted; the one-parameter adjustment developed here does not address that additional violation.

How the Imbalance Shifts the IV Contrast

Write the population reduced-form contrast as \(\theta_Y=E[Y\mid Z=1]-E[Y\mid Z=0]\), and the population first-stage contrast as \(\theta_D=E[D\mid Z=1]-E[D\mid Z=0]\). Since \(D=D(Z)\), independence of \(Z\) from the treatment-uptake potentials gives \(\theta_D=\pi\). Exclusion and the constant-effect model give \(Y=Y(0)+\tau D\). Therefore:

The second equality uses the stated independence of assignment and the treatment-uptake potentials. Thus, if \(\pi\ne 0\), the usual Wald ratio satisfies:

This is the key sensitivity result. The outcome imbalance adds \(\delta\) to the reduced form and, after division by the first stage, adds \(\delta/\pi\) to the IV ratio. A small first stage magnifies a given imbalance. If \(\delta\) is specified, the sensitivity-adjusted effect is \((\theta_Y-\delta)/\theta_D\). These are population identities under the model; in applications, sample contrasts estimate the population contrasts and have sampling uncertainty as well.

A bound can be stated as \(\delta\in[L,U]\), in outcome units. For a positive first stage, the corresponding adjusted-effect range is:

The order of the endpoints reverses because a larger positive imbalance is subtracted from the reduced form. If the first stage is negative, the endpoint order must be reversed accordingly. This range reflects the chosen sensitivity bound, not a confidence interval; sampling uncertainty requires separate statistical inference.

Turning a Bound into an Effect Range

Often the imbalance cannot be specified exactly. Instead, analysts choose a substantively plausible range and report how the adjusted estimate varies over it. The range should be justified in outcome units, for example by comparing it with observed outcome differences between groups known to differ in prognostic factors. Such comparisons can calibrate a sensitivity analysis, but they do not prove that the instrument imbalance is within the chosen bound.

Tipping Points and Substantive Conclusions

A tipping point is the value of the sensitivity parameter at which a chosen conclusion changes. For a target effect \(\tau_0\), solve the sensitivity equation \(\widehat{\tau}(\delta)=\tau_0\) for \(\delta\):

This calculation is useful because it puts the required violation on the outcome scale. Rather than saying only that the result is “sensitive,” one can report how large the imbalance would have to be to reach a specified threshold, such as zero effect or a clinically meaningful effect.

Assumptions, Scope, and Common Mistakes

The calculation isolates one particular threat: assignment dependence in untreated potential outcomes while the treatment-uptake potentials remain independent of assignment. It also relies on binary treatment, exclusion, a constant treatment effect, consistency, no interference, and a nonzero first stage. If treatment effects vary across people, a single \(\delta\) for \(Y(0)\) need not capture all ways outcome dependence can alter the reduced form. If the first stage is affected by assignment dependence in \(D(0)\) or \(D(1)\), adjusting only the numerator is not enough. Those cases require a richer model or additional sensitivity parameters.

The constant-effect assumption matters for interpretation. Under these assumptions, the adjusted ratio identifies the common effect \(\tau\). One cannot simply drop constant effects and call the ratio a complier average causal effect: such an interpretation needs its own assumptions, including binary treatment and appropriate instrument independence. Here the purpose is narrower—to show how a specified outcome imbalance shifts an IV ratio under a clearly defined model.

• Wrong reasoning: “The assignment groups differ in outcome by 1.60 points, so that is the treatment effect.” The reduced form includes both the treatment-induced contrast and \(\delta\). The treatment effect under the model is \((\theta_Y-\delta)/\theta_D\), not \(\theta_Y\) alone.
• Wrong reasoning: “Subtracting any chosen number makes the estimate robust.” The adjusted estimate is conditional on a substantively justified imbalance value or bound. Sensitivity analysis displays dependence on that choice; it does not verify the bound.
• Wrong reasoning: “The same correction works if assignment also predicts treatment-uptake potentials.” In that case the observed first stage may not equal \(\pi\). A numerator-only adjustment does not repair a biased denominator.
• Wrong reasoning: “The resulting sensitivity range is a confidence interval.” It varies the assumed imbalance while holding the sample contrasts fixed. A confidence interval additionally reflects sampling variability.