Sensitivity Analysis for Direct Effects of the Instrument

Why Parameterize a Direct Effect?

A binary instrument can change treatment uptake and, through a separate pathway, change the outcome directly. For example, an encouragement might affect an outcome through information or attention as well as by changing whether a person receives treatment. If that direct pathway is present, the usual Wald ratio does not isolate the effect of treatment uptake. A sensitivity analysis makes the potential distortion explicit by assigning a value to the direct effect and showing how the implied treatment effect changes.

The aim here is a specific decomposition for a binary instrument. We retain random assignment of the instrument and monotonicity, but relax exclusion using a simple additive model: the instrument shifts the outcome by the same amount at either treatment level. This gives a transparent adjustment to the Wald ratio. The direct-effect value is a sensitivity parameter; the observed instrument and outcome contrasts alone do not identify it.

Setup and Assumptions

Let \(Z\in\{0,1\}\) be the instrument, \(D\in\{0,1\}\) the treatment, and \(Y\) the outcome. Write \(D(z)\) for the treatment that would be received under instrument value \(z\), and \(Y(d,z)\) for the outcome under treatment \(d\) and instrument value \(z\). The observed variables satisfy consistency: \(D=D(Z)\) and \(Y=Y(D,Z)\). Assume there is no interference between units, so one unit’s instrument or treatment does not change another unit’s outcome.

The assumptions used for the decomposition are:

• Instrument independence: \(Z\) is randomized, or otherwise independent of the potential treatments and outcomes \(\{D(0),D(1),Y(d,z)\}\).
• Relevance: the instrument changes treatment uptake, so \(\Delta_D=E[D\mid Z=1]-E[D\mid Z=0]\ne 0\).
• Monotonicity: \(D(1)\ge D(0)\) for every unit. Thus there are no defiers, people who would take treatment under \(Z=0\) but not under \(Z=1\).
• Additive direct-effect model: for every unit and either treatment level, \(Y(d,1)=Y(d,0)+\delta\), where \(\delta\) is a common direct effect of changing the instrument from 0 to 1, holding treatment fixed.

The last condition relaxes exclusion. Exclusion would require \(Y(d,1)=Y(d,0)\), which is the special case \(\delta=0\). The additive model is deliberately restrictive: it assumes the direct effect is the same for all units and does not depend on treatment. It is useful because it yields a simple, interpretable sensitivity calculation. The units of \(\delta\) are the same as the units of \(Y\).

Deriving the Adjusted Wald Estimand

Under monotonicity, units fall into three principal strata: compliers, with \(D(0)=0,D(1)=1\); always-takers, with \(D(0)=D(1)=1\); and never-takers, with \(D(0)=D(1)=0\). Let \(\pi_C=P(D(0)=0,D(1)=1)\) be the proportion of compliers, and define their average treatment effect at \(Z=0\) as \(\tau_C=E[Y(1,0)-Y(0,0)\mid D(0)=0,D(1)=1]\).

Random assignment and consistency let us write the observed reduced form as the population average contrast in outcomes under the two instrument assignments:

Add and subtract \(Y(D(1),0)\) inside this expectation. The first difference below is the direct instrument shift at the treatment level \(D(1)\); the additive model makes it equal to \(\delta\) for every unit. The remaining difference is the treatment change at \(Z=0\):

For always-takers and never-takers, \(D(1)=D(0)\), so their contribution to the second expectation is zero. For compliers, \(D(1)=1\) and \(D(0)=0\), so their contribution is \(Y(1,0)-Y(0,0)\). Averaging across the population therefore gives \(\pi_C\tau_C\). The same principal-strata argument gives \(\Delta_D=E[D(1)-D(0)]=\pi_C\). Consequently:

This is the key result: for a specified direct effect \(\delta\), subtract it from the reduced form before dividing by the first stage. If \(\delta=0\), the expression reduces to the familiar Wald ratio and identifies the complier average treatment effect under the assumptions above. If \(\delta\ne0\), the unadjusted Wald ratio instead equals \[ \frac{\Delta_Y}{\Delta_D} =\tau_C+\frac{\delta}{\pi_C}. \] Thus the bias relative to the complier effect is \(\delta/\pi_C\). A small direct effect can matter substantially when the first stage—and hence the complier share—is small.

Worked Sensitivity Calculations

How to Use and Interpret the Sensitivity Analysis

A useful analysis reports the reduced form, first stage, and the range of direct-effect values considered, then gives the adjusted estimate for each value. The values of \(\delta\) should be expressed in outcome units and justified substantively—for example, using knowledge about the instrument’s non-treatment pathways. The sensitivity calculation does not tell us which value is correct. Rather, it shows how conclusions change if those pathways produce different outcome shifts.

The denominator matters as much as the numerator. For two assumed direct effects \(\delta_1\) and \(\delta_2\), the corresponding adjusted estimates differ by:

When \(\Delta_D\) is small, a given change in the assumed direct effect produces a larger change in the adjusted estimate. This algebra explains why even modest exclusion violations can have substantial impact with a weak first stage. It does not, by itself, provide a confidence interval or account for sampling uncertainty; the calculations above concern how the estimand changes as the sensitivity parameter changes.

The additive model also has limits. If the direct effect varies across units or differs between \(D=0\) and \(D=1\), one common \(\delta\) may not capture the relevant outcome contrast. In that case the simple subtraction need not recover the complier effect. The model should therefore be treated as a clearly stated sensitivity assumption, not as a consequence of randomization.

Common Mistake

• Wrong reasoning: “The instrument was randomized, so the Wald ratio must be a treatment effect.” Randomization supports independence of the instrument, but does not rule out a direct pathway from the instrument to the outcome.
• Correction: Randomization alone does not guarantee exclusion. Under the additive direct-effect model, the Wald ratio is \(\tau_C+\delta/\pi_C\), not simply \(\tau_C\). State the assumed \(\delta\), subtract it from the reduced form, and divide by the first stage before interpreting the adjusted result.
• Another error: Subtracting \(\delta\) from the Wald ratio rather than from the reduced form. The correct adjustment is \((\Delta_Y-\delta)/\Delta_D\); the direct effect is in outcome units, while the Wald ratio is already scaled by the first stage.