Instrument-Specific Complier Populations

Why the Instrument Defines the Complier Group

An instrumental-variable estimate does not automatically describe the people who received treatment, everyone eligible for an intervention, or a fixed group called “the compliers.” A complier is defined by a counterfactual response: the person would take treatment under one instrument value and not take it under another. Change the instrument, or change which two of its values are contrasted, and the people who meet that definition can change.

This matters when an intervention can be encouraged in more than one way. A reminder, a transportation voucher, and an appointment offered at a different time might each change who receives care. Even if every encouragement is randomized and changes treatment uptake, each may move treatment for a different subset of people. The corresponding Wald effects therefore need not refer to the same population or have the same value.

Definition: For instrument \(J\), let \(D^J(z)\) denote a person’s potential binary treatment status under instrument value \(z\). For a specified contrast from \(z_0\) to \(z_1\), that person is a complier for instrument \(J\) and contrast \((z_0,z_1)\) if \(D^J(z_0)=0\) and \(D^J(z_1)=1\). This definition is contrast-specific. It does not, by itself, say whether the person complies with a different instrument or a different contrast of the same instrument.

The direction of the contrast matters. For a contrast oriented so that \(z_1\) is the more encouraging value, the usual monotonicity assumption is \(D^J(z_1)\geq D^J(z_0)\) for every person. It rules out defiers for that contrast: people who would take treatment at \(z_0\) but not at \(z_1\). Under this assumption, everyone whose treatment status changes across the contrast is a complier. People whose status does not change are not compliers for that contrast, whether they always take treatment or never take it.

What a Contrast-Specific Wald Effect Represents

To make the interpretation precise, suppose \(Z^J\) is randomly assigned to one of two values, \(z_0\) or \(z_1\), and treatment is binary. Write \(Y(1)\) and \(Y(0)\) for the potential outcomes under treatment and no treatment, with the exclusion restriction: changing \(Z^J\) can affect the outcome only by changing treatment. Also assume assignment is independent of the relevant potential treatment statuses and outcomes, there is no interference between people, and the first stage is nonzero.

For this contrast, the Wald ratio is the difference in mean outcomes between assignment to \(z_1\) and \(z_0\), divided by the difference in mean treatment receipt between those assignments. Under the assumptions just stated and monotonicity for this contrast, it identifies the average treatment effect among that contrast’s compliers. That is a local average treatment effect (LATE), local to the specified instrument and contrast.

\text{Wald}_{J}(z_0,z_1) = \frac{E[Y\mid Z^J=z_1]-E[Y\mid Z^J=z_0]} {E[D\mid Z^J=z_1]-E[D\mid Z^J=z_0]} = E\!\left[Y(1)-Y(0)\mid D^J(z_0)=0,\ D^J(z_1)=1\right].

The assumptions are needed for this interpretation, not merely for the arithmetic. Random assignment makes the assignment-group differences reflect the instrument contrast rather than systematic differences between groups. Exclusion rules out a direct effect of the instrument on the outcome. Monotonicity ensures that the first stage is not a mixture of treatment increases for some people and treatment decreases for others. A nonzero first stage ensures the ratio is defined and that the contrast changes treatment receipt.

A Short Justification of the Population Interpretation

Let \(\Delta D_i=D_i^J(z_1)-D_i^J(z_0)\) and \(\Delta Y_i=Y_i^J(z_1)-Y_i^J(z_0)\) denote a person’s changes in treatment and outcome across the contrast. By exclusion, the outcome change is the treatment effect multiplied by the change in treatment:

\Delta Y_i = \{Y_i(1)-Y_i(0)\}\Delta D_i.

Independence of assignment allows the observed mean differences to identify the population averages of these potential changes. Under monotonicity, \(\Delta D_i\) is 1 for compliers and 0 for everyone else. Consequently, the average change in treatment is the proportion of compliers, while the average change in outcome is the total of their treatment effects divided by the population size. Taking the ratio cancels that population-size factor and averages effects over compliers:

\frac{E[\Delta Y]}{E[\Delta D]} = \frac{E[\{Y(1)-Y(0)\}\Delta D]}{E[\Delta D]} = E[Y(1)-Y(0)\mid \Delta D=1].

This justification also shows what the ratio does not establish. It does not say that compliers under another instrument have the same treatment effects, or even that they are the same people. That would require additional information or assumptions about how the instruments affect treatment uptake.

Worked Example: Two Encouragements, Overlapping but Different Compliers

Consider a hypothetical population in which two alternative randomized encouragements could be used: a clinic reminder (instrument A) or a transportation voucher (instrument B). The table summarizes the potential treatment responses and assumed treatment effects for four groups. The encouragements are evaluated separately, each against no encouragement.

Group	Population share	Complier under A?	Complier under B?	Mean treatment effect
Complies with both	0.10	Yes	Yes	0.50
Complies only with A	0.20	Yes	No	0.20
Complies only with B	0.10	No	Yes	0.80
Complies with neither	0.60	No	No	Not used in either LATE

Assume that each encouragement is randomized, affects the outcome only through treatment, and satisfies monotonicity relative to its own no-encouragement contrast. The first stage for A is \(0.10+0.20=0.30\), and the first stage for B is \(0.10+0.10=0.20\). The overlap does not make the two complier groups identical: A also moves the A-only group, and B also moves the B-only group.

For A, the average treatment effect among compliers is

\text{LATE}_A = \frac{(0.10)(0.50)+(0.20)(0.20)}{0.30} = \frac{0.05+0.04}{0.30} = 0.30.

The outcome contrast for A is therefore \(0.30\times0.30=0.09\). For B, it is

\text{LATE}_B = \frac{(0.10)(0.50)+(0.10)(0.80)}{0.20} = \frac{0.05+0.08}{0.20} = 0.65,

so the outcome contrast for B is \(0.20\times0.65=0.13\). If the outcome is a health score measured in points, these effects are 0.30 and 0.65 points per treated person, respectively. The difference is not a contradiction: the two valid ratios average effects over different complier populations.

Different Contrasts of One Instrument Can Target Different People

Instrument-specificity is not limited to comparing different encouragement programs. If an instrument has three or more levels, two different contrasts of that same instrument can define different complier groups. A higher level might move people who are particularly responsive to a small change, while a further increase moves a different set of people.

Worked Example: Low and High Access Contrasts

Suppose a hypothetical randomized access program has three levels: no additional access (0), standard access (1), and enhanced access (2). The population has three response groups. Sixty percent never take treatment at any level; 20% start taking it at level 1 and continue at level 2; and 20% start only at level 2. Assume treatment status is nondecreasing across levels for every person, and the program has no effect on the outcome except through treatment.

For the contrast from 0 to 1, only the 20% who start treatment at level 1 are compliers. For the contrast from 1 to 2, only the 20% who start at level 2 are compliers. If the average treatment effects in these groups are 0.10 and 0.50 outcome units, respectively, the outcome contrasts are \(0.20\times0.10=0.02\) and \(0.20\times0.50=0.10\). Dividing each outcome contrast by its first stage of 0.20 gives Wald effects of 0.10 and 0.50 outcome units.

For the contrast from 0 to 2, both groups change treatment status. The complier share is \(0.20+0.20=0.40\), and the Wald effect is

\text{Wald}(0,2) = \frac{(0.20)(0.10)+(0.20)(0.50)}{0.40} = \frac{0.12}{0.40} = 0.30 \text{ outcome units}.

The 0-to-2 contrast identifies an average over the combined group, whereas the two smaller contrasts each identify an average over one response group. The assumptions must apply to the contrast being interpreted; for example, monotonicity for the 0-to-2 contrast alone does not establish that its two component contrasts each satisfy all the same assumptions.

Equal First Stages Do Not Mean Equal Complier Populations

A first stage measures how much an instrument contrast changes treatment uptake on average. It does not identify which individuals changed treatment. Two contrasts can have the same first-stage size and still affect disjoint groups. Even when their complier groups overlap, the remaining members may have different treatment effects.

Worked Example: Same First Stage, Different Local Effects

Suppose two hypothetical, separately randomized instruments each increase treatment receipt by 0.25 in their respective study populations. For instrument C, assume the mean outcome contrast is 0.05; for instrument D, assume it is 0.15. In both settings, assume random assignment, exclusion, monotonicity for the relevant contrast, and a nonzero first stage.

The Wald effects are

\text{Wald}_C=\frac{0.05}{0.25}=0.20, \qquad \text{Wald}_D=\frac{0.15}{0.25}=0.60.

If the outcome is measured in points, the estimates are 0.20 and 0.60 points per treated person, for the compliers of C and D respectively. Equal first stages show that the contrasts moved the same proportion of their populations into treatment. They do not show that they moved the same individuals, nor that the average treatment effect among those individuals was the same.

Interpreting and Reporting Instrument-Specific Effects

A careful interpretation names the instrument, the contrast, and the target group. For example: “Under the stated IV assumptions, the Wald ratio for assignment to a voucher rather than no voucher estimates the average treatment effect among people whose treatment receipt was changed by that voucher contrast.” This is more informative than saying simply “the IV estimate is the treatment effect.”

When comparing instruments, treat each contrast as a separate causal design. Its assumptions and first stage need to be considered on their own. A strong first stage for one instrument does not establish relevance for another; likewise, validity of one instrument does not establish validity of a different one. Even when both are valid, their LATEs can differ because they may concern different people, different treatment-effect distributions, or both.

In some applications, the target population itself is a substantive concern. A reminder may mainly affect people who already have access but are likely to forget; a transportation intervention may mainly affect people whose access is constrained. The labels are illustrative, not assumptions about who will respond. The key point is that uptake mechanisms shape the complier group, so conclusions should not be generalized beyond that group without additional justification.

Common Mistake: Treating “Compliers” as a Universal Group

Wrong reasoning: “Both instruments increase treatment, so their compliers are the same people.” Increasing treatment on average does not establish individual-level overlap.
Correction: Define potential treatment status for each instrument and name the exact contrast. The complier group is the set whose treatment status changes across that particular contrast.
Wrong reasoning: “The two first stages are equal, so their LATEs refer to the same population.” A first stage is a difference in average treatment uptake, not a description of individual identities.
Correction: Equal first stages imply equal complier proportions under monotonicity, not equal complier membership or equal effects.
Wrong reasoning: “A Wald ratio for one valid instrument gives the effect for everyone, so another valid instrument should give the same estimate.”
Correction: Each valid ratio targets its own contrast-specific complier effect. Equality across instruments requires additional conditions; it does not follow from instrument validity alone.

Key takeaway: Always attach “complier” to a specific instrument and instrument contrast. A Wald ratio identifies an average treatment effect for the people whose treatment status that contrast changes, not a universal population effect.

Check Your Understanding

Use the definition and contrast-specific interpretation to answer these questions.

What counterfactual treatment responses make a person a complier for a specified contrast from \(z_0\) to \(z_1\)?
Two randomized encouragements have the same first stage. What does that tell you about their complier populations under monotonicity, and what does it not tell you?
An instrument has levels 0, 1, and 2. Explain why its 0-to-1 and 1-to-2 contrasts can have different compliers.
A Wald ratio is 0.4 outcome units per treated person. State a suitable interpretation that names the target group without claiming an effect for everyone.
Why does random assignment alone not guarantee that a Wald ratio identifies a complier average treatment effect?