Who Are the Compliers in a LATE

Who Is the LATE About?

An instrumental-variable estimate is often summarized as an effect of treatment, but its target is not automatically everyone in a study. Under the assumptions that identify a local average treatment effect (LATE), the relevant population consists of people whose treatment receipt would change if the instrument changed. These people are called compliers. Their membership is defined by two potential treatment decisions, only one of which can be observed for any individual.

This distinction matters when translating an estimate into a substantive conclusion. A LATE may describe the effect for people reached by a particular encouragement or access change, rather than the effect for all participants. Whether those people resemble the rest of the study population is an empirical and design-specific question, not a consequence of the IV assumptions.

The other possible compliance types clarify what the definition does and does not mean. An always-taker has \(D(1)=D(0)=1\); a never-taker has \(D(1)=D(0)=0\); and a defier has \(D(1)=0,D(0)=1\). The monotonicity assumption rules out defiers, so that an instrument coded in the encouraging direction does not cause anyone to move away from treatment while causing others to move toward it. Relevance requires the instrument to change treatment receipt for at least some units.

For the usual LATE interpretation, one also needs instrument independence (unconditional in a randomized design, or conditional on stated covariates in a conditional design) and exclusion: the instrument affects the outcome only through treatment receipt. These assumptions support the causal interpretation of the IV contrast. They do not say that compliers have the same characteristics as the full study population, or that every participant has the same chance of being a complier.

Latent Types and Observed Groups Are Different

For each individual, only the treatment receipt under the assigned instrument value is observed. If \(Z=1\), the data reveal \(D(1)\), but not \(D(0)\); if \(Z=0\), they reveal \(D(0)\), but not \(D(1)\). Consequently, a person's compliance type is generally not known, even in a randomized study.

Monotonicity allows some observed assignments to rule out types. A unit observed with \(Z=1,D=0\) must be a never-taker: it did not take treatment even when encouraged, and cannot be a defier under monotonicity. A unit observed with \(Z=0,D=1\) must be an always-taker. But \(Z=1,D=1\) could indicate an always-taker or a complier, and \(Z=0,D=0\) could indicate a never-taker or a complier. Thus the observed treated group is not the complier group, nor is the observed untreated group.

Describing Which Participants Are Compliers

Although individual compliance types are latent, covariate patterns can reveal how the complier population is distributed across observed characteristics. For a discrete covariate \(X\), suppose instrument assignment is independent of the potential treatment receipts within levels of \(X\), and monotonicity holds within those levels. Define the conditional first-stage contrast \(\Delta_D(x)=E[D\mid Z=1,X=x]-E[D\mid Z=0,X=x]\). Then \(\Delta_D(x)\) is the complier share among units with \(X=x\). The share of all compliers who have characteristic value \(x\) is proportional to both the prevalence of that characteristic and its conditional first stage.

The reasoning follows directly from the definition. Within covariate level \(x\), the difference in treatment rates equals the average difference in potential treatment receipt. With no defiers, \(D(1)-D(0)\) equals 1 for compliers and 0 for everyone else. Therefore \(\Delta_D(x)=P(C\mid X=x)\). Multiplying by \(P(X=x)\) gives the joint share of the population that both has \(X=x\) and is a complier. Dividing by the total complier share normalizes these joint shares to obtain the distribution among compliers.

Why the Target Population Need Not Resemble Everyone

Compliance is defined by a response to the instrument, not by a general trait such as willingness, health status, or treatment eligibility. It depends on the particular instrument contrast and treatment decision in the study. An encouragement may change uptake mainly for participants who face a particular barrier; a different instrument may reach a different subset. Even with a valid instrument, always-takers and never-takers can differ from compliers in observed characteristics, unobserved characteristics, or both.

The next example isolates this point: two instruments can have the same overall first stage while changing treatment receipt in different subgroups. Each instrument can therefore define a different complier population. A matching estimate or a comparison of observed treated and untreated participants answers a different question unless additional assumptions make its target coincide with this complier target.

Diagnostics, Limitations, and a Common Mistake

Observed covariates can be useful for assessing whether the complier population is concentrated in particular subgroups. Estimate treatment rates by instrument level within substantively important covariate strata, then examine the conditional first-stage contrasts. A markedly larger contrast in one stratum signals that compliers are more prevalent there, subject to the assumptions and sampling uncertainty. A near-zero contrast means the data provide little evidence that the instrument changes treatment receipt in that stratum; a conditional LATE there will also be poorly supported or unidentified.

A negative conditional contrast when \(Z=1\) is coded as the encouraging state is a warning, not proof of defiers. It may reflect sampling variability, a coding or implementation problem, or a failure of conditional independence or monotonicity. Monotonicity cannot generally be verified by observing each person's two potential treatment decisions, because one of those decisions is missing. Likewise, a positive overall first stage does not guarantee relevance in every covariate stratum. Composition estimates become unstable when first-stage contrasts are imprecise, especially if they are used to divide by a small overall first stage.

The main limitation is conceptual as well as statistical: compliance type is counterfactual and usually cannot be observed person by person. The LATE is not automatically the population average treatment effect (ATE), the effect among those who actually received treatment (ATT), or an effect for a policy population that the instrument does not reach. Those targets require their own identifying assumptions. A careful report states the instrument contrast, the assumptions used, and the population of instrument-responsive units to which the result pertains.

• Common mistake: “The IV estimate is the treatment effect for everyone because the instrument was randomized.”
• Correction: Randomization supports independence of the instrument, but with exclusion and monotonicity the Wald estimand is an effect for compliers. Random assignment does not make compliers representative of all participants.
• Another mistake: “Everyone treated in the encouraged arm is a complier.”
• Correction: That group can include always-takers as well as compliers. Compliance types depend on both potential treatment decisions, and generally only one is observed.