Interpreting Complier Effects Alongside Encouragement Effects

Two Causal Questions, Two Effects

In a randomized encouragement study, an invitation, reminder, or other offer may change whether people receive a treatment. It is useful to distinguish the effect of offering the encouragement from the effect of receiving the treatment. The first describes what happens under a policy of making the offer; the second, in the instrumental-variable setting, describes the treatment effect among people whose treatment receipt is changed by that offer.

These effects need not have the same numerical value. The offer may change treatment receipt for only a fraction of the population, so its average effect on outcomes can be smaller than the treatment effect among that fraction. The connection is precise under standard instrumental-variable assumptions, but the two effects remain answers to different questions.

Here \(Y(d)\) is the outcome that would be observed if treatment receipt were set to \(d\), and \(D(z)\) is the treatment receipt that would occur under assignment \(z\). The encouragement effect is often called the intention-to-treat effect on the outcome. It compares assignment groups, whether or not every person follows the offer. The complier effect instead compares treatment with no treatment for the subgroup whose receipt responds to assignment.

The difference is not merely terminological. A health agency deciding whether to send an invitation needs to know its effect on outcomes in the population offered the invitation. A question about the benefit of treatment for people induced to take it concerns the complier effect. Neither effect, by itself, answers every policy question: for example, the complier effect is not automatically the treatment effect for people who would take treatment regardless of an offer.

Assumptions Behind the Relationship

For a binary instrument and binary treatment, the familiar relationship between the two effects requires assumptions. First, assignment to encouragement is randomized, so \(Z\) is independent of the potential treatment receipts and outcomes. Second, exclusion requires that assignment affect the outcome only through treatment receipt: changing \(Z\) while holding \(D\) fixed does not change \(Y\). Third, monotonicity requires \(D(1)\geq D(0)\) for every person, ruling out people who would take treatment without the offer but not take it when offered. Finally, the encouragement must change treatment receipt for some people; this is the relevance condition.

Under monotonicity, people are either compliers, always-takers, or never-takers: compliers take treatment only when encouraged; always-takers take it either way; never-takers do not take it either way. The first stage is the effect of assignment on treatment receipt, \(E[D\mid Z=1]-E[D\mid Z=0]\). In the population, under randomization and monotonicity, it equals the proportion of compliers. Exclusion ensures that assignment does not change outcomes for always-takers or never-takers, because their treatment receipt does not change.

This result identifies a local effect for compliers, not an average treatment effect for everyone. The encouragement effect itself is the average causal effect of assignment in the study population. It incorporates the fact that many people may not change their treatment receipt when offered encouragement.

Why the Effects Have This Relationship

The identity follows by considering each person’s treatment receipt under both assignments. Randomization lets us interpret differences between observed assignment groups as differences in the corresponding potential outcomes. Under exclusion, the potential outcome under assignment \(z\) is the outcome under the treatment receipt that assignment produces: \(Y(D(z))\). Therefore the encouragement effect is the average of \(Y(D(1))-Y(D(0))\).

For an always-taker, \(D(1)=D(0)=1\), so this difference is zero. For a never-taker, \(D(1)=D(0)=0\), so it is also zero. For a complier, \(D(1)=1\) and \(D(0)=0\), so the difference is \(Y(1)-Y(0)\). Averaging across the population gives:

The first stage follows the same type-by-type comparison. For always-takers and never-takers, \(D(1)-D(0)=0\); for compliers, it equals 1. Hence:

When \(P(C)>0\), dividing the first identity by the second yields the complier effect. This derivation also explains why the encouragement effect is scaled by the share whose treatment receipt actually responds. If the complier effect is nonzero, a larger first stage produces a larger encouragement effect in absolute value when the complier effect is held fixed. If the complier effect is zero, the encouragement effect is zero for any first stage.

Worked Examples

Interpreting and Reporting Both Results

When both effects are relevant, report them separately, name their target populations, and state their scales. The encouragement effect answers, “What is the effect of assigning the offer in this study population?” The complier effect answers, “What is the treatment effect for people whose receipt is changed by assignment, under the instrumental-variable assumptions?” Reporting only the Wald ratio can hide how small or large the offer’s actual population impact is. Reporting only the encouragement effect does not reveal the estimated treatment effect among compliers.

Their units depend on the outcome. If \(Y\) is a score, both effects are in score points. If \(Y\) is an indicator, both are risk differences, usually expressed as percentage points. The first stage is a difference in treatment probabilities; multiplying it by the complier effect gives the encouragement effect on the outcome scale. The Wald ratio divides by that probability difference and returns to the outcome scale.

These interpretations are causal only to the extent that the assumptions are credible. Randomization supports independence of assignment from potential outcomes and treatment receipt, but it does not by itself establish exclusion or monotonicity. The numerical equality between a reduced form and a first stage times a Wald ratio is arithmetic; it does not prove that the ratio has the desired causal interpretation.

Common Mistake

Wrong reasoning: “The offer effect and the complier effect are both effects of the same program, so they should be the same number.” This overlooks that one is averaged over the assignment population and the other over compliers, and that the first stage scales the former.

Correction: Under the stated assumptions, encouragement effect = first stage \(\times\) complier effect. The offer effect reflects both the size of the treatment effect among compliers and the fraction whose receipt changes. A small offer effect can occur with a large complier effect if the first stage is small. Conversely, a common nonzero complier effect across two studies implies different offer effects when their first stages differ. A zero complier effect is the important exception: it implies a zero offer effect at any first stage.