Why an Encouragement Is Not the Treatment

Two Different Interventions

An encouragement design assigns or delivers an offer intended to change whether people receive a treatment. The offer might be an invitation, eligibility notice, reminder, or transportation voucher. But assignment to an offer and receipt of treatment are different events. Some people offered treatment do not take it, and some people not offered treatment obtain it elsewhere. Consequently, comparing outcomes by offer status answers a different question from comparing outcomes by treatment receipt.

The first comparison concerns the effect of the offer or encouragement policy. The second concerns the effect of receiving treatment. Under suitable instrumental-variable assumptions, the latter may be identified for compliers, but it is not generally equal to the former. In particular, an outcome difference between offer groups is not automatically an effect per person whose treatment status was changed.

This notation allows an offer to affect an outcome by routes other than treatment receipt. For example, an invitation might provide information or reassurance that changes an outcome even for someone who does not take up the treatment. The exclusion restriction rules out such direct effects: it says that \(Y(d,z)=Y(d)\) for each \(d\) and \(z\). Without exclusion, treatment and offer must remain separate in the outcome notation.

What a Randomized Offer Contrast Identifies

Suppose offer assignment is randomized and the observed variables follow consistency: each person’s observed receipt equals \(D(Z)\), and the observed outcome equals \(Y(D(Z),Z)\). Also suppose there is no interference, so one person’s offer or treatment does not change another person’s outcome. Randomization must make \(Z\) independent of the relevant potential outcomes and receipt statuses; in the general case, this means independence of \(Z\) from \(D(z)\) and \(Y(d,z)\) for all \(d,z\). Then the difference in average observed outcomes by assigned offer identifies the average causal effect of the offer policy:

This is an intention-to-treat contrast: it compares people according to assigned offer, whether or not they took up treatment. It can include changes caused by treatment receipt, by the offer itself, or by both. Randomization makes the offer groups comparable in expectation; it does not make offer assignment identical to treatment receipt.

Why the Offer Contrast Is Not Generally the LATE

The LATE is the average effect of treatment receipt among compliers: people who would receive treatment if offered it and would not receive it otherwise. When the exclusion restriction holds, outcomes depend on \(d\), not directly on \(z\), so write them as \(Y(d)\). Under random assignment, consistency, no interference, exclusion, and monotonicity \(D(1)\ge D(0)\), the offer contrast has a particular relationship to the LATE.

Monotonicity rules out people who would take treatment when not offered but refuse it when offered. The remaining response types are always-takers, never-takers, and compliers. Always-takers receive treatment under either assignment; never-takers receive it under neither. Only compliers change treatment status with the offer. The key distinction is that the offer contrast averages outcome changes across the full population, while the LATE averages treatment effects among compliers alone.

and

Thus the outcome contrast is the complier share multiplied by the LATE, not generally the LATE itself. The Wald ratio divides the outcome contrast by the uptake contrast to recover the LATE when the uptake contrast is positive. Here the important point is that the numerator alone is an offer effect, not the per-complier treatment effect.

A Short Derivation

Under exclusion, a person’s potential outcome under assignment \(z\) is \(Y(D(z))\). Random assignment and consistency let us express the offer-group difference as the population average of \(Y(D(1))-Y(D(0))\). Under monotonicity, always-takers have \(D(1)=D(0)=1\), and never-takers have \(D(1)=D(0)=0\). For either type, the outcome difference across offer assignments is zero because treatment status does not change and exclusion removes any direct offer effect. For compliers, \(D(1)=1\) and \(D(0)=0\), so their outcome difference is \(Y(1)-Y(0)\). Averaging these contributions gives

The same reasoning for treatment receipt gives zero change for always-takers and never-takers, and a one-unit change for each complier. Therefore the average receipt difference is \(P(\text{complier})\). These two identities explain both why the offer contrast is not generally the LATE and why dividing by the uptake contrast can recover the complier average effect under the stated assumptions.

When Assignment Is Not Randomized

A contrast by offer status need not be causal if assignment depends on people’s characteristics or expected outcomes. In that case, the offered and non-offered groups may differ even before the offer’s effects are considered. The following hypothetical example separates this selection difference from the effect of the offer.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting an encouragement contrast, check what was actually assigned and what was actually received. Confirm that the offer was implemented as intended, that receipt was measured comparably across assignment groups, and that outcome follow-up was not selectively lost. Examine the uptake rates by assignment: little or no uptake difference means the offer scarcely changed receipt, so it cannot provide much information about a treatment effect through this design. Also consider whether the offer itself could change outcomes through information, attention, costs, stigma, or access. Such routes challenge exclusion, even when assignment is randomized.

A direct randomized-treatment trial, when feasible and implemented with high adherence, targets the effect of assigning treatment and can avoid the distinction between offer and receipt. In contrast, an encouragement design preserves an offer-based randomized comparison but typically estimates an offer effect directly; interpreting a treatment effect for compliers requires additional assumptions. Comparing people who actually received treatment with those who did not is another tempting alternative, but receipt is often influenced by health, motivation, access, or prognosis. That comparison can be confounded and does not inherit the protection of random assignment to the offer.

• Mistake: “The offer groups were randomized, so their outcome difference is the treatment effect.” Randomization identifies the offer-assignment effect, not automatically the effect of treatment receipt.
• Mistake: “The outcome contrast is the LATE because only compliers respond.” The contrast averages over the full population. Under the LATE assumptions it equals the complier share times the LATE.
• Mistake: “Exclusion is harmless because assignment was randomized.” Randomization can identify the causal effect of the offer even when the offer has a direct pathway; that very pathway prevents treating the offer contrast as an effect mediated only by treatment.
• Mistake: “A larger outcome difference proves a stronger treatment effect.” The offer effect can grow because the offer changes uptake for more people, because it has a direct effect, or because the treatment effect differs. The outcome contrast alone does not distinguish these explanations.