An Encouragement Design With Treatment Noncompliance

Why Encouragement Designs Need Two Effects

In an encouragement design, participants are randomly assigned an offer, reminder, referral, or other encouragement intended to increase treatment uptake. Assignment is randomized, but treatment itself is not necessarily received by everyone assigned the encouragement, and some participants in the comparison group may obtain treatment anyway. This incomplete uptake is called noncompliance.

The first causal question is therefore about assignment: what is the effect of being assigned the encouragement? A second question concerns treatment: what is the effect of receiving treatment for the participants whose uptake is changed by the encouragement? These are different estimands. Randomization directly supports the first; interpreting the second requires additional instrumental-variable assumptions.

Definition: Let \(Z=1\) denote assignment to encouragement and \(Z=0\) assignment to the comparison condition. Let \(D=1\) indicate treatment uptake and \(D=0\) no uptake, and let \(Y\) be the outcome. The assignment effect (or intention-to-treat effect) is the difference in mean outcomes by assigned group, \(E(Y\mid Z=1)-E(Y\mid Z=0)\). The first-stage effect is the difference in treatment uptake, \(E(D\mid Z=1)-E(D\mid Z=0)\). The complier average causal effect (CACE) is the average effect of treatment among participants whose uptake is changed by assignment.

For a binary outcome, the assignment effect is a risk difference, measured in percentage points or as a proportion. For a quantitative outcome, it is a difference in means, measured in the outcome's units. The first-stage effect is a difference in uptake proportions. If the assumptions below hold and the first stage is nonzero, the Wald estimate divides the assignment effect by the first-stage effect.

Assumptions for a Complier Effect

Use potential outcomes to state what the assumptions mean. Write \(D(z)\) for the treatment uptake a participant would have under assignment \(z\), and \(Y(d,z)\) for the outcome under treatment status \(d\) and assignment \(z\). The observed values are \(D=D(Z)\) and \(Y=Y(D(Z),Z)\). A participant's compliance type is determined by the pair \((D(0),D(1))\).

Random assignment: \(Z\) is independent of the potential uptake and outcome values. In a randomized study this follows from the assignment procedure, subject to correct implementation and follow-up that does not undermine the comparison.
Consistency and no interference: each participant's observed uptake and outcome agree with the potential values under their actual assignment and uptake, and one participant's assignment or treatment does not affect another participant's outcome.
Relevance: assignment changes uptake in the study population, so \(E(D\mid Z=1)-E(D\mid Z=0)\ne 0\). A larger first stage generally provides more information about a treatment effect than a very small first stage.
Exclusion restriction: assignment affects the outcome only through treatment uptake. Formally, \(Y(d,1)=Y(d,0)\) for each treatment status \(d\). An offer that also changes access to other care or provides a direct benefit may violate this condition.
Monotonicity: no participant does the opposite of the encouragement's intended uptake effect: \(D(1)\ge D(0)\) for every participant. This rules out defiers, who would take treatment when not encouraged but not take it when encouraged.

Randomization identifies the assignment effect without exclusion or monotonicity. Relevance, exclusion, and monotonicity are needed to interpret the Wald ratio as the CACE. The ratio is not automatically a population average treatment effect: it concerns compliers, a group defined by how uptake responds to assignment.

Why the Wald Ratio Is the Complier Effect

There are four possible compliance types. Compliers have \((D(0),D(1))=(0,1)\); always-takers have \((1,1)\); never-takers have \((0,0)\); and defiers have \((1,0)\). Under monotonicity, there are no defiers. Under exclusion, always-takers and never-takers have no outcome change caused by assignment, because their treatment status does not change and assignment has no other route to the outcome.

Let \(C\) denote membership in the complier group. Random assignment makes the expected outcome difference between assigned groups equal to the average effect of changing assignment across the population. With exclusion, only compliers change treatment status, so the assignment effect is their population share times their average treatment effect. The first-stage difference is the same complier share: the other two types do not change uptake.

E(Y\mid Z=1)-E(Y\mid Z=0) = P(C)\,E\{Y(1)-Y(0)\mid C\}, \qquad E(D\mid Z=1)-E(D\mid Z=0)=P(C).

Dividing the first equality by the second gives the Wald identification result:

\frac{E(Y\mid Z=1)-E(Y\mid Z=0)} {E(D\mid Z=1)-E(D\mid Z=0)} =E\{Y(1)-Y(0)\mid C\}.

This is a population result under the assumptions; replacing expectations with observed sample means and proportions gives an estimate. With binary \(Y\), the CACE is a risk difference among compliers. With continuous \(Y\), it is a mean difference in the outcome's units among compliers.

Worked Example: A Binary Outcome With Incomplete Uptake

Suppose a hypothetical randomized study assigns 200 participants to a reminder and 200 to usual information. Treatment is a preventive service, and the outcome is an adverse event during follow-up. In the reminder group, 130 participants take up treatment and 26 have the event. In the comparison group, 50 take up treatment and 36 have the event.

Assigned group	Participants	Treatment uptake	Adverse events
Reminder	200	130	26
Usual information	200	50	36

Assignment effect. The event risks are \(26/200=0.130\) and \(36/200=0.180\). Thus the estimated assignment effect is \(0.130-0.180=-0.050\), a 5-percentage-point lower event risk under assignment to the reminder. Because assignment is randomized, this estimates the causal effect of assignment, assuming outcomes are measured comparably and follow-up is adequately handled.

First stage. Uptake is \(130/200=0.650\) in the reminder group and \(50/200=0.250\) in the comparison group. The first-stage estimate is \(0.650-0.250=0.400\), a 40-percentage-point increase in treatment uptake.

Wald estimate. The ratio is

\widehat{\text{CACE}} =\frac{-0.050}{0.400} =-0.125.

Under random assignment, consistency and no interference, relevance, exclusion, and monotonicity, this estimates a 12.5-percentage-point reduction in event risk among compliers. The assignment effect itself remains a 5-percentage-point reduction in the full randomized population. The two estimates answer different questions and should not be reported as interchangeable.

Checking What the Data and Design Support

A sound analysis checks the design and the observed data separately. The randomized assignment protocol supports the claim that \(Z\) is independent of potential outcomes; baseline comparisons can reveal implementation or data problems, but balance on measured variables does not prove randomization or prove the absence of unmeasured imbalance. Check that assignment groups are defined as randomized, outcome denominators are correct, and missing outcomes or exclusions have not created a selected comparison.

The observed first stage is a direct diagnostic of instrument relevance. Report both uptake proportions and their difference, rather than only the Wald ratio. If the difference is near zero, the ratio is unstable: small changes in the observed assignment effect can produce large changes in the ratio. A positive, substantial first stage is evidence that assignment changed uptake, but it does not establish exclusion or monotonicity.

Exclusion and monotonicity are substantive design assumptions, not facts that can generally be verified from the observed group summaries. Ask whether the encouragement could alter the outcome through another route, such as extra clinical contact, information, or access to services besides the treatment being studied. Ask whether any participants might be discouraged from taking treatment that they otherwise would have received. These questions require knowledge of how the encouragement and treatment operate.

Worked Example: A Continuous Outcome

In a second hypothetical randomized study, assignment to an appointment-navigation call is intended to increase attendance at a treatment program. There are 120 participants in each group. In the call group, 84 attend and the mean outcome is 72 points; in the comparison group, 36 attend and the mean is 68 points. Assume higher scores indicate better functioning.

Assignment effect: \(72-68=4\) points. This is the estimated causal effect of assignment to the call on mean functioning, given valid randomization and outcome follow-up.

First stage: \(84/120-36/120=0.700-0.300=0.400\), a 40-percentage-point increase in attendance.

Wald estimate:

\widehat{\text{CACE}} =\frac{72-68}{0.700-0.300} =\frac{4}{0.400} =10\text{ points}.

If the call affects functioning only by changing attendance, there are no defiers, and the other assumptions hold, the estimate is a 10-point average treatment effect among compliers. It is not necessarily the effect for all participants, and the 4-point assignment effect remains the relevant estimate for the policy of offering the call to this study population.

Worked Example: A Direct Effect Makes the Ratio Misleading

Consider a third hypothetical randomized study of a counseling program. Assignment to encouragement also provides participants with a transport voucher, which can improve the outcome independently of counseling. In 100 participants per group, counseling uptake is 70% with encouragement and 50% without it. The success proportions are 60% and 54%, respectively.

The assignment effect is \(0.60-0.54=0.06\), or 6 percentage points. The first stage is \(0.70-0.50=0.20\), or 20 percentage points. The numerical Wald ratio is

\frac{0.06}{0.20}=0.30.

The 6-percentage-point difference can still estimate the causal effect of assignment under valid randomization and outcome measurement. But the ratio cannot be interpreted as a 30-percentage-point CACE if the voucher itself affects success: exclusion is violated, because assignment can affect the outcome without changing counseling uptake. The calculation is arithmetically correct; its treatment-effect interpretation is not justified by this design.

Choosing the Right Interpretation

The intention-to-treat analysis compares participants according to randomized assignment, regardless of whether they take up treatment. It preserves the randomized comparison and answers the policy question about offering the encouragement. The Wald analysis uses assignment as an instrument for uptake to target a complier effect, but depends on the additional assumptions. A per-protocol comparison instead groups participants by what they actually did. Because uptake is not randomized, that comparison can be confounded by health, motivation, access, or other factors related to both uptake and outcome.

In practice, report the assignment effect and first-stage effect alongside any Wald estimate. State units and the target population for each. When the first stage is weak, ordinary ratio estimates can be especially uncertain; use appropriate instrumental-variable uncertainty methods rather than treating the ratio as precise because it has a convenient numerical value. The introductory material on instrumental variables provides the broader framework; here the central design lesson is that random encouragement alone does not establish every condition needed for a complier-effect claim.

Common mistake: “Treatment was randomized, so comparing people who received treatment with people who did not gives the treatment effect.” In an encouragement design, assignment is randomized, not necessarily uptake. The randomized comparison estimates the effect of assignment. A treatment effect for compliers requires the first-stage ratio and its additional assumptions; comparing actual recipients directly can reintroduce confounding.

Check Your Understanding

For each question, identify the relevant estimand or calculation before reading the solution.

A study has uptake of 80% in the encouraged group and 50% in the comparison group. What is the estimated first-stage effect?
The assignment effect on a continuous outcome is \(-3\) points, and the first stage is \(0.30\). Calculate the Wald estimate and state its conditional interpretation.
Why can a randomized assignment effect be interpreted causally even if exclusion fails?
Suppose some participants would take the treatment only without encouragement and would refuse it if encouraged. Which assumption is threatened, and why does that matter?
A report gives a large Wald estimate but does not report the uptake proportions. What diagnostic information is missing?

Solutions. (1) \(0.80-0.50=0.30\), a 30-percentage-point increase in uptake. (2) \((-3)/0.30=-10\) points. Under the IV assumptions, this estimates a 10-point reduction in the outcome among compliers; it is not automatically the effect for everyone. (3) Random assignment identifies the effect of assignment by comparing randomized groups. Exclusion is needed to attribute that assignment effect solely to treatment uptake, not to estimate the assignment effect itself. (4) Monotonicity is threatened because these participants are defiers. Without monotonicity, the first stage no longer equals the complier share, and the simple ratio need not equal the CACE. (5) The first-stage magnitude and its uncertainty are missing. Without them, readers cannot assess instrument relevance or whether a small denominator may make the ratio unstable.