Introduction
In a randomized encouragement study, assignment may offer access, a referral, or an incentive for treatment without forcing anyone to take it. A particularly clear design is one-sided noncompliance: treatment cannot be received without encouragement, although some encouraged people may still decline. This structure rules out some compliance types and simplifies the relation between the treatment take-up contrast and the proportion of people whose treatment is changed by encouragement.
The restriction is about what could happen to each person under either assignment, not merely about the treatment rates observed in a sample. That distinction matters: an observed zero treatment rate among controls is evidence to examine, but the design and its treatment-access rules must support the stronger claim that no one could receive treatment without encouragement.
Potential Treatment and the Possible Compliance Types
Let \(Z_i\) indicate encouragement assignment for person \(i\), with \(Z_i=1\) for encouragement and \(Z_i=0\) for no encouragement. Let \(D_i(z)\) be the treatment that person \(i\) would receive if assigned \(Z_i=z\), where \(D_i(z)\) is binary. A person's compliance type is the pair \((D_i(0),D_i(1))\).
There are four possible pairs before imposing design restrictions. Under one-sided noncompliance, any type with \(D_i(0)=1\) is ruled out. The remaining types are:
| Compliance type | \(D_i(0)\) | \(D_i(1)\) | Meaning under one-sided access |
|---|---|---|---|
| Never-taker | 0 | 0 | Does not receive treatment with or without encouragement |
| Complier | 0 | 1 | Receives treatment when encouraged, not otherwise |
| Always-taker | 1 | 1 | Ruled out because treatment is unavailable without encouragement |
| Defier | 1 | 0 | Ruled out because treatment is unavailable without encouragement |
Consequently, one-sided noncompliance implies \(D_i(1)\ge D_i(0)\) for every person. In this design, the no-defiers condition follows from the stronger access restriction; it is not an additional restriction needed to exclude defiers. The restriction does not eliminate never-takers. If some encouraged people refuse treatment, they are never-takers, not failures of randomization.
Worked Example: Classifying Types in an Access-Limited Program
Suppose a hypothetical trial randomly assigns 600 patients either an invitation to attend a treatment clinic or usual information. The clinic accepts only patients with an invitation, so a patient without an invitation cannot receive the treatment through the study. Among those invited, some attend and some do not. Classify the possible compliance types.
For every patient, the access rule gives \(D_i(0)=0\). If a patient would attend when invited, then \((D_i(0),D_i(1))=(0,1)\), so that patient is a complier. If the patient would not attend even when invited, then \((D_i(0),D_i(1))=(0,0)\), so that patient is a never-taker.
An always-taker would have \(D_i(0)=1\), and a defier would also have \(D_i(0)=1\). Both contradict the stated access rule, so neither is possible in this design. The number of compliers cannot be read off from the invited patients alone as a complete list of individual types; it is a population proportion that can be related to the randomized-group take-up contrast.
From One-Sided Access to the Wald Estimand
Write \(Y_i(d)\) for the outcome person \(i\) would have under treatment status \(d\), for \(d=0,1\). For a causal interpretation of the Wald ratio, the key assumptions must be stated. Encouragement is randomized, so \(Z_i\) is independent of the potential treatment values and potential outcomes. The observed treatment is \(D_i=D_i(Z_i)\), and the observed outcome is \(Y_i=Y_i(D_i)\); the latter notation invokes the exclusion restriction that encouragement affects the outcome only through treatment. Outcomes and treatment are well-defined under each assignment, and one person's assignment does not change another person's outcome or treatment. Finally, the one-sided restriction holds and the encouragement changes treatment receipt for a positive proportion of people, so the first-stage contrast is not zero.
The population Wald estimand is the effect of encouragement on the mean outcome divided by its effect on the mean treatment receipt:
Under the assumptions above, this ratio identifies the average treatment effect among compliers, often called the complier LATE. To see why, let \(p_C=P(D_i(0)=0,D_i(1)=1)\) be the proportion of compliers. Since the only possible types are compliers and never-takers, \(D_i(1)-D_i(0)\) equals 1 for a complier and 0 for a never-taker. Random assignment therefore gives the first-stage contrast \(E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]=p_C\).
For the outcome contrast, random assignment and exclusion let us write it as \(E[Y_i(D_i(1))-Y_i(D_i(0))]\). For a never-taker, both potential treatment values are zero, so this difference is \(Y_i(0)-Y_i(0)=0\). For a complier, the difference is \(Y_i(1)-Y_i(0)\). Thus the outcome contrast is \(p_C E[Y_i(1)-Y_i(0)\mid C]\). Dividing by the first stage \(p_C>0\) yields:
This result does not say that the ratio is the average treatment effect for everyone in the study. It is the average treatment effect for the people whose treatment status is changed by encouragement. In a one-sided design, those people are precisely the compliers.
Worked Example: Calculating the Wald Effect
Consider a hypothetical randomized study of an invitation to a rehabilitation program. There are 500 people in each assignment group. Treatment is the program, and the outcome is days of symptoms over a fixed follow-up period; fewer days are better. The mean treatment receipt is 0 among the 500 people not invited and 0.36 among the 500 invited. The mean outcome is 9.3 days without an invitation and 8.1 days with an invitation. Assume the access rule ensures no treatment without invitation.
The first-stage contrast is:
The encouragement-group treatment rate is \(180/500=0.36\), and the control-group rate is \(0/500=0\). Under one-sided access, the population first stage corresponds to the complier proportion; in this sample, the observed treatment-rate contrast is 0.36.
The reduced-form outcome contrast, taking invited minus not invited, is:
The sample Wald estimate is therefore:
Subject to the randomized-assignment, exclusion, and other stated assumptions, this estimate indicates that treatment reduces symptoms by an average of about 3.33 days among compliers. It is not an estimate of the effect for all 1,000 people: never-takers' treatment status is not changed by the invitation. The calculation also does not establish that the assumptions are true; for example, the invitation must not affect symptoms through some route other than program participation.
What the First Stage Tells You—and What It Does Not
In a one-sided design, the population first-stage contrast simplifies to \(E[D_i\mid Z_i=1]\), because \(E[D_i\mid Z_i=0]=0\). It is also the proportion of compliers. This is useful for describing how strongly encouragement changes treatment receipt. A small first stage means that the Wald ratio divides by a small number, so relatively small changes in the outcome contrast can produce large changes in the estimate.
The simplification depends on the one-sided restriction. If people in the non-encouraged group can obtain treatment through another route, the first stage is instead the difference between the two treatment rates. It is no longer simply the encouraged group's treatment rate, and one cannot use the one-sided classification to conclude that only compliers and never-takers are possible.
Worked Example: Checking Whether One-Sided Access Fits the Data
Suppose a hypothetical randomized study reports that 9 of 300 people assigned to no encouragement received treatment, while 93 of 300 people assigned encouragement received it. Calculate the observed rates and assess the one-sided description.
The no-encouragement rate is \(9/300=0.030\), or 3.0%. The encouraged rate is \(93/300=0.310\), or 31.0%. Their observed difference is:
The control group has observed treatment receipt, so the data do not fit a strict claim that nobody can receive treatment without encouragement. If the study protocol allowed outside access, the design itself also fails to guarantee \(D_i(0)=0\). It would be incorrect to call the 31.0% treatment rate the complier proportion under one-sided access or to classify all participants as only compliers and never-takers. The 0.280 rate difference is the observed first stage, but a causal interpretation of its Wald ratio would require evaluating the appropriate assumptions for the actual design.
A zero observed control-group rate would not, by itself, prove that \(D_i(0)=0\) for every person in the target population. The rule should be supported by how treatment access was implemented, and the observed data should be checked for violations such as outside receipt or recording errors.
Common Mistakes and Practical Checks
- Confusing one-sided access with guaranteed take-up. The restriction says no one is treated without encouragement; it does not say everyone encouraged receives treatment. Encouraged nonparticipants can be never-takers.
- Calling the encouraged treatment rate the complier proportion without checking the control rate. That equality follows only when the population control treatment rate is zero and assignment is randomized. If control participants can receive treatment, use the treatment-rate difference as the first stage and do not impose the one-sided type classification.
- Interpreting the Wald ratio as an effect on everyone. Under the stated assumptions, it is an effect among compliers. Generalizing beyond them requires additional justification.
- Treating a randomized encouragement as proof of exclusion. Randomization supports independence of assignment from potential outcomes and treatment. It does not rule out a direct effect of encouragement on the outcome, such as an invitation changing health behavior even without treatment.
Check Your Understanding
Use the potential-treatment notation and distinguish design restrictions from conclusions based on sample rates.
- If \(D_i(0)=0\) for everyone, which of the four compliance types remain possible, and why?
- Does one-sided noncompliance mean every person assigned encouragement receives treatment? Explain.
- In a one-sided randomized design, the encouraged treatment rate is 0.28 and the control treatment rate is 0. What does the population first stage represent?
- A hypothetical study has an outcome contrast of 0.9 points and a first-stage contrast of 0.30. Calculate the Wald ratio and state the assumptions needed to interpret it as a causal effect.
- Why does observing no treated people in a finite control sample not, by itself, prove one-sided access for the population?