Why Treatment Transitions Matter
In an encouragement study, assignment may change the chance that a person receives a treatment without determining receipt perfectly. Some people take the treatment when encouraged; others do not. Some may obtain it even without encouragement. To interpret these patterns, it helps to ask a counterfactual question about each person: what treatment would this person receive under each possible assignment?
The answer defines a compliance type. One-sided noncompliance restricts which answers are possible: no one receives treatment without encouragement. Two-sided noncompliance allows uptake in either assignment arm. These restrictions change which treatment transitions can occur and what can be inferred from differences in observed uptake.
The table of compliance types is a conceptual classification of people, not usually a table that can be directly assembled from observed records. Random assignment reveals group-level uptake rates, but it does not reveal both potential treatment receipts for each individual.
One-Sided Noncompliance
In the usual one-sided setting, treatment is unavailable, prohibited, or otherwise impossible without encouragement. This is the restriction \(D(0)=0\) for everyone. A person who is encouraged may still decline treatment, so \(D(1)\) can be either zero or one.
| Potential receipt without encouragement, \(D(0)\) | \(D(1)=0\): no receipt with encouragement | \(D(1)=1\): receipt with encouragement |
|---|---|---|
| \(D(0)=0\) | Never-takers: \((0,0)\) | Compliers: \((0,1)\) |
| \(D(0)=1\) | Impossible under one-sided noncompliance | Impossible under one-sided noncompliance |
Thus only two types are possible. Never-takers receive no treatment under either assignment. Compliers receive treatment if encouraged but not otherwise. There can be no always-takers, because that would require \(D(0)=1\), and no defiers, because that would also require treatment receipt without encouragement.
Worked Example: Building a One-Sided Compliance Table
Suppose a hypothetical program randomly assigns 100 people to an encouragement and 100 to no encouragement. Treatment is available only through the program, so assume the one-sided restriction holds. In a hypothetical roster that lists both potential receipts, 35 people have type \((0,1)\) and 165 have type \((0,0)\). Construct the table and determine the uptake rates in each assignment arm.
Place each person according to \((D(0),D(1))\). All 200 people have \(D(0)=0\); the 35 compliers have \(D(1)=1\), and the 165 never-takers have \(D(1)=0\).
| Type | Number | Receipt if \(Z=0\) | Receipt if \(Z=1\) |
|---|---|---|---|
| Never-takers \((0,0)\) | 165 | 0 | 0 |
| Compliers \((0,1)\) | 35 | 0 | 1 |
| Total | 200 | 0 | 35 |
If assigned no encouragement, nobody receives treatment, so the uptake rate is \(0/100=0\). If assigned encouragement, 35 of the 200-person roster would receive treatment. With equal random assignment, the expected uptake rate in that arm is \(35/200=0.175\), or 17.5%. In an actual experiment, the observed arm rate varies with the realized assignment; the latent roster itself is not observed.
Under random assignment, the one-sided restriction has a useful implication for the population proportion of compliers. Random assignment makes the encouraged arm representative of the population’s \(D(1)\) values, and one-sidedness sets everyone’s \(D(0)\) to zero. Therefore:
The observed uptake rate in the encouraged arm estimates the complier proportion. The no-encouragement arm should have zero uptake if the restriction is exact and receipt is measured correctly. An observed treatment receipt in that arm may point to a violation of one-sidedness, a treatment-access route not accounted for, or a measurement or implementation problem.
Two-Sided Noncompliance
In a two-sided setting, people can receive treatment in either assignment arm. For example, an encouragement may increase access without preventing people in the no-encouragement arm from obtaining treatment elsewhere. All four pairs of potential receipts are possible.
| \(D(0)\) | \(D(1)=0\): no receipt with encouragement | \(D(1)=1\): receipt with encouragement |
|---|---|---|
| \(D(0)=0\) | Never-takers: \((0,0)\) | Compliers: \((0,1)\) |
| \(D(0)=1\) | Defiers: \((1,0)\) | Always-takers: \((1,1)\) |
Always-takers receive treatment regardless of assignment; never-takers do not. Compliers change from no treatment to treatment when encouraged. Defiers change in the opposite direction: they receive treatment without encouragement but not when encouraged. The phrase two-sided noncompliance means that treatment receipt can occur in either assignment arm; it does not mean that every one of the four types must be present.
Worked Example: Different Compliance Tables, Same Uptake Rates
Consider two hypothetical populations of 100 people each. In both, the treatment uptake rate is 40% under \(Z=0\) and 60% under \(Z=1\). The following latent type counts are possible.
| Type | Population A | Population B | Receipt pair |
|---|---|---|---|
| Always-takers | 40 | 30 | \((1,1)\) |
| Never-takers | 40 | 30 | \((0,0)\) |
| Compliers | 20 | 30 | \((0,1)\) |
| Defiers | 0 | 10 | \((1,0)\) |
For Population A, 40 always-takers receive treatment under \(Z=0\), giving \(40/100=0.40\). Under \(Z=1\), the 40 always-takers and 20 compliers receive it, giving \((40+20)/100=0.60\).
For Population B, uptake under \(Z=0\) comes from 30 always-takers and 10 defiers: \((30+10)/100=0.40\). Under \(Z=1\), the 30 always-takers and 30 compliers receive treatment: \((30+30)/100=0.60\). Both populations therefore have the same observed uptake rates, despite having different compliance-type compositions.
The example shows why two arm-level uptake rates do not, by themselves, reveal the shares of all four types. In particular, their difference, \(0.60-0.40=0.20\), is a net change in uptake across assignment arms, not a direct count of every individual whose treatment receipt changes.
What an Encouragement Study Estimates
Random assignment supports a causal comparison of outcomes by assignment, subject to appropriate implementation and outcome measurement. That assignment contrast concerns the effect of being offered or assigned the encouragement, whether or not each person follows it. Imperfect uptake does not erase the meaning of this assignment effect.
A different question is what the outcome contrast says about receiving the treatment. That interpretation depends on the relationship between assignment, treatment uptake, and outcome. For example, if encouragement can affect the outcome through a route other than treatment receipt, an outcome difference between assignment arms includes that route as well. Even when such a direct route is ruled out, the pattern of possible treatment transitions matters: one-sided noncompliance permits only never-takers and compliers, whereas two-sided noncompliance permits all four types unless further restrictions are imposed.
Worked Example: Interpreting Uptake and Outcome Differences
In a hypothetical randomized encouragement study, 200 people are assigned to each arm. The treatment uptake rates are 45% under encouragement and 30% without encouragement. The mean outcome is 7.6 days in the encouraged arm and 8.2 days in the other arm, where lower values are better. Assume assignment was randomized, outcomes are measured consistently, and there is no interference between people.
The difference in uptake rates is \(0.45-0.30=0.15\), or 15 percentage points. The outcome difference by assignment is \(7.6-8.2=-0.6\) days. Its direct interpretation is that assignment to encouragement reduced the average outcome by 0.6 days in this hypothetical study.
Because uptake is positive even without encouragement, the one-sided restriction does not fit these data as stated. Under two-sided noncompliance, the 15-percentage-point uptake difference does not establish that exactly 15% of people are compliers: the difference can reflect both people who increase uptake and people whose uptake moves in the opposite direction. The assignment contrast remains an encouragement effect; interpreting it as an effect of treatment receipt would require additional assumptions about treatment transitions and how encouragement affects the outcome.
A Short Justification: Why One-Sidedness Identifies the Complier Share
The result for one-sided noncompliance follows directly from the definition of the types. If everyone has \(D(0)=0\), then the only people with \(D(1)=1\) are compliers. Random assignment ensures that, in expectation, the encouraged group has the population distribution of \(D(1)\). Thus its uptake probability equals the population probability of being a complier:
The first equality uses consistency: people assigned \(Z=1\) have observed receipt \(D=D(1)\). The second uses random assignment, which makes assignment independent of the potential receipts. The final equality uses the one-sided restriction \(D(0)=0\). Without that restriction, people with \(D(1)=1\) could include both compliers and always-takers, so the encouraged-arm uptake rate would not equal the complier share.
Common Mistakes and Practical Checks
- Calling any increase in uptake a complier count. In a two-sided setting, the difference in uptake rates is a net difference across arms. It need not equal the proportion of people who move from no treatment to treatment.
- Assuming noncompliance means only compliers and never-takers. That two-type table requires one-sidedness. If people can receive treatment without encouragement, always-takers and potentially defiers must be considered.
- Treating compliance type as observed. A person’s \(D(0)\) and \(D(1)\) cannot both be observed in the same study. Type tables classify potential behavior; arm-level rates do not reveal every person’s type.
- Calling an assignment effect a treatment effect. A randomized comparison by encouragement estimates the effect of assignment to encouragement. Further assumptions are needed to interpret that contrast as an effect of treatment receipt.
Check Your Understanding
Answer each question by referring to potential treatment receipts under both assignments.
- Under one-sided noncompliance, which compliance types are possible, and what restriction rules out the other two?
- In a two-sided setting, what type has potential receipts \((D(0),D(1))=(1,0)\)?
- Two randomized arms have uptake rates of 25% and 40%. Why is the 15-percentage-point difference not automatically the proportion of compliers in a two-sided setting?
- What does a randomized comparison of mean outcomes by encouragement assignment estimate directly?
- If the no-encouragement arm has positive treatment uptake, what one-sided assumption is contradicted, assuming uptake is measured correctly?