Randomized Encouragement With Incomplete Uptake

Why Incomplete Uptake Requires More Than an Offer Rate

In a randomized encouragement design, the instrument \(Z\) indicates whether a person is assigned an offer or encouragement, while \(D\) indicates whether that person actually receives the treatment. These are different variables: an offered person may decline, and someone assigned no offer may obtain the treatment elsewhere. Consequently, the treatment rate among those offered is not, by itself, the proportion who comply with the offer.

The treatment patterns in both randomized groups help describe the latent compliance types. The central idea is to compare the probability of treatment receipt under assignment to an offer with the probability under assignment to no offer. With random assignment, those two group rates estimate the two population probabilities that define the types. With monotonicity, their difference identifies the complier share.

Definition: Let \(D(1)\) be a person's potential treatment receipt if assigned an offer, and let \(D(0)\) be that person's potential treatment receipt if assigned no offer. Each is binary. The pair \((D(1),D(0))\) defines the person's compliance type: always-takers have \((1,1)\), never-takers have \((0,0)\), compliers have \((1,0)\), and defiers have \((0,1)\).

A person’s type is defined by both potential receipts, but only one receipt is observed for each person. If \(Z=1\), the observed receipt is \(D(1)\); if \(Z=0\), it is \(D(0)\). Therefore, types are not generally identified person by person from the observed data. The goal here is to identify or estimate their population shares from randomized-group treatment rates.

Assumptions and the Share Calculation

Write \(q_1=P(D=1\mid Z=1)\) for the treatment-receipt rate in the offer group and \(q_0=P(D=1\mid Z=0)\) for the rate in the no-offer group. To interpret these observed rates as the potential-receipt probabilities \(P(D(1)=1)\) and \(P(D(0)=1)\), assignment must be randomized, treatment receipt must be recorded consistently with the assigned condition, and one person's assignment must not change another person's treatment receipt. In particular, randomization gives \(Z\) independence from the potential treatment receipts.

The key additional assumption for the simple three-type calculation is monotonicity: no person would receive treatment when assigned no offer but not receive it when assigned an offer. In symbols, \(D(1)\geq D(0)\) for every person. This rules out defiers. It does not say that everyone accepts an offer, or that nobody in the no-offer group receives treatment.

Result: Under randomized assignment and monotonicity, the population shares are \(\pi_{AT}=q_0\), \(\pi_{NT}=1-q_1\), and \(\pi_C=q_1-q_0\), where \(AT\), \(NT\), and \(C\) denote always-takers, never-takers, and compliers. The difference \(q_1-q_0\) must be nonnegative in the population.

The result follows by relating each assignment group's uptake rate to the types. In the no-offer condition, always-takers and defiers would receive treatment. Under monotonicity, there are no defiers, so the receipt rate in that condition is the always-taker share. In the offer condition, always-takers and compliers receive treatment; subtracting the always-taker share from that rate leaves the complier share. Everyone who does not receive treatment under the offer is a never-taker.

A Short Derivation

Random assignment lets us use the observed conditional rates to represent the marginal potential-receipt rates: \(q_1=P(D(1)=1)\) and \(q_0=P(D(0)=1)\). Let \(\pi_{AT},\pi_{NT},\pi_C,\pi_D\) denote the four type shares. Counting types with potential receipt equal to one gives

q_1=\pi_{AT}+\pi_C,\qquad q_0=\pi_{AT}+\pi_D.

Monotonicity sets \(\pi_D=0\), so \(q_0=\pi_{AT}\). Substituting this into the first equation gives \(\pi_C=q_1-q_0\). The four shares sum to one, and with no defiers the remaining share is \(\pi_{NT}=1-\pi_{AT}-\pi_C=1-q_1\). This also shows why, without monotonicity, the difference \(q_1-q_0\) equals \(\pi_C-\pi_D\), not the complier share alone. Uptake rates alone cannot separate compliers from defiers in that case.

Worked Example: Uptake in Both Randomized Groups

Suppose a hypothetical encouragement study randomly assigns 400 people to receive an offer and 400 to receive no offer. Of those offered, 280 receive the treatment; of those not offered, 80 obtain it elsewhere. The observed receipt rates are

\hat q_1=\frac{280}{400}=0.70,\qquad \hat q_0=\frac{80}{400}=0.20.

Assume assignment was implemented as randomized, treatment receipt is measured in the same way in both groups, there is no interference, and monotonicity is substantively plausible. Then the estimated shares are

\hat\pi_{AT}=0.20,\qquad \hat\pi_{NT}=1-0.70=0.30,\qquad \hat\pi_C=0.70-0.20=0.50.

The estimates sum to \(0.20+0.30+0.50=1.00\). In this hypothetical sample, the estimated always-taker share is 20%, the never-taker share is 30%, and the complier share is 50%. In particular, the 70% uptake rate among those offered is not the complier share: it includes both always-takers and compliers. These are sample estimates of population shares, not exact classifications of the 800 individuals.

Worked Example: Reading the Two Groups Separately

Consider a second hypothetical randomized study. Among 200 people assigned no offer, 50 receive the treatment. Among 300 assigned an offer, 210 receive it. Suppose randomization, consistent measurement, no interference, and monotonicity are credible.

The rates are \(\hat q_0=50/200=0.25\) and \(\hat q_1=210/300=0.70\). The estimated shares are \(\hat\pi_{AT}=0.25\), \(\hat\pi_{NT}=1-0.70=0.30\), and \(\hat\pi_C=0.70-0.25=0.45\). These shares offer a useful check on how each assignment group’s receipt rate is composed. For the offer group, the estimated 70% receiving treatment consists of 25% always-takers plus 45% compliers. For the no-offer group, the estimated 25% receiving treatment consists of always-takers; compliers do not receive treatment under no offer.

As a count-based check, applying these estimated proportions to the 300-person offer group gives about \(0.25(300)=75\) always-takers and \(0.45(300)=135\) compliers, or \(75+135=210\) treatment recipients. Applying them to the 200-person no-offer group gives about \(0.25(200)=50\) always-takers receiving treatment. These calculations explain the group rates under the model; they do not reveal which particular participants are compliers.

Diagnostics and What a Contradictory Pattern Means

With population probabilities and monotonicity, \(q_1-q_0\) cannot be negative: it is the complier share. In sample data, however, the estimated difference can be negative because the group rates vary from sample to sample. A negative sample difference should prompt both a check of the data and an assessment of sampling uncertainty; it is not automatically proof that monotonicity fails. If a well-measured population pattern has a lower treatment rate under the offer, then the combination of randomized assignment and no defiers cannot account for that pattern.

Worked Example: A Negative Sample Difference

In a hypothetical study, 84 of 200 people assigned an offer receive treatment, while 94 of 200 assigned no offer receive it. Thus \(\hat q_1=84/200=0.42\), \(\hat q_0=94/200=0.47\), and the estimated difference is \(-0.05\). Plugging this into the monotonicity formula would give a negative estimated complier share, which is impossible as a population proportion.

The rates are estimates, so quantify their sampling uncertainty before treating the pattern as a population contradiction. For independent randomized groups, an approximate standard error for the difference in receipt rates is

SE(\hat q_1-\hat q_0) =\sqrt{\frac{0.42(1-0.42)}{200}+\frac{0.47(1-0.47)}{200}} =\sqrt{0.0024635} \approx 0.04963.

An approximate 95% interval for the difference is \(-0.05\pm1.96(0.04963)\), or about \((-0.147,\,0.047)\). Because this interval includes zero, the observed negative difference is not, by itself, strong evidence that the population difference is negative. The interval is an uncertainty summary for the difference in rates, not a valid interval for each compliance-type share. A researcher should also check randomization, denominators, missing receipt data, and whether the offer could plausibly reduce anyone’s treatment receipt.

Further diagnostics concern the design and interpretation, not just the arithmetic. Confirm that \(Z\) really was randomized, that treatment receipt has the same definition in both groups, and that group-specific denominators reflect the intended analysis population. Check whether the offer could discourage treatment receipt for some people; if so, monotonicity may not be credible. Finally, remember that group rates identify shares only at the population level under the assumptions. They do not label individuals’ types.

What These Shares Do—and Do Not—Establish

The compliance-share calculation uses assignment and treatment receipt. It does not require the exclusion restriction, because no outcome \(Y\) appears in the calculation. It also does not, by itself, establish a treatment effect. Interpreting outcome contrasts using compliance types requires additional assumptions, including assumptions about how assignment may affect outcomes; those are separate from identifying the uptake pattern.

The alternatives answer different questions. An assignment-based comparison preserves the randomized groups and describes the effect of being assigned the offer, regardless of uptake. A comparison of people who received treatment with people who did not is instead a comparison by observed behavior; uptake may be related to characteristics that also predict outcomes. The compliance-share calculation describes how receipt responds to assignment under its assumptions. It should not be mistaken for either an outcome-effect estimate or a person-level classification.

A common mistake is to call everyone in the offer group who receives treatment a complier. That reasoning ignores people who would receive treatment even without an offer. For example, if 70% receive treatment when offered and 20% receive it without an offer, the complier share is estimated as \(70\%-20\%=50\%\), not 70%. Under monotonicity, the no-offer receipt rate accounts for always-takers, while the offer-group nonreceipt rate accounts for never-takers.

The estimates have practical limitations. Sampling variation can make estimated shares imprecise or push the estimated complier share below zero. Weak differences in uptake leave the complier share small and make the offer a limited shifter of treatment receipt. And the assumptions are not verified merely because the arithmetic produces shares between zero and one: a plausible monotonicity argument and a sound randomization process remain essential.

Key takeaway: With randomized assignment and no defiers, the no-offer treatment rate estimates the always-taker share, the offer-group nonreceipt rate estimates the never-taker share, and the difference in treatment rates estimates the complier share. These are population-share results, not individual labels or outcome effects.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a randomized design, treatment receipt is 65% in the offer group and 15% in the no-offer group. Under monotonicity, what are the estimated shares of always-takers, never-takers, and compliers?
Why is treatment receipt in the no-offer group informative about always-takers, and what assumption is needed for that interpretation?
What does a negative estimated difference in uptake rates imply immediately? What additional check should be made before concluding that the population assumptions are contradicted?
Do randomized assignment and monotonicity alone identify the effect of treatment receipt on an outcome? Explain briefly.

Solutions. (1) The estimates are \(\hat\pi_{AT}=0.15\), \(\hat\pi_{NT}=1-0.65=0.35\), and \(\hat\pi_C=0.65-0.15=0.50\); they sum to one. (2) Under monotonicity, anyone who receives treatment without the offer must also receive it with the offer, so that person is an always-taker. Random assignment is also needed to interpret the observed no-offer rate as the population potential-receipt rate under no offer. (3) The sample estimate cannot be a valid negative population share under monotonicity, but sampling variation can produce a negative sample difference. Check the data and quantify uncertainty before treating it as evidence against the population pattern. (4) No. These assumptions identify compliance-type shares from treatment receipt; identifying an outcome effect requires additional assumptions about outcomes and assignment.