Why Monotonicity Matters
In an encouragement experiment, assignment may change whether a person receives a treatment without determining receipt perfectly. The causal effect of assignment is therefore not automatically the effect of treatment. The complier average causal effect (CACE) addresses this distinction: it describes the treatment effect for people whose treatment receipt is changed by assignment. Its interpretation depends on assumptions about how assignment affects receipt, especially monotonicity.
Monotonicity rules out a particular response: a person who would receive treatment if assigned to the control condition but would not receive it if assigned to the treatment condition. Such a person is called a defier. Ruling out defiers permits the difference in treatment uptake between assignment groups to represent the share of compliers. It does not, however, reveal which particular people are compliers.
Compliance Types and the Monotonicity Assumption
Let \(Z_i\in\{0,1\}\) denote the randomized assignment for unit \(i\), where \(Z_i=1\) is the encouragement or offer and \(Z_i=0\) is control. Let \(D_i(z)\in\{0,1\}\) be the treatment receipt that unit \(i\) would have under assignment \(z\). The pair \((D_i(0),D_i(1))\) defines that unit’s compliance type.
These types concern potential treatment receipt, not observed behavior alone. For each person, only \(D_i(Z_i)\) is observed; the receipt that would have occurred under the other assignment is counterfactual. Consequently, an observed person with \(Z_i=1,D_i=1\) might be an always-taker or a complier. Monotonicity is an assumption about all units’ potential responses, not a classification that can generally be checked person by person in the data.
The CACE also needs assumptions that connect assignment to outcomes. Write \(Y_i(d)\) for the outcome that unit \(i\) would have under treatment receipt \(d\). In the standard binary-instrument setup, the relevant conditions are:
- Random assignment: \(Z_i\) is independent of the potential receipts and outcomes in the target population.
- Exclusion: assignment affects the outcome only through treatment receipt, so the outcome under assignment \(z\) is \(Y_i(D_i(z))\).
- Monotonicity: \(D_i(1)\geq D_i(0)\) for every unit.
- Relevance: assignment changes treatment receipt on average, so \(E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]\neq 0\).
- Consistent treatment and no interference: the recorded treatment corresponds to the treatment being studied, and one unit’s assignment does not change another unit’s outcome.
Under these conditions, the CACE is the average treatment effect among compliers:
The expectation is over the complier population. It is not generally the average effect for all units, for those who actually received treatment, or for everyone who was offered treatment.
Worked Example: What Monotonicity Says About the Complier Share
Suppose a hypothetical randomized encouragement study has 200 people in each assignment group. Treatment receipt is recorded for everyone: 120 of 200 people assigned to encouragement receive treatment, compared with 60 of 200 assigned to control. Assume random assignment, consistent measurement of receipt, and monotonicity.
The receipt rates are \(120/200=0.60\) and \(60/200=0.30\), so the first-stage difference is:
With monotonicity, only compliers change their receipt when assignment changes: their receipt changes from 0 to 1. Always-takers and never-takers have no change, and defiers are ruled out. Thus the difference of 0.30 estimates a complier share of 30% in the target population. Random assignment supports interpreting the group rates as averages of the potential receipt rates.
This does not mean that the 30% of people treated only when encouraged can be picked out from the observed records. For example, someone assigned to encouragement who receives treatment could be a complier or an always-taker. The assumption and the observed rates identify a population proportion, not each individual’s latent type.
Why the Wald Ratio Is a Complier Effect
The assignment effect on the outcome is the reduced-form contrast \(E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0]\). The assignment effect on treatment receipt is the first-stage contrast. Under the assumptions above, their ratio equals the CACE. The key is that, with monotonicity, only compliers change treatment receipt across the two assignment states.
A short derivation shows where monotonicity enters. Let \(p_C\) be the proportion of compliers, and let \(p_F\) be the proportion of defiers. Because \(D_i(1)-D_i(0)\) equals 1 for a complier, \(-1\) for a defier, and 0 for either always-takers or never-takers, random assignment gives:
Under exclusion and random assignment, always-takers and never-takers have the same treatment receipt under either assignment, so they contribute no assignment-induced outcome difference. Compliers contribute their treatment effect, while defiers contribute the negative of theirs. Writing \(\tau_F=E[Y_i(1)-Y_i(0)\mid\text{defier}]\), the reduced-form contrast is:
Under monotonicity, \(p_F=0\). The first-stage contrast is then \(p_C\), and the reduced-form contrast is \(p_C\tau_C\). Relevance ensures \(p_C>0\), so dividing the latter by the former gives \(\tau_C\). This derivation also makes clear that a nonzero first stage alone does not establish monotonicity.
Worked Example: Interpreting an Estimated Complier Effect
Consider a hypothetical randomized encouragement trial with a continuous outcome measured in points. The mean outcome is 68.4 points in the encouraged group and 62.4 points in the control group. As in the previous hypothetical setup, receipt rates are 0.60 and 0.30. Suppose assignment was randomized; treatment receipt is well-defined; there is no interference; and exclusion and monotonicity are credible.
The reduced-form contrast is \(68.4-62.4=6.0\) points. The first-stage contrast is \(0.60-0.30=0.30\). Substituting into the Wald ratio gives:
Under the stated assumptions, the estimated treatment effect for compliers is an average increase of 20.0 points. The numerator is an effect of assignment, in outcome points; the denominator is the change in the probability of treatment receipt, a proportion. Dividing yields an effect in outcome points per change of one in treatment probability, with the interpretation given by the CACE assumptions.
This is not automatically the effect for all people in the trial. The result also depends on the assumptions: randomization justifies the assignment-group contrasts, while exclusion and monotonicity justify interpreting their ratio as a treatment effect among compliers.
What Changes If Defiers Are Plausible?
If defiers exist, the first stage is \(p_C-p_F\), not the complier share. The reduced form is \(p_C\tau_C-p_F\tau_F\). Consequently, the Wald ratio generally combines effects from two different groups and is not the CACE. Its value can differ substantially from the complier effect, even when assignment is randomized and exclusion holds.
Worked Example: A Wald Ratio With Defiers
Suppose, hypothetically, 40% of a population are compliers and 10% are defiers; the remaining 50% are always-takers or never-takers. Assume random assignment, exclusion, consistent treatment, and no interference, but do not assume monotonicity. Suppose the average treatment effect is 10 outcome units for compliers and 4 outcome units for defiers.
The first-stage difference is the complier share minus the defier share:
The reduced-form difference is the complier contribution minus the defier contribution:
Therefore, the Wald ratio is \(3.6/0.30=12\) outcome units. It is not the complier effect of 10 units. The extra value arises because defiers enter the numerator and denominator with negative signs; the ratio here is not even a straightforward average of the two group effects.
Monotonicity is usually not empirically testable from the assignment, receipt, and outcome data alone: each person’s two potential receipt states are not both observed. A positive first stage says that encouragement increases receipt on average. It does not rule out some people moving in the opposite direction. The substantive question is whether there is a credible reason that assignment cannot reduce receipt for any unit.
Treatment Versions Can Change the Estimand
Monotonicity is not the only condition needed for the usual interpretation. The treatment must also be sufficiently well-defined. If \(D_i=1\) combines materially different versions of treatment, then \(Y_i(1)\) may not refer to one common intervention. Moreover, if assignment changes which version a person receives, assignment may affect outcomes through that version even after conditioning on the binary indicator \(D_i\). In that case, exclusion for the binary treatment can fail.
Worked Example: An Encouragement That Changes Treatment Version
Suppose a hypothetical study evaluates whether students attend any tutoring, recorded as \(D=1\). Students randomly assigned an encouragement are offered weekly one-to-one tutoring, while students who seek tutoring without the encouragement usually attend a short group session. Assume encouragement raises the probability of attending, and there are no defiers.
Monotonicity may be plausible for the binary indicator “attends any tutoring.” But the encouraged and unencouraged students classified as treated may receive different amounts and forms of tutoring. If those versions have different effects, the binary-treatment contrast does not compare receipt of one common intervention. Assignment may also affect outcomes through tutoring intensity or format, so the exclusion restriction for binary \(D\) is not justified merely by observing that assignment changes attendance.
A careful analysis would define the intervention and estimand to reflect the versions that matter, or justify why the versions can be treated as equivalent for the scientific question. Without that clarification, calling the Wald ratio “the effect of tutoring among compliers” overstates what the design identifies.
Common Mistakes and Practical Interpretation
- Calling the observed treatment group the complier group. Receipt under one assignment does not reveal the counterfactual receipt under the other assignment. The CACE concerns a latent subgroup.
- Claiming that a positive first stage proves monotonicity. A positive average change is compatible with some defiers. Monotonicity requires no unit to respond in the opposite direction.
- Reporting the CACE as an effect for everyone. The estimand is an average effect for compliers, whose membership depends on the assignment mechanism and population.
- Using the ratio without checking treatment meaning. If assignment changes treatment versions or affects outcomes through another route, the binary-treatment exclusion restriction may fail.
The central interpretive lesson is that monotonicity gives the first stage its role as the complier proportion and lets the reduced form be scaled to a complier effect. It does not make the assumptions automatic, identify every complier, or turn a context-dependent effect into a population-wide treatment effect. In an application, state the target population, describe why defiers are unlikely, and define treatment versions clearly before interpreting the ratio.
Check Your Understanding
Answer each question by connecting the assumption to the quantity it identifies.
- What potential receipt pattern defines a defier, and what inequality rules that pattern out?
- Under monotonicity, what does a first-stage difference of 0.25 identify? Does it identify which individuals are compliers?
- State the assumptions that allow the Wald ratio to be interpreted as the complier average causal effect.
- If 30% are compliers and 10% are defiers, what is the first-stage difference in terms of these shares?
- How could an encouragement that changes treatment intensity undermine the usual CACE interpretation, even if there are no defiers?