Why Monotonicity Is a Unit-Level Assumption
An instrument may encourage or facilitate treatment, but people can respond differently to the same encouragement. Some may take up treatment only when encouraged; others may take it either way or neither way. The key question for monotonicity is more specific: could the instrument make any individual less likely to receive treatment?
To make that question precise, let \(Z\) be a binary instrument, with \(Z=1\) denoting the instrument’s active or encouraging condition and \(Z=0\) its comparison condition. Let \(D_i(z)\) be unit \(i\)’s potential treatment status under assignment \(z\), where \(D_i(z)\) is either 0 or 1. Each unit has two such potential statuses, \(D_i(0)\) and \(D_i(1)\), even though in a study only the status under the unit’s actual assignment is observed.
The inequality applies to each unit’s pair of potential treatment statuses, not merely to an average or to a tendency in the population. Because treatment is binary, the possible pairs are \((D_i(0),D_i(1))=(0,0),(0,1),(1,1)\), and \((1,0)\). Monotonicity rules out exactly the last pair.
Principal Strata: The Four Response Types
The four possible pairs define principal strata, groups formed by each unit’s potential response to the two instrument assignments. The names describe treatment response, not observed baseline characteristics. A unit belongs to one stratum whether or not both of its potential statuses can be seen in the data.
| Type | \(D_i(0)\) | \(D_i(1)\) | Response to \(Z=1\) |
|---|---|---|---|
| Never-taker | 0 | 0 | Does not receive treatment under either assignment |
| Complier | 0 | 1 | Receives treatment only under the encouraging assignment |
| Always-taker | 1 | 1 | Receives treatment under either assignment |
| Defier | 1 | 0 | Receives treatment only under the comparison assignment |
The names are relative to the chosen direction of the instrument. If \(Z=1\) is defined as a stronger encouragement, a complier moves toward treatment when assigned \(Z=1\), while a defier moves away from it. If the instrument’s coding or direction is reversed, the interpretation of the labels must be reversed too. Monotonicity is not a claim that everyone complies: it allows never-takers and always-takers as well as compliers.
This assumption concerns treatment response only. It does not say that the instrument is assigned independently of potential outcomes, that the instrument affects outcomes only through treatment, or that the instrument changes treatment for anyone. Those are separate questions. Monotonicity also places no restriction on how treatment affects outcomes: two units of the same principal stratum may have different outcome responses to treatment.
A Short Algebraic Check
For binary treatment, subtracting one potential treatment status from the other gives a simple way to see what monotonicity allows. The four possible types have the following differences:
Thus \(D_i(1) \ge D_i(0)\) for every unit if and only if no unit has difference \(-1\), which is exactly the no-defiers condition. This equivalence depends on treatment being binary and on the two assignment levels being ordered as \(0\) and \(1\). It does not require every unit to respond in the same way.
Worked Examples
Worked Example: Classifying a Hypothetical Response Schedule
Suppose, for illustration, that the potential treatment statuses of 80 hypothetical people were fully known. The response schedule contains 12 always-takers, 28 never-takers, 40 compliers, and no defiers. Does this schedule satisfy monotonicity, and what are the treatment rates under the two assignments?
Check the unit-level condition. The only type that violates \(D_i(1) \ge D_i(0)\) is a defier. There are zero defiers in this hypothetical schedule, so every unit satisfies the condition.
Under \(Z=0\), the 12 always-takers receive treatment and the other 68 do not. The treatment rate is \(12/80=0.150\), or 15.0%. Under \(Z=1\), the 12 always-takers and 40 compliers receive treatment, giving \(52/80=0.650\), or 65.0%.
The calculation describes this stipulated full response schedule. In an ordinary study, each person is observed under only one assignment, so the two potential treatment statuses for that person are not both directly observed.
Worked Example: A Positive Average Change With Some Defiers
Consider a second hypothetical response schedule for 60 people: 6 always-takers, 18 never-takers, 34 compliers, and 2 defiers. Calculate the treatment rates under each assignment and assess monotonicity.
Under \(Z=0\), the 6 always-takers and 2 defiers receive treatment. Therefore the rate is \((6+2)/60=8/60=0.1333\), or 13.3%. Under \(Z=1\), the 6 always-takers and 34 compliers receive treatment. The rate is \((6+34)/60=40/60=0.6667\), or 66.7%.
The rate is higher under \(Z=1\), but the schedule still violates monotonicity: its 2 defiers have \((D_i(0),D_i(1))=(1,0)\). The increases among compliers outweigh the decreases among defiers in the overall comparison. This illustrates why a favorable average pattern is not the same claim as \(D_i(1) \ge D_i(0)\) for every unit.
The example is a logical illustration using known potential statuses, not a method for determining each person’s response type from the usual observed data. Its point is that the definition of no defiers is individual-level.
Worked Example: Assessing an Invitation to a Treatment Program
Suppose a health system studies whether an invitation increases attendance at a cessation program. Define \(Z=1\) as receiving the invitation and \(D=1\) as attending that program. In a hypothetical population of 100 people, imagine 10 always-takers, 50 never-takers, 35 compliers, and 5 people who would attend without an invitation but would not attend if invited. Does the invitation support a no-defiers claim under this treatment definition?
The last group consists of defiers: each has \((D_i(0),D_i(1))=(1,0)\). Therefore the stipulated response schedule does not satisfy monotonicity. For comparison, the treatment rate without invitation is \((10+5)/100=0.150\), while the rate with invitation is \((10+35)/100=0.450\). The higher invited-group rate does not remove the five defiers from the assumed schedule.
Whether this pattern is substantively credible depends on how the invitation and treatment are defined. If an invitation can create confusion, discourage attendance, or displace an alternative route into this specific program, a defier response may be conceivable. If the intervention simply adds access without changing anyone’s other opportunities or decisions, no defiers may be more plausible. The mechanism is what needs to be assessed; the arithmetic alone cannot establish the assumption.
When Is No Defiers Substantively Plausible?
A no-defiers claim is most credible when the instrument has a clear one-directional mechanism and the treatment is defined in a way that matches that mechanism. For example, an offer that provides an additional opportunity to receive a particular service may plausibly increase uptake without taking away other opportunities. Even there, plausibility depends on the details: an offer might impose burdens, alter perceptions, or replace a different route to treatment.
The question is not whether the instrument was intended to encourage treatment, or whether average uptake is higher under encouragement. It is whether there is a credible account in which some individuals would be less likely to receive the defined treatment because they were assigned the encouragement. To assess this, examine the assignment mechanism, the treatment definition, possible competing pathways, and how people might react. A broad treatment definition can make substitution less likely in some settings; a narrow definition may make it easier for one option to displace another.
Monotonicity is a substantive assumption about the response structure. It does not follow just because assignment was randomized, because the instrument is relevant on average, or because an exclusion restriction is considered plausible. Those properties address different features of the design. A researcher should explain why defier responses are unlikely in context, rather than treating no defiers as a consequence of study design alone.
Common Mistake: Treating an Average Increase as Proof
- Wrong reasoning: “Treatment uptake is higher when \(Z=1\), so nobody can be a defier.”
- Correction: The average can rise even when some individuals have \(D_i(1)=0\) and \(D_i(0)=1\). Monotonicity rules out those individual response pairs, not merely a negative average contrast.
- Wrong reasoning: “The instrument is randomized, so no one can respond negatively to it.”
- Correction: Random assignment concerns how \(Z\) is allocated. It does not, by itself, restrict how each person’s treatment status would differ across the two assignments.
- Wrong reasoning: “No defiers means everyone takes treatment when encouraged.”
- Correction: Never-takers and always-takers are allowed. The assumption excludes only defiers.
Check Your Understanding
Use the potential treatment statuses to answer each question.
- What principal stratum does a unit with \(D_i(0)=0\) and \(D_i(1)=1\) belong to?
- Which potential treatment-status pair violates monotonicity?
- A hypothetical population has 20 compliers and 3 defiers. Does a larger number of compliers make the no-defiers assumption true?
- Why does random assignment of \(Z\) not establish \(D_i(1) \ge D_i(0)\) for every unit?
- Give one feature of an encouragement mechanism that could make a defier response more plausible.