Monotonicity and the Absence of Defiers

Why Monotonicity Is a Unit-Level Assumption

An instrument is useful when its values shift treatment receipt, but a positive shift on average does not say how every unit responds. Some units might take treatment only when encouraged, while others might do the opposite. Monotonicity addresses this unit-level question: it rules out responses in the opposite direction to the instrument’s coded increase. Its substantive content is therefore stronger than saying that treatment receipt is more common in one instrument group than another.

Use a binary instrument \(Z\in\{0,1\}\) and a binary treatment \(D\in\{0,1\}\). For unit \(i\), let \(D_i(z)\) denote the treatment that unit would receive if the instrument were set to \(z\). Only one of these two potential treatment values is observed for a unit, because the unit receives one instrument value. The distinction between the two potential values is essential: monotonicity compares what the same unit would do under both instrument values.

Definition: Monotonicity, with the instrument coded so that \(Z=1\) is the higher or more treatment-promoting value, is the unit-level condition \(D_i(1)\geq D_i(0)\) for every unit \(i\). With binary treatment, it rules out units for whom \(D_i(1)=0\) and \(D_i(0)=1\), called defiers. It permits compliers \((D_i(0),D_i(1))=(0,1)\), never-takers \((0,0)\), and always-takers \((1,1)\).

The ordered pair here is always written as \((D_i(0),D_i(1))\): treatment receipt under \(Z=0\), followed by receipt under \(Z=1\). A complier changes from not receiving to receiving treatment when the instrument changes from 0 to 1. A never-taker receives treatment under neither value, and an always-taker receives it under both. A defier changes in the reverse direction. These labels describe potential responses, not observed labels that can usually be assigned to individuals.

What the Restriction Does—and Does Not—Say

For a binary treatment, subtracting the two potential treatment values gives \(D_i(1)-D_i(0)\in\{-1,0,1\}\). Monotonicity requires that this difference be nonnegative for each unit. The only possibilities are therefore 0 and 1: the unit’s treatment receipt either stays the same or increases. A difference of \(-1\), which occurs precisely for a defier, is excluded.

D_i(1)-D_i(0)\geq 0\ \text{for every }i \quad\Longleftrightarrow\quad (D_i(0),D_i(1))\in\{(0,0),(0,1),(1,1)\}

This is a restriction on each unit’s pair of potential treatment receipts. It is not a claim that every unit takes treatment when \(Z=1\), nor that the instrument affects everyone. Monotonicity allows many units to have no treatment response at all: both always-takers and never-takers have \(D_i(1)=D_i(0)\). It also does not require the same response for every unit.

The assumption depends on the instrument’s coding and substantive meaning. If \(Z=1\) means “offered a subsidy,” monotonicity says that the offer does not cause any unit who would otherwise take treatment to refuse it. If the coding instead makes \(Z=1\) the less encouraging condition, the inequality may need to be reversed. One can recode the instrument, but cannot make the substantive assumption true merely by changing labels: the proposed direction must match the behavior the setting supports.

Worked Example: An Offer That Only Adds an Option

Suppose a hypothetical program offers eligible patients a free ride to a treatment appointment when \(Z=1\), while \(Z=0\) provides no ride. Assume treatment receipt means attending that appointment. Consider three hypothetical patients with potential receipts \((D_i(0),D_i(1))\): patient A has \((0,1)\), patient B has \((0,0)\), and patient C has \((1,1)\).

For patient A, the offer changes receipt from \(D_A(0)=0\) to \(D_A(1)=1\); A is a complier. Patient B does not attend under either condition and is a never-taker. Patient C attends under either condition and is an always-taker. For each patient, the value under the offer is at least the value without it: \(1\geq 0\) for A, \(0\geq 0\) for B, and \(1\geq 1\) for C.

In this simplified setting, monotonicity is plausible if receiving the ride cannot make anyone less likely to attend—for example, because patients remain free to use their usual transport and the offer creates no burdensome eligibility procedure. Those details matter. If accepting the ride requires cancelling another arrangement, or the offer creates an administrative obstacle that deters attendance, a patient with \((1,0)\) is possible. The description “encouragement” alone does not prove that the offer cannot deter anyone.

A Positive Average Shift Does Not Rule Out Defiers

A common temptation is to infer monotonicity from a positive first-stage contrast: treatment receipt is more frequent when \(Z=1\) than when \(Z=0\). But this comparison averages over units. Compliers increase treatment receipt, defiers decrease it, and the average can be positive even when both types exist. Thus, a positive average shift does not establish the unit-level restriction.

Worked Example: A Positive First Stage With Defiers Present

Consider a hypothetical population of 100 units. Suppose 40 are compliers with potential receipts \((0,1)\), 10 are defiers with \((1,0)\), 30 are never-takers with \((0,0)\), and 20 are always-takers with \((1,1)\). These are stipulated potential responses for the example, not classifications that would ordinarily be observed for each person.

Under \(Z=0\), treatment receipt is 1 for the 10 defiers and 20 always-takers, so the receipt rate is \(\frac{10+20}{100}=0.30\). Under \(Z=1\), receipt is 1 for the 40 compliers and 20 always-takers, so the rate is \(\frac{40+20}{100}=0.60\). The average contrast is \(0.60-0.30=0.30\), a positive 30-percentage-point shift.

Nevertheless, the 10 defiers violate monotonicity because each has \(D_i(1)=0<1=D_i(0)\). The arithmetic shows why the observed direction of the average contrast cannot certify that no individual moves in the opposite direction. It establishes a net shift in this hypothetical population, not a universal statement about potential treatment receipt.

The same point follows directly from the individual differences. If the population is finite and each unit has equal weight, the average difference is the proportion of compliers minus the proportion of defiers:

\frac{1}{N}\sum_{i=1}^{N}\{D_i(1)-D_i(0)\} = \text{proportion of compliers} - \text{proportion of defiers}.

This equality follows because a complier contributes \(1\), a defier contributes \(-1\), and either type with equal potential receipts contributes \(0\). A positive average only says the complier proportion exceeds the defier proportion. It does not imply that the defier proportion is zero.

Worked Example: Checking a Proposed Set of Responses

A researcher describes four hypothetical units by the pairs \((D_i(0),D_i(1))\): \((0,1)\), \((1,1)\), \((0,0)\), and \((1,0)\). Determine whether monotonicity holds for all four and identify the response type of each.

For \((0,1)\), receipt rises from 0 to 1, so this unit is a complier and satisfies \(D_i(1)\geq D_i(0)\). For \((1,1)\), receipt stays at 1, so the unit is an always-taker and satisfies the condition. For \((0,0)\), receipt stays at 0, so the unit is a never-taker and also satisfies it.

For \((1,0)\), receipt falls from 1 under \(Z=0\) to 0 under \(Z=1\). This unit is a defier, since \(D_i(1)=0<1=D_i(0)\). Therefore, monotonicity does not hold for all four units. One defier is enough to violate the universal assumption, regardless of the responses of the other three.

When the Assumption Is Credible, and How to Assess It

Monotonicity is most credible when the instrument operates as a one-directional change in access or encouragement and there is no plausible mechanism by which that change discourages treatment receipt for some units. For instance, an offer that adds a cost-free treatment option while leaving all other options unchanged may support the assumption. A policy change that imposes new paperwork, alters access to alternatives, or changes what treatment means for different units may not.

The assessment should focus on potential responses at the unit level. Ask whether any unit could reasonably receive treatment under \(Z=0\) but not under \(Z=1\), and investigate mechanisms that could produce that reversal. Also define treatment and instrument precisely: an ambiguous treatment, inconsistent timing, or multiple versions of the instrument makes the relevant potential receipt unclear. A well-defined \(D_i(z)\) is necessary before its ordering can be assessed.

Observed data generally do not reveal both \(D_i(0)\) and \(D_i(1)\) for the same unit. Consequently, monotonicity is not normally verifiable by comparing observed treatment rates alone. Those rates can diagnose whether there is an average shift, but they do not expose each unit’s missing potential receipt. Subject-matter knowledge, program design, and careful consideration of possible deterrence are central to defending the assumption.

Monotonicity is distinct from other instrumental-variable conditions. Relevance concerns whether the instrument shifts treatment receipt on average; monotonicity rules out individual shifts in the opposite direction. Independence concerns relationships between the instrument and potential outcomes or receipts, while exclusion concerns whether the instrument affects the outcome other than through treatment. Evidence or reasoning for one condition does not establish the others.

Common Pitfalls and Alternatives

Confusing an average direction with a universal direction. A higher treatment rate under \(Z=1\) does not rule out defiers; positive shifts among some units can outweigh negative shifts among others.
Assuming an encouragement cannot discourage anyone. That may be plausible in a particular design, but it is a substantive claim to examine, not a consequence of the word “encouragement.”
Reversing the ordered pair. Keep the convention \((D_i(0),D_i(1))\) explicit. Under that convention, \((1,0)\) is the defier type.
Treating monotonicity as an outcome restriction. The condition concerns potential treatment receipt, not whether treatment helps or harms a unit.

The main alternative is to avoid imposing the no-defiers restriction when the setting does not support it, and instead use an analysis that permits opposing responses. That choice can change what is identified and may yield less specific conclusions. A weaker statement about average treatment uptake is not a substitute for unit-level monotonicity: it is a different condition. The appropriate choice depends on the design and the scientific question, not on which assumption produces a more convenient result.

Key takeaway: With \(Z=1\) coded as the more treatment-promoting value, monotonicity means \(D_i(1)\geq D_i(0)\) for every unit. It excludes defiers but permits compliers, never-takers, and always-takers. A positive average treatment shift alone cannot establish this assumption.

Check Your Understanding

Answer each question, then compare your reasoning with the solution shown.

For the pair \((D_i(0),D_i(1))=(1,0)\), identify the response type and say whether monotonicity holds.
Solution: The unit is a defier: treatment receipt falls from 1 to 0 when the instrument changes from 0 to 1. Monotonicity fails for this unit because \(D_i(1)=0<1=D_i(0)\).
In a hypothetical population, 25% are compliers and 5% are defiers, while all other units have no change in receipt. What is the average contrast in potential treatment receipt, and does it establish monotonicity?
Solution: The average contrast is \(0.25-0.05=0.20\), or a 20-percentage-point increase. It does not establish monotonicity: the positive contrast coexists with a 5% defier share.
A binary instrument is recoded so that \(Z=1\) means “no offer” and \(Z=0\) means “offer.” If the offer cannot reduce anyone’s treatment receipt, what direction does the monotonicity inequality take in this coding?
Solution: “Offer cannot reduce receipt” means potential receipt under the offer is at least receipt without it. In the stated coding, the offer is \(Z=0\), so \(D_i(0)\geq D_i(1)\) for every unit. Equivalently, the usual \(D_i(1)\geq D_i(0)\) direction would apply after coding the offer as \(Z=1\).
Why can a researcher not usually identify each person’s compliance type from observed \(Z\) and \(D\) alone?
Solution: The observed data reveal \(D_i(Z_i)\), the receipt under the instrument value actually assigned, but not the receipt under the other value. Classifying a unit’s type requires the pair \((D_i(0),D_i(1))\), so one potential receipt is missing for that unit.
An intervention adds a treatment option but also requires a lengthy approval process that may deter some eligible people. What should be examined before asserting monotonicity?
Solution: Examine whether the approval burden could cause any unit who would receive treatment without the intervention not to receive it with the intervention. If that response \((1,0)\) is plausible, the no-defiers assumption is not automatically credible, even if the intervention increases receipt on average.