Monotonicity Violations and Defiers

Why Individual Responses to Encouragement Matter

An instrumental-variable interpretation can depend on how each person would respond to either level of an instrument. For example, an encouragement may increase treatment uptake for some people while causing others to avoid treatment. The latter response is not merely a weaker form of compliance: it is a distinct counterfactual pattern that changes what the usual Wald ratio represents.

To isolate this issue, consider a binary instrument \(Z\), a binary treatment \(D\), and an outcome \(Y\). Think of \(Z=1\) as receiving an encouragement and \(Z=0\) as not receiving it. Assume a fixed version of treatment, so the issue here is the direction of uptake response rather than differences between versions.

A defier takes treatment without the encouragement but would not take it if encouraged. The label describes a comparison between two potential responses for the same person; it does not mean that the person’s observed behavior is irrational or that the instrument was assigned incorrectly.

Monotonicity and Its Consequences

The standard monotonicity assumption is an individual-level restriction on potential uptake:

Because uptake is binary, this condition permits always-takers, never-takers, and compliers, but it excludes the pattern \(D(1)=0,D(0)=1\). Monotonicity therefore rules out defiers by assumption. It is stronger than saying that encouragement increases uptake on average: a positive average increase allows some people’s uptake to move in the opposite direction.

For the usual Wald interpretation, monotonicity is considered alongside other instrumental-variable assumptions. The instrument is independent of potential uptake and potential outcomes, as under random assignment; the observed treatment satisfies \(D=D(Z)\); and the observed outcome satisfies \(Y=Y(D)\), the exclusion restriction that the instrument affects the outcome only through treatment. Consistency and no interference are also assumed. Finally, the instrument must change the treatment rate, so \(E[D\mid Z=1]-E[D\mid Z=0]\ne 0\). These conditions are stated here to clarify which part of the interpretation is at issue: defiers violate monotonicity, not necessarily random assignment or exclusion.

Let \(p_C\) and \(p_D\) be the population proportions of compliers and defiers. Under random assignment and consistency, the first stage—the difference in treatment uptake rates by instrument assignment—is

Always-takers and never-takers contribute no difference between the two assignment groups. Compliers contribute \(+1\) to their individual uptake difference, and defiers contribute \(-1\). Thus a positive first stage means \(p_C>p_D\), not \(p_D=0\).

How Defiers Change the Wald Ratio

Write \(Y(1)\) and \(Y(0)\) for the potential outcome under treatment and no treatment, and let \(\tau_C=E[Y(1)-Y(0)\mid C]\) and \(\tau_D=E[Y(1)-Y(0)\mid D_f]\) denote the average treatment effects among compliers and defiers. Here \(C\) and \(D_f\) denote the corresponding principal strata. Under independence, exclusion, and consistency, the reduced form is

A complier’s treatment changes from 0 to 1 as the instrument changes from 0 to 1, so the person contributes \(Y(1)-Y(0)\) to the outcome contrast. A defier’s treatment changes from 1 to 0, so the contribution is \(Y(0)-Y(1)\), the negative of that treatment effect. Always-takers and never-takers have unchanged treatment and contribute zero to this contrast under exclusion.

When the first stage is nonzero, dividing the reduced form by the first stage gives the Wald ratio:

If \(p_D=0\), this ratio equals \(\tau_C\), the complier average treatment effect. If \(p_D>0\), it is generally not that effect. When \(p_C>p_D\), it can be written as

The defier effect enters with a negative weight. Those weights do not form an ordinary nonnegative weighted average, so the ratio can fall outside the range of the two stratum-specific effects. If \(p_C=p_D\), the first stage is zero and the Wald ratio is undefined; the reduced form need not be zero. A zero denominator does not produce a complier effect.

Defiers Can Reverse or Distort the Interpretation

The sign and size of the Wald ratio depend on both the treatment effects and the balance of compliers and defiers. A negative Wald ratio does not necessarily mean that treatment harms compliers; it may result from a negative first stage combined with positive effects in both response groups.

Even when the first stage is positive, defiers can make the ratio depart substantially from the complier effect. For instance, the first example’s ratio of 5 exceeds both stratum-specific effects, 4 and 1. This happens because, with \(p_C>p_D>0\), the defier effect is subtracted rather than added as a positive-weight component.

Justifying the Decomposition

The reduced-form identity follows by comparing potential outcomes across the two instrument assignments. Under random assignment, the mean outcome in each assignment group equals the corresponding population mean of \(Y(D(z))\). Exclusion and consistency let us write the observed outcome as \(Y(D(z))\). The difference is therefore

Partition the population into its four uptake types. For always-takers, \(D(1)=D(0)=1\), so the individual outcome difference is \(Y(1)-Y(1)=0\). For never-takers, it is \(Y(0)-Y(0)=0\). For compliers, it is \(Y(1)-Y(0)\); averaging over their population share contributes \(p_C\tau_C\). For defiers, it is \(Y(0)-Y(1)\); averaging contributes \(-p_D\tau_D\). Adding the four contributions gives \(p_C\tau_C-p_D\tau_D\). The corresponding uptake difference is \(+1\) for each complier, \(-1\) for each defier, and 0 for the other types, yielding \(p_C-p_D\).

Common Mistake and Practical Meaning

A common mistake is to reason, “The encouragement increases treatment uptake overall, so nobody can be a defier; therefore the Wald ratio is the complier effect.” The first clause establishes only \(p_C-p_D>0\), not \(p_D=0\). For the second conclusion, one needs the individual-level monotonicity assumption, as well as the other instrumental-variable assumptions. The assumption should be defended using the mechanism and setting, not inferred from a positive average first stage alone.

For example, an encouragement might increase uptake for most people but provoke reactance in a smaller group, leading them to avoid treatment. Whether that mechanism is plausible depends on the specific intervention and population. Monotonicity may be a reasonable working assumption in some settings, but it is not a consequence of random assignment and cannot generally be verified by comparing observed uptake rates.