Defiers and the Consequences of Nonmonotonicity

Why Defiers Matter

With a binary instrument and a binary treatment, individuals can respond to the instrument in different ways. Some take treatment only when encouraged; others do the reverse. The second group consists of defiers. Their responses matter because their contributions to the first stage and reduced form have the opposite sign from compliers’ contributions. As a result, the Wald ratio need not be the average treatment effect for compliers.

To isolate this consequence, suppose the instrument \(Z\) and treatment \(D\) are binary. Write \(D(z)\) for the treatment an individual would receive if the instrument were set to \(z\), and \(Y(d)\) for the outcome under treatment level \(d\). Assume consistency, so the observed treatment and outcome are \(D=D(Z)\) and \(Y=Y(D)\); exclusion, so the instrument affects the outcome only through treatment; and independence, so \(Z\) is independent of the joint potential values \(\{D(0),D(1),Y(0),Y(1)\}\). These assumptions let us express contrasts by instrument assignment in terms of potential values. The algebra below does not assume monotonicity.

How Defiers Enter the First Stage

The population first stage is the difference in mean treatment across instrument levels, \(E[D\mid Z=1]-E[D\mid Z=0]\). By independence and consistency, the mean treatment in each assignment group identifies the corresponding mean potential treatment. Thus the first stage equals \(E[D(1)-D(0)]\). The table shows that compliers add one, defiers subtract one, and always-takers and never-takers add zero.

This equation says that the first stage is a net change in treatment uptake. It is not the proportion of people whose treatment changes unless there are no defiers. A positive first stage means compliers outnumber defiers, not that defiers are absent. If \(p_C=p_F\), their changes cancel and the first stage is zero, even though many individuals may change treatment in response to the instrument.

How Defiers Enter the Reduced Form

Define an individual’s treatment effect on the outcome as \(\tau_i=Y_i(1)-Y_i(0)\). Under exclusion, the potential outcome under instrument level \(z\) is \(Y(D(z))\). A complier’s outcome changes from \(Y(0)\) to \(Y(1)\) as \(Z\) changes from 0 to 1, so the change is \(\tau_i\). For a defier, treatment changes from 1 to 0, so the outcome change is \(-\tau_i\). The outcome does not change with the instrument for always-takers or never-takers, because their treatment remains fixed.

Let \(\tau_C=E[Y(1)-Y(0)\mid C]\) when \(p_C>0\), and let \(\tau_F=E[Y(1)-Y(0)\mid F]\) when \(p_F>0\). If a stratum has zero proportion, its weighted contribution is defined to be zero; its conditional mean treatment effect need not be defined. Equivalently, the expression can be written using stratum indicators, which is valid whether or not either stratum is present.

Here \(\mathbf{1}_C\) and \(\mathbf{1}_F\) are indicators of membership in the complier and defier strata. In the last line, \(p_C\tau_C\) means \(E[\mathbf{1}_C\{Y(1)-Y(0)\}]\), and is zero if \(p_C=0\); \(p_F\tau_F\) is interpreted in the same way. This convention avoids assigning a conditional average effect to an absent stratum.

What the Wald Ratio Represents

When the first stage is nonzero, the Wald ratio is the reduced-form contrast divided by the first-stage contrast. Combining the expressions above gives the relevant algebra.

If there are no defiers and there is a positive proportion of compliers, this reduces to \(\tau_C\), the average treatment effect among compliers. When defiers are present, the ratio instead combines treatment effects from both groups, with the defier contribution subtracted in both numerator and denominator. It is not generally \(\tau_C\), nor is it generally an ordinary population average of the two stratum effects. If the first stage is zero, the Wald ratio is undefined.

A special case makes the contrast clear: if there are no compliers but there are defiers, the ratio—when defined—equals \(\tau_F\), not a complier effect. More generally, the ratio can differ substantially from the complier effect, and can even have the opposite sign. The next examples show these possibilities with hypothetical values.

Derivation and a Key Interpretation Check

The principal-strata derivation can be summarized directly. For each individual, the treatment contrast \(D(1)-D(0)\) is \(1\) for a complier, \(-1\) for a defier, and \(0\) otherwise. Averaging gives \(p_C-p_F\). Under exclusion, the outcome contrast is \(Y(D(1))-Y(D(0))\). It equals \(Y(1)-Y(0)\) for compliers, its negative for defiers, and zero for always-takers and never-takers. Averaging therefore gives the signed reduced form \(p_C\tau_C-p_F\tau_F\). Dividing the second average by the first yields the Wald expression whenever the denominator is nonzero.

This argument also identifies exactly what the no-defiers condition contributes. If \(p_F=0\) and \(p_C>0\), then \(\Delta_D=p_C\), \(\Delta_Y=p_C\tau_C\), and the ratio is \(\tau_C\). If \(p_F>0\), the formula includes a defier term; the ratio equals \(\tau_C\) only under special additional relationships, not merely because the first stage is positive. A positive first stage alone establishes \(p_C>p_F\), not \(p_F=0\).

Common Mistake

Wrong reasoning: “The instrument increases treatment on average, so the Wald ratio is the average effect for people whose treatment it changes; those people are compliers.” This silently assumes that nobody moves in the opposite direction. A positive first stage only means the complier proportion exceeds the defier proportion. Defiers can still be present and can contribute strongly to the reduced form.

Correction: Under the stated independence and exclusion assumptions, first compute the signed contrasts: \(\Delta_D=p_C-p_F\) and \(\Delta_Y=p_C\tau_C-p_F\tau_F\). The ratio identifies the complier average effect under no defiers and a nonzero complier proportion. With defiers, use the signed-ratio formula and do not label it a complier effect without further justification. Also check that the first stage is nonzero before forming the ratio.