Bounds with Limited Defiers in Noncompliance

Why Limited Defiance Changes the IV Argument

Random assignment makes the assigned groups comparable, but it does not guarantee that everyone receives the treatment to which they are assigned. When some participants take the opposite treatment, the difference in outcomes by assignment combines effects for two groups moving in opposite directions. A bound on the size of the defier group can still restrict the treatment effect for compliers, but it generally does not recover a single effect.

The target here is the complier average treatment effect: the average effect of treatment for people who receive treatment if assigned to it and do not receive it otherwise. The result is a moment-based identified region, not automatically a sharp region using every detail of the observed outcome distribution.

Definition: Let \(Z\in\{0,1\}\) denote randomized assignment and \(D\in\{0,1\}\) denote treatment received. Write \(D(z)\) for the treatment a person would receive under assignment \(z\), and \(Y(d)\) for the outcome under treatment status \(d\). A complier has \(D(1)=1,D(0)=0\); a defier has \(D(1)=0,D(0)=1\). The complier average treatment effect is \(\tau_C=E[Y(1)-Y(0)\mid D(1)=1,D(0)=0]\).

Assumptions and Observable Contrasts

Suppose assignment is randomized, so \(Z\) is independent of the potential treatment and outcome values. Assume consistency, no interference between individuals, and the exclusion restriction: assignment affects the outcome only through treatment received. Thus the observed outcome is \(Y=Y(D)\). Treatment receipt is binary, and outcomes lie in \([0,1]\), so any average treatment effect by type lies in \([-1,1]\). Finally, assume the first stage is positive: assignment to treatment increases the treatment-receipt rate.

Define the observed treatment rates \(p_z=P(D=1\mid Z=z)\), the first stage \(d=p_1-p_0>0\), and the reduced-form outcome contrast \(\Delta_Y=E[Y\mid Z=1]-E[Y\mid Z=0]\). Let \(\pi_C\) and \(\pi_F\) be the population shares of compliers and defiers. The shares of always-takers and never-takers are denoted by \(\pi_A\) and \(\pi_N\).

The first stage is the net share moved into treatment by assignment. Compliers move into treatment, while defiers move out. Therefore \(d=\pi_C-\pi_F\). If the defier share is \(f=\pi_F\), then \(\pi_C=d+f\). Treatment rates also imply \(\pi_A=p_0-f\) and \(\pi_N=1-p_1-f\). Nonnegative type shares require \(f\leq p_0\) and \(f\leq1-p_1\).

Moment restriction: If \(\tau_F=E[Y(1)-Y(0)\mid\text{defier}]\), random assignment and exclusion imply \(\Delta_Y=\pi_C\tau_C-\pi_F\tau_F\). The minus sign appears because defiers receive treatment under control assignment and control under treatment assignment.

Deriving the Region Under a Defier Cap

Suppose substantive knowledge gives an upper bound \(\delta\) on the population share of defiers. The largest feasible defier share is the cap, limited by the observed treatment rates:

0\leq f\leq F,\qquad F=\min\{\delta,\ p_0,\ 1-p_1\}.

For any feasible \(f\), the reduced-form equation becomes \(\Delta_Y=(d+f)\tau_C-f\tau_F\). Because outcomes lie in \([0,1]\), \(-1\leq\tau_F\leq1\). Rearranging gives a range for \(\tau_C\) at that value of \(f\). Taking the union over feasible defier shares yields the moment-based region:

\mathcal I_{\mathrm{mom}} = \bigcup_{0\leq f\leq F} \left[ \max\left\{-1,\frac{\Delta_Y-f}{d+f}\right\}, \min\left\{1,\frac{\Delta_Y+f}{d+f}\right\} \right].

The endpoints follow directly: the smallest compatible \(\tau_C\) uses the largest possible \(\tau_F\), namely \(1\); the largest compatible \(\tau_C\) uses \(\tau_F=-1\). The outer limits \(-1\) and \(1\) enforce the possible range of a treatment effect when outcomes lie in \([0,1]\). The region uses the reduced-form mean equation and outcome bounds; restrictions from the full observed outcome distributions can sometimes narrow it further.

When the cap is zero, \(f=0\), and the region reduces to the Wald ratio \(\Delta_Y/d\), provided that ratio is compatible with the outcome bounds. With a positive cap, the Wald ratio alone is no longer the complier effect: the observed reduced form may include an opposing contribution from defiers.

Worked Example: A Positive Reduced Form With Some Possible Defiers

Consider hypothetical randomized-assignment data with \(p_0=0.25\), \(p_1=0.55\), and \(\Delta_Y=0.12\). Suppose the known defier cap is \(\delta=0.10\). The first stage is \(d=0.55-0.25=0.30\), and the feasible maximum defier share is \(F=\min(0.10,0.25,0.45)=0.10\). These values satisfy the stated rate constraints.

For a given \(f\), the lower and upper endpoints before clipping are \((0.12-f)/(0.30+f)\) and \((0.12+f)/(0.30+f)\). The lower expression decreases over this range, since its derivative is \(-0.42/(0.30+f)^2<0\). The upper expression increases, since its derivative is \(0.18/(0.30+f)^2>0\). Thus the union has endpoints at \(f=0.10\):

\left[\frac{0.12-0.10}{0.30+0.10}, \frac{0.12+0.10}{0.30+0.10}\right] = [0.0500,\ 0.550].

Under these assumptions and moments, the complier average treatment effect is restricted to this moment-based region: from 0.0500 to 0.550 on the outcome scale. It is not pinned down at the Wald ratio \(0.12/0.30=0.400\).

How a Tighter Cap Changes the Region

A smaller upper bound on the defier share removes some values of \(f\) from the union. Since fewer mixtures of complier and defier effects can explain the same reduced form, the resulting region can narrow. The next calculation holds the hypothetical observed rates and outcome contrast fixed and changes only the cap.

Worked Example: Tightening the Defier Cap

Keep \(p_0=0.25\), \(p_1=0.55\), and \(\Delta_Y=0.12\), but suppose the cap is \(\delta=0.02\). Now \(F=\min(0.02,0.25,0.45)=0.02\). The same derivatives show that the lower endpoint decreases and the upper endpoint increases as \(f\) runs from zero to \(F\). Substitution gives:

\left[\frac{0.12-0.02}{0.30+0.02}, \frac{0.12+0.02}{0.30+0.02}\right] = [0.3125,\ 0.4375].

The region is now narrower than \([0.0500,0.550]\). The cap does not assert that the defier share equals 0.02; it permits any feasible share from zero through 0.02. If the cap were zero, the moment equation would instead give the single Wald value \(0.400\).

When the Reduced Form Is Negative

A positive first stage does not imply a positive outcome contrast. With defiers, a negative reduced form can be compatible with a negative complier effect, or with a small positive complier effect offset by a sufficiently negative effect among defiers. The cap determines how much such offsetting is allowed by the moment equation.

Worked Example: A Region That Includes Zero

For a different hypothetical case, let \(p_0=0.30\), \(p_1=0.55\), and \(\Delta_Y=-0.06\), with defier cap \(\delta=0.08\). The first stage is \(d=0.25\), and \(F=\min(0.08,0.30,0.45)=0.08\). At \(f=0.08\), the lower endpoint is \((-0.06-0.08)/(0.25+0.08)=-0.4242\), and the upper endpoint is \((-0.06+0.08)/(0.25+0.08)=0.06061\). The lower expression decreases with \(f\), because its derivative is \(-(0.25-0.06)/(0.25+f)^2<0\); the upper expression increases, because its derivative is \((0.25+0.06)/(0.25+f)^2>0\). Hence the region is approximately \([-0.424,\ 0.0606]\).

This region includes zero. The observed negative outcome contrast alone therefore does not establish that the treatment effect for compliers is negative under the stated defier allowance.

Interpretation and Common Pitfalls

These bounds concern the average treatment effect among compliers, not the population average treatment effect. Assignment randomization identifies contrasts by assignment; it does not make everyone a complier, and the exclusion restriction does not make treatment receipt random. The role of the defier cap is to limit one source of ambiguity in interpreting the reduced form.

Using the Wald ratio as though there were no defiers. The ratio \(\Delta_Y/d\) generally mixes complier and defier effects when \(f>0\). Use the reduced-form equation with both terms, or impose a zero-defier assumption explicitly.
Treating the cap as an effect bound. The cap bounds a population share, not the size of an individual treatment effect. Here the effect bound \([-1,1]\) follows separately from the outcome range \([0,1]\).
Calling the moment-based region sharp without checking the data used. The calculations use the treatment rates, reduced-form mean, and outcome range. Additional information in the observed outcome distributions may impose further restrictions.
Interpreting a region as a confidence interval. This is an identification region at the population level, not a sampling uncertainty interval. Estimating its endpoints from finite data introduces an additional inference problem.

The main practical lesson is that a transparent sensitivity parameter can make an otherwise strong interpretation conditional: report the defier cap, the resulting region, and how that region changes under alternative plausible caps.

Check Your Understanding

Use the definitions and moment restriction above to answer the following.

Why does the defier term enter the reduced-form outcome contrast with a minus sign?
If \(p_0=0.20\), \(p_1=0.50\), and \(\delta=0.15\), what is the largest feasible defier share \(F\)?
For a fixed feasible \(f\), which value of \(\tau_F\in[-1,1]\) gives the lowest compatible value of \(\tau_C\)?
What does the moment-based region become when the defier cap is zero?
Why is a moment-based region not necessarily sharp when the complete observed outcome distributions are available?