Sensitivity Bounds for a Nonzero Defier Share

Why a Small Defier Share Matters

In a randomized encouragement study, assignment can affect whether a person takes up treatment. The familiar Wald ratio has a straightforward complier-effect interpretation when the encouragement never makes anyone less likely to take treatment. If some people do the opposite, however, the reduced-form outcome difference combines effects for both compliers and defiers. A limit on the proportion of defiers can still be informative: it constrains how much the defiers’ effects can shift the complier-effect estimate.

This tutorial derives a sensitivity interval for the complier average treatment effect under a binary encouragement and a binary outcome. The key input is an upper bound on the population share of defiers. The result is a bound, not an estimate that assumes the defier share is known exactly.

Setup and Assumptions

Let $Z$ be a binary, randomized encouragement, with $Z=1$ denoting encouragement and $Z=0$ its absence. Let $D$ be a binary treatment received, and let $Y$ be a binary outcome. Write $D(z)$ for the treatment uptake that would occur under assignment $z$, and $Y(d)$ for the outcome under treatment status $d$. The observed variables obey consistency: $D=D(Z)$ and $Y=Y(D)$. Assume assignment is independent of the potential treatment and outcome values, and assume exclusion: assignment affects the outcome only through treatment received.

These assumptions give the observed differences $d=P(D=1\mid Z=1)-P(D=1\mid Z=0)$ and $r=E(Y\mid Z=1)-E(Y\mid Z=0)$ their causal interpretation as a first stage and a reduced form. Here we focus on a positive first stage, $d>0$. If the encouragement has a negative first stage, the assignment labels can be reversed before applying the analysis.

Definition: A complier has $D(1)=1,D(0)=0$; an always-taker has $D(1)=D(0)=1$; a never-taker has $D(1)=D(0)=0$; and a defier has $D(1)=0,D(0)=1$. Let $p_C$ and $p_F$ denote the population proportions of compliers and defiers. Let $\tau_C=E[Y(1)-Y(0)\mid\text{complier}]$ be the complier average treatment effect, and define $\tau_F$ analogously for defiers.

Suppose substantive knowledge or a sensitivity analysis supports $p_F\leq\delta$, where $\delta$ is a chosen upper limit between zero and one. This is not the assumption that there are no defiers: it allows a nonzero defier share, up to the specified limit. Because $Y$ is binary, each individual treatment effect lies between $-1$ and $1$, so $-1\leq\tau_C,\tau_F\leq1$.

From the Reduced Form to Bounds

Under random assignment and exclusion, always-takers and never-takers have no assignment-induced change in treatment received, so they contribute zero to the reduced-form contrast. Compliers move from untreated to treated when $Z$ changes from zero to one, while defiers move from treated to untreated. Consequently,

d=p_C-p_F,\qquad r=p_C\tau_C-p_F\tau_F.

The first equation means that a positive first stage identifies the difference between the complier and defier shares, not the complier share alone. Set $f=p_F$. Then $p_C=d+f$, and rearranging the reduced-form equation gives

\tau_C=\frac{r+f\tau_F}{d+f}.

For a fixed possible defier share $f$, the most negative value consistent with $-1\leq\tau_F\leq1$ uses $\tau_F=-1$; the most positive uses $\tau_F=1$. Thus

\frac{r-f}{d+f}\leq\tau_C\leq\frac{r+f}{d+f}.

Key result: Let $p_0=P(D=1\mid Z=0)$ and $p_1=P(D=1\mid Z=1)$, so $d=p_1-p_0>0$. The uptake margins require $p_F\leq p_0$ and $p_F\leq1-p_1$. Therefore define the largest defier share allowed by both the sensitivity limit and the uptake data as $F=\min(\delta,p_0,1-p_1)$. A valid, potentially conservative bound is $$ \max\left(-1,\min\left\{\frac{r}{d},\frac{r-F}{d+F}\right\}\right) \leq\tau_C\leq \min\left(1,\max\left\{\frac{r}{d},\frac{r+F}{d+F}\right\}\right). $$

To see why the uptake restrictions hold, $p_0$ includes all always-takers and defiers, while $1-p_1$ includes all never-takers and defiers. Also, $p_C=d+f$ cannot exceed one. The stated restrictions follow directly from the two observed uptake rates. For each $f$ between zero and $F$, the lower candidate is $(r-f)/(d+f)$ and the upper candidate is $(r+f)/(d+f)$. Each is a ratio of linear functions with a positive denominator, so it is monotone or constant over that range; its extreme occurs at an endpoint. Evaluating at $f=0$ and $f=F$, then intersecting with the logically necessary range $[-1,1]$, gives the displayed bound.

A common case is $|r|\leq d$. Then the lower candidate decreases as $f$ increases, and the upper candidate increases. The bound simplifies to

\max\left(-1,\frac{r-F}{d+F}\right) \leq\tau_C\leq \min\left(1,\frac{r+F}{d+F}\right).

This shows directly how allowing more defiers can widen the interval. When $\delta=0$, $F=0$, and the interval collapses to the Wald ratio $r/d$. As the allowed share increases, the interval does not necessarily widen at every step if the uptake margins cap $F$, but it cannot become more informative merely by allowing a larger defier share.

These are moment-based bounds: they use the first-stage and reduced-form contrasts and the range of binary-outcome treatment effects. The full observed outcome distributions can impose further constraints, so the interval need not be sharp. A necessary compatibility check is $|r|\leq d+2F$: with a defier share $f$, the absolute reduced form cannot exceed $p_C+p_F=d+2f$. If this check fails, the stated assumptions and defier-share limit cannot jointly explain the observed contrasts.

Worked Sensitivity Analyses

Worked Example: A Modest Defier Limit Leaves the Sign Unresolved

In a hypothetical randomized encouragement study, treatment uptake is 35% without encouragement and 55% with encouragement. The binary outcome rate is 20% without encouragement and 28% with encouragement. Suppose the maximum defier share is set to $\delta=0.10$.

The first stage and reduced form are $d=0.55-0.35=0.20$ and $r=0.28-0.20=0.08$. The uptake constraints give $p_F\leq0.35$ and $p_F\leq1-0.55=0.45$, so $F=\min(0.10,0.35,0.45)=0.10$. Since $|r|=0.08\leq0.20=d$, use the simplified bound:

\frac{0.08-0.10}{0.20+0.10} \leq\tau_C\leq \frac{0.08+0.10}{0.20+0.10}, \qquad -0.0667\leq\tau_C\leq0.600.

The data and the chosen defier limit are compatible, since $0.08\leq0.20+2(0.10)=0.40$. The resulting interval allows both a slightly negative and a positive complier effect. Under these assumptions, this sensitivity analysis does not determine the sign of the complier average treatment effect.

Worked Example: A Negative Reduced Form Can Still Give a Wide Interval

In a second hypothetical study, uptake is 25% in the unencouraged group and 40% in the encouraged group. The outcome rates are 31% and 28%, respectively. Set the maximum defier share to $\delta=0.08$.

Here $d=0.40-0.25=0.15$, $r=0.28-0.31=-0.03$, and $F=\min(0.08,0.25,0.60)=0.08$. As $|r|\leq d$, the bounds are

\frac{-0.03-0.08}{0.15+0.08} \leq\tau_C\leq \frac{-0.03+0.08}{0.15+0.08}, \qquad -0.478\leq\tau_C\leq0.217.

With no defiers, the Wald ratio would be $-0.03/0.15=-0.200$. Allowing up to 8% defiers makes the interval much wider and includes positive effects. The negative reduced form alone therefore does not establish that treatment harms compliers under this sensitivity limit.

Worked Example: Check the General Bound When the Reduced Form Exceeds the First Stage

In a third hypothetical study, uptake is 30% without encouragement and 50% with encouragement. Outcome rates are 15% and 42%. Suppose $\delta=0.10$. The outcome contrast is larger than the first stage, so the simplified expression for $|r|\leq d$ does not apply.

Calculate $d=0.50-0.30=0.20$, $r=0.42-0.15=0.27$, and $F=\min(0.10,0.30,0.50)=0.10$. The endpoint candidates are

\frac{r}{d}=\frac{0.27}{0.20}=1.35,\qquad \frac{r-F}{d+F}=\frac{0.27-0.10}{0.20+0.10}=0.567, $$ $$ \frac{r+F}{d+F}=\frac{0.27+0.10}{0.20+0.10}=1.233.

The lower bound is the smaller lower candidate, $0.567$. The larger upper candidate is $1.35$, but a binary-outcome treatment effect cannot exceed 1. Thus the valid interval is $0.567\leq\tau_C\leq1$. The compatibility check also passes: $0.27\leq0.20+2(0.10)=0.40$. This example illustrates why the general endpoint formula and the logical effect range matter when the reduced form exceeds the first stage.

How to Read the Sensitivity Interval

The limit $\delta$ is an assumption varied by the analyst, not a quantity automatically learned from the two uptake rates. The uptake rates do restrict how large the defier share could be, but they do not usually identify it. A useful report therefore states the assumed limit, the resulting effective cap $F$, and the corresponding interval for $\tau_C$. Showing results over several plausible values makes clear how dependent the conclusion is on the allowance for defiers.

For example, in the first study, reducing the limit from 10% to 4% gives $F=0.04$, with bounds $(0.08-0.04)/(0.20+0.04)=0.167$ and $(0.08+0.04)/(0.20+0.04)=0.500$. The interval is then entirely positive. Raising the limit back to 10% changes it to $[-0.0667,0.600]$, so the sign is no longer established. The conclusion is robustly positive only under the smaller defier-share limit.

Compute the observed contrasts.
Find the uptake difference $d=p_1-p_0$ and outcome difference $r=E(Y\mid Z=1)-E(Y\mid Z=0)$, preserving the same encouragement ordering.

Set the feasible defier cap.
For a proposed limit $\delta$, use $F=\min(\delta,p_0,1-p_1)$. If $d\leq0$, reverse the assignment labels or reformulate the analysis.

Calculate and interpret the interval.
Use the general endpoint formula, check compatibility, and report whether the interval includes zero. Repeat for other substantively plausible values of $\delta$.

Common Mistake

A frequent error is to treat the Wald ratio $r/d$ as the complier effect even after allowing defiers, or to subtract the defier share from the numerator without accounting for its contribution to the first stage. The correct identity is $d=p_C-p_F$, so $p_C=d+p_F$, while $r=p_C\tau_C-p_F\tau_F$. Both equations are needed to isolate $\tau_C$. A second error is to interpret the sensitivity interval as a confidence interval: it describes uncertainty about the causal effect under a specified defier-share limit, not sampling uncertainty. Sampling uncertainty requires additional statistical inference.

Key takeaway: With a positive first stage, exclusion, randomized encouragement, a binary outcome, and at most $F$ defiers, the reduced form combines complier and defier effects. Restricting the defier share constrains that mixture and yields a sensitivity interval for the complier effect. A wider allowed defier share generally means weaker conclusions.

Check Your Understanding

Use the definitions and bounds above to answer these questions.

Why is the first stage $d$ equal to $p_C-p_F$, rather than to the complier proportion alone?
If $d=0.18$, $r=0.06$, and $F=0.05$, calculate the simplified bounds for $\tau_C$. Does the interval include zero?
For uptake rates $p_0=0.20$ and $p_1=0.65$, what is the largest defier share permitted by the uptake margins?
Why can the Wald ratio fail to equal the complier average treatment effect when the defier share is nonzero?
What does it mean if the compatibility check $|r|\leq d+2F$ fails?