Monotone Treatment Response for Sharper IV Bounds

Why Monotone Treatment Response Can Sharpen Bounds

An instrumental variable can help bound the effect of a treatment when treatment receipt is not randomized. Even with a randomized instrument, however, the data may not reveal the treatment effect for every person: some people always take treatment, some never do, and others change their treatment status in response to the instrument. When the outcome is binary, an additional assumption can narrow the range of possible effects: treatment does not lower anyone’s outcome.

This is an assumption about how treatment affects outcomes, not about how the instrument affects treatment uptake. That distinction matters. Instrument monotonicity rules out people who would take treatment when encouraged but not take it when unencouraged; monotone treatment response rules out people whose outcome would be worse under treatment. Neither assumption implies the other.

Definition: Let $D\in\{0,1\}$ denote treatment receipt, $Z\in\{0,1\}$ a randomized instrument, and $Y(d)\in\{0,1\}$ the outcome that would be observed under treatment level $d$. Monotone treatment response (MTR) means $Y(1)\geq Y(0)$ for every unit. Thus each unit’s treatment effect $Y(1)-Y(0)$ is either 0 or 1.

The target here is the population average treatment effect, $E[Y(1)-Y(0)]$, expressed as a risk difference. The outcome is binary, so the effect is in percentage points when reported as a percentage. The aim is to bound this population quantity, not to claim that the instrument identifies a point estimate for everyone.

Assumptions and the Instrument-Induced Groups

The formulas below use a binary instrument and binary treatment. The instrument is randomized, so its assignment is independent of the potential treatment and outcome values. We also require consistency, no interference between units, and the exclusion restriction: the instrument affects the outcome only through treatment. Finally, we assume instrument monotonicity, $D(1)\geq D(0)$, where $D(z)$ is the treatment that would be received under instrument level $z$. There must be a positive first stage, so treatment uptake differs between the two instrument arms.

Under instrument monotonicity, units fall into three groups. Always-takers take treatment at either instrument level; never-takers take treatment at neither level; and compliers take treatment when $Z=1$ but not when $Z=0$. Write their population proportions as $a$, $n$, and $c$, respectively. These proportions are determined by uptake rates:

a=P(D=1\mid Z=0),\qquad n=P(D=0\mid Z=1),\qquad c=P(D=1\mid Z=1)-P(D=1\mid Z=0).

The groups reveal different potential outcomes. Among always-takers, observed outcomes reveal $Y(1)$; among never-takers, they reveal $Y(0)$. For compliers, outcomes in the $Z=0$ arm reveal $Y(0)$, and outcomes in the $Z=1$ arm reveal $Y(1)$. Because the groups are not directly observed, those complier outcome rates must be recovered by subtracting the known always-taker or never-taker contributions from the relevant observed cells.

Key result: Define $m_A=P(Y=1\mid Z=0,D=1)$ and $m_N=P(Y=1\mid Z=1,D=0)$. When $c>0$, the complier outcome rates are $$ q_{C0}=\frac{P(Y=1,D=0\mid Z=0)-n m_N}{c}, \qquad q_{C1}=\frac{P(Y=1,D=1\mid Z=1)-a m_A}{c}. $$ Here $q_{C0}=P(Y(0)=1\mid C)$ and $q_{C1}=P(Y(1)=1\mid C)$, with $C$ denoting the complier group.

The rates must be between 0 and 1. In addition, MTR requires $q_{C1}\geq q_{C0}$: among compliers, the proportion with outcome 1 under treatment cannot be smaller than the proportion with outcome 1 without treatment. A failure of this inequality means the observed population distributions cannot all be reconciled with the stated assumptions.

Deriving the MTR Bounds

For compliers, both potential-outcome means are identified by the randomized instrument and exclusion restriction, so their mean treatment effect is $q_{C1}-q_{C0}$. For always-takers and never-takers, only one of the two potential outcomes is observed. MTR nevertheless restricts the missing outcome and therefore limits each group’s possible contribution.

\begin{aligned} 0 &\leq E[Y(1)-Y(0)\mid N] \leq 1-m_N,\\ 0 &\leq E[Y(1)-Y(0)\mid A] \leq m_A. \end{aligned}

For a never-taker, $Y(0)$ is observed. If it is 1, MTR forces $Y(1)=1$, so that person’s effect is 0. If it is 0, the missing $Y(1)$ can be either 0 or 1, giving an effect between 0 and 1. The fraction of never-takers with $Y(0)=0$ is $1-m_N$, which gives the upper bound $1-m_N$. For an always-taker, $Y(1)$ is observed. If it is 0, MTR forces $Y(0)=0$; if it is 1, $Y(0)$ can be 0 or 1. Hence the group’s effect is between 0 and $m_A$.

The population average is the sum of each group’s average effect weighted by its population share. The resulting bounds are:

c(q_{C1}-q_{C0}) \ \leq\ E[Y(1)-Y(0)] \ \leq\ c(q_{C1}-q_{C0})+n(1-m_N)+a m_A.

These bounds are sharp under the assumptions: each endpoint can be attained by assigning the unobserved potential outcomes of always-takers and never-takers in a way consistent with MTR and the observed outcomes. For the lower endpoint, set each such unit’s missing outcome equal to the observed outcome. For the upper endpoint, set the missing outcome to 1 for never-takers with observed $Y(0)=0$, and to 0 for always-takers with observed $Y(1)=1$. For compliers, MTR is compatible with their two binary outcome rates precisely when $q_{C1}\geq q_{C0}$; a joint assignment of their two potential outcomes then exists, and their average effect is fixed at $q_{C1}-q_{C0}$.

Worked Example: Calculating the Sharpened Bounds

Suppose a hypothetical randomized encouragement study has the following population distributions. In the $Z=0$ arm, 30% receive treatment, and 80% of those treated have $Y=1$; among the 70% untreated, 30% have $Y=1$. In the $Z=1$ arm, 20% are untreated, and 25% of those untreated have $Y=1$; among the 80% treated, 65% have $Y=1$. Assume randomization, exclusion, consistency, no interference, and instrument monotonicity.

First, the group shares are $a=0.30$, $n=0.20$, and $c=0.80-0.30=0.50$. The always-taker outcome rate is $m_A=0.80$, and the never-taker outcome rate is $m_N=0.25$.

In the $Z=0$ arm, the joint probability of being untreated and having outcome 1 is $0.70(0.30)=0.21$. The never-takers contribute $0.20(0.25)=0.05$ of that probability. Therefore:

q_{C0}=\frac{0.21-0.05}{0.50} =\frac{0.16}{0.50}=0.32.

In the $Z=1$ arm, the joint probability of being treated and having outcome 1 is $0.80(0.65)=0.52$. The always-takers contribute $0.30(0.80)=0.24$. Thus:

q_{C1}=\frac{0.52-0.24}{0.50} =\frac{0.28}{0.50}=0.56.

The MTR compatibility check passes because $0.56\geq0.32$. The complier contribution to the population effect is $0.50(0.56-0.32)=0.12$. The upper endpoint adds the possible effects among never-takers and always-takers:

\begin{aligned} \text{Lower bound}&=0.50(0.56-0.32)=0.12,\\ \text{Upper bound}&=0.12+0.20(1-0.25)+0.30(0.80)\\ &=0.12+0.15+0.24=0.51. \end{aligned}

Under the assumptions, the population average treatment effect is bounded between 0.12 and 0.51, or between 12 and 51 percentage points on the binary outcome scale. These are bounds, not a 95% confidence interval, and the calculation alone does not account for sampling uncertainty.

How Much Does the Assumption Tighten the Bounds?

The same example illustrates what MTR adds. Without MTR, the always-takers’ missing $Y(0)$ could be either 0 or 1, so their average effect could range from $m_A-1=-0.20$ to $m_A=0.80$. The never-takers’ missing $Y(1)$ could likewise be either 0 or 1, giving effects from $-m_N=-0.25$ to $1-m_N=0.75$. Keeping the identified complier effect, the bounds without MTR are:

\begin{aligned} \text{Lower bound}&=0.12+0.30(-0.20)+0.20(-0.25)=0.01,\\ \text{Upper bound}&=0.12+0.30(0.80)+0.20(0.75)=0.51. \end{aligned}

Adding MTR raises the lower bound from 0.01 to 0.12 in this example, while leaving the upper bound unchanged. This is not guaranteed in every application to be an equally large gain, but the direction is clear: MTR rules out negative effects in the always-taker and never-taker groups. It can therefore raise the lower endpoint, while the largest permitted effects in those groups remain possible.

Worked Example: Checking Compatibility With MTR

Consider another hypothetical randomized study. In each instrument arm, 25% are always-takers and 25% are never-takers, so the treatment uptake rates are 25% under $Z=0$ and 75% under $Z=1$, with $c=0.50$. Suppose $m_A=0.60$, $m_N=0.20$, the outcome-1 rate among untreated units under $Z=0$ is 0.35, and the outcome-1 rate among treated units under $Z=1$ is 0.40.

The recovered complier rate under no treatment is:

q_{C0}=\frac{0.75(0.35)-0.25(0.20)}{0.50} =\frac{0.2625-0.05}{0.50}=0.425.

The recovered complier rate under treatment is:

q_{C1}=\frac{0.75(0.40)-0.25(0.60)}{0.50} =\frac{0.30-0.15}{0.50}=0.30.

Both rates lie between 0 and 1, but $q_{C1}<q_{C0}$. That contradicts MTR for the complier group: the treated outcome rate would be lower than the untreated outcome rate. Consequently, there is no MTR bound interval calculated from these population values that is compatible with all the stated assumptions. In real samples, estimated rates can violate the inequality due to sampling variation, so the discrepancy is a model check rather than conclusive evidence that treatment harms individuals.

Common Pitfalls and Interpretation

Confusing the two monotonicity assumptions. Instrument monotonicity concerns $D(1)\geq D(0)$; MTR concerns $Y(1)\geq Y(0)$. The first defines the three uptake groups, while the second constrains missing outcomes within those groups.
Using an outcome scale without considering its meaning. MTR is a substantive claim about the direction of every individual treatment effect. For an outcome where smaller values are better, such as symptom severity, the inequality may need to be reversed if the outcome is coded so that larger values mean worse health. The assumed direction must match the chosen coding.
Treating the bound as a point estimate. A range such as 0.12 to 0.51 leaves meaningful uncertainty about the population effect. The lower endpoint is not necessarily the true effect, and the midpoint has no special identification status.
Ignoring compatibility checks. Before reporting bounds, verify that recovered rates are valid probabilities and that $q_{C1}\geq q_{C0}$. If the check fails, do not simply truncate the rates or report an interval as if the assumptions held.

The assumption can be useful when it is defensible on scientific grounds, but it may be implausible for treatments that can cause adverse outcomes for some individuals. It should be stated explicitly and assessed in context, rather than adopted solely because it produces narrower bounds. In empirical work, uncertainty in the estimated endpoint values also requires separate statistical treatment.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

In one sentence, distinguish instrument monotonicity from monotone treatment response.
Why does the $Z=0,D=0$ cell contain both never-takers and compliers under instrument monotonicity?
If $q_{C0}=0.28$ and $q_{C1}=0.46$, what is the complier contribution to the population average effect when $c=0.40$?
For never-takers with $m_N=0.35$, what is the largest average treatment effect permitted by MTR within that group, and why?
What does it mean if the recovered complier rates satisfy $q_{C1}<q_{C0}$?

Monotone Treatment Response for Sharper IV Bounds

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