Worst-Case Bounds Under Arbitrary Attrition

Why Attrition Changes the Target

When outcomes are missing for some participants, the observed outcome mean in an assigned treatment arm describes its responders, not necessarily everyone assigned to that arm. If the chance of response can depend on the unobserved outcome itself, the responder mean need not represent the mean outcome for nonresponders. Worst-case bounds make no assumption that removes this dependence. Instead, they use the known range of possible outcomes to determine how far the full-population treatment effect could vary.

The bounds are useful precisely because they do not require attrition to be independent of treatment or potential outcomes, nor do they require a monotone response pattern across treatment arms. Their cost is that they can be wide. Their width reflects how much the unobserved outcomes could matter given the outcome support and the amount of missingness.

Setup and Assumptions

Consider a binary treatment $D\in\{0,1\}$ assigned at random in a target population. For each person, let $Y(d)$ be the outcome that would be observed under treatment status $d$, and let $R(d)$ indicate whether that outcome would be recorded under $d$. The observed outcome is available only when the person is assigned to $d$ and $R(d)=1$. The target is the average treatment effect $\tau=E[Y(1)-Y(0)]$.

Assume treatment assignment is independent of the vector of potential outcomes and response indicators, $(Y(0),Y(1),R(0),R(1))$. This is the randomization condition that allows the outcomes and response patterns observed in each assigned arm to reveal the corresponding population quantities. Also assume consistency: for a person assigned $D=d$, the response indicator equals $R(d)$, and when $R(d)=1$, the recorded outcome equals $Y(d)$. Finally, assume both potential outcomes lie in a known, finite support interval $[L,U]$, with $L\leq U$.

Definition: For treatment status $d$, the response rate is $q_d=P(R=1\mid D=d)$. When $q_d>0$, the responder mean is $m_d=E[Y\mid D=d,R=1]$. The treatment-specific population mean is $\mu_d=E[Y(d)]$, and the average treatment effect is $\tau=\mu_1-\mu_0$. Attrition is arbitrary here: $R(d)$ may depend on $Y(d)$, on treatment status, and on other potential outcomes.

Under random assignment and consistency, the observed response rate identifies $P(R(d)=1)$, and, when that rate is positive, the observed responder mean identifies $E[Y(d)\mid R(d)=1]$. These facts do not identify the mean among nonresponders. The support restriction supplies the only information used to bound that missing part.

Worst-Case Bounds for One Treatment Arm

A convenient derivation uses the response indicator to avoid conditional expectations for groups that may have zero probability. For each $d$, define $s_d=E[R(d)Y(d)]$. This is identified from the assigned arm: it is the average of the outcome multiplied by the response indicator, treating the product as zero for nonresponders. Since $Y(d)\in[L,U]$, the unobserved contribution from nonresponders must lie between $L$ and $U$ times their population share.

\mu_d =E[R(d)Y(d)]+E[(1-R(d))Y(d)].

The first term equals $s_d$, while $P(R(d)=0)=1-q_d$. Therefore $s_d+(1-q_d)L\leq\mu_d\leq s_d+(1-q_d)U$. When $q_d>0$, $s_d=q_dm_d$, yielding the familiar formula in terms of the responder mean. The unconditional expression in $s_d$ also covers boundary cases without requiring a conditional mean where none exists.

Theorem: Under random assignment, consistency, and the known support restriction $Y(d)\in[L,U]$, the treatment-specific mean satisfies $$ s_d+(1-q_d)L\ \leq\ \mu_d\ \leq\ s_d+(1-q_d)U, \qquad s_d=E[R(d)Y(d)]. $$ If $q_d>0$, then $s_d=q_dm_d$. In particular, if $q_d=1$, then $\mu_d=s_d=m_d$, so both bounds equal the observed mean. If $q_d=0$, then $s_d=0$, and the bounds reduce to $L\leq\mu_d\leq U$; no responder mean is defined in that arm.

For $0<q_d<1$, the same result can be read as a decomposition into the responder and nonresponder groups:

\mu_d =q_dm_d+(1-q_d)E[Y(d)\mid R(d)=0].

This conditional-mean decomposition is used only when both groups have positive probability. The nonresponder mean lies in $[L,U]$, so substituting its lower and upper support limits gives the stated bounds. If $q_d=1$, there are no nonresponders: $\mu_d=m_d$ directly. If $q_d=0$, there are no responders: the arm's mean is bounded by the support alone. Thus neither boundary case relies on an undefined conditional expectation.

From Arm Means to Treatment-Effect Bounds

Write the arm-specific bounds as $[\underline{\mu}_d,\overline{\mu}_d]$. Since $\tau=\mu_1-\mu_0$, a lower value of the treatment effect pairs the smallest feasible treated mean with the largest feasible control mean. An upper value pairs the largest treated mean with the smallest control mean.

\underline{\tau} =\underline{\mu}_1-\overline{\mu}_0, \qquad \overline{\tau} =\overline{\mu}_1-\underline{\mu}_0.

With a common known support $[L,U]$, this gives $\underline{\tau}=s_1+(1-q_1)L-s_0-(1-q_0)U$ and $\overline{\tau}=s_1+(1-q_1)U-s_0-(1-q_0)L$. When both response rates are positive, the $s_d$ terms can be replaced by $q_dm_d$. If a treatment-specific support is known instead, use its lower and upper limits for that arm in the corresponding arm bounds.

Worked Example: Continuous Outcomes on a Bounded Scale

Suppose a hypothetical randomized study has outcomes known to lie between 0 and 100 points. In the treatment arm, 800 of 1,000 assigned participants respond, and their mean score is 70. In the control arm, 900 of 1,000 respond, and their mean is 60. Compute the worst-case bounds for the average treatment effect. Treat these summaries as empirical inputs to the bound calculation, not as a confidence interval.

The response rates and responder means are $q_1=0.8$, $m_1=70$, $q_0=0.9$, and $m_0=60$. Both response rates are between zero and one, so the responder-mean formula applies to both arms. The treated mean is bounded by

\underline{\mu}_1=0.8(70)+0.2(0)=56, \qquad \overline{\mu}_1=0.8(70)+0.2(100)=76.

The control mean is bounded by

\underline{\mu}_0=0.9(60)+0.1(0)=54, \qquad \overline{\mu}_0=0.9(60)+0.1(100)=64.

Subtracting in the correct order gives $\underline{\tau}=56-64=-8$ points and $\overline{\tau}=76-54=22$ points. Thus the worst-case identified interval is $[-8,22]$ points. The responder means favor treatment by 10 points, but the available information does not determine the sign of the full-population treatment effect: the missing outcomes could make it negative or positive while remaining within the stated support.

Why the Bounds Are Sharp

A bound is sharp when its endpoints can actually be attained by some distribution of the unobserved outcomes that agrees with the observed information and the assumptions. Here, for each treatment arm, the observed responder outcomes are held fixed. To attain the lower endpoint for $\mu_d$, assign every nonresponder outcome the value $L$; to attain the upper endpoint, assign every nonresponder outcome the value $U$. Both completions satisfy the support restriction and preserve the observed response rate and responder outcomes.

For the lower endpoint of $\tau$, choose the lower completion in the treated arm and the upper completion in the control arm. For the upper endpoint, reverse those choices. Random assignment imposes no restriction on the unobserved pairing of the two treatment-specific potential-outcome and response variables, so these arm-specific completions can be coupled into a joint population of potential outcomes. Intermediate values can also be attained by assigning nonresponders values between $L$ and $U$. Thus the treatment-effect interval is not merely a conservative outer range under these assumptions; no narrower interval is justified by the observed response rates, responder outcomes, and support restriction alone.

Worked Example: Binary Outcomes

Suppose a hypothetical randomized study has a binary outcome, so the known support is $[0,1]$. The treatment response rate is $0.75$, and the responder mean is $0.60$. The control response rate is $0.90$, and the responder mean is $0.40$. Calculate the treatment-effect bounds in probability points.

In the treatment arm, the lowest possible mean sets every missing outcome to 0, and the highest sets every missing outcome to 1:

\underline{\mu}_1=0.75(0.60)+0.25(0)=0.45, \qquad \overline{\mu}_1=0.75(0.60)+0.25(1)=0.70.

For control, the corresponding calculations are

\underline{\mu}_0=0.90(0.40)+0.10(0)=0.36, \qquad \overline{\mu}_0=0.90(0.40)+0.10(1)=0.46.

Therefore $\underline{\tau}=0.45-0.46=-0.01$ and $\overline{\tau}=0.70-0.36=0.34$. The treatment effect is bounded between $-0.01$ and $0.34$, or between $-1$ and $34$ percentage points. This range includes zero, so the observed summaries and support alone do not establish that the average effect is positive.

Worked Example: Full Response in One Arm

Consider another hypothetical randomized study with outcomes in $[0,20]$. Every participant assigned treatment responds, and the treatment-arm mean is 12. In control, 60% respond and their mean is 8. Find the treatment-effect bounds.

Because $q_1=1$, there are no missing treatment outcomes and the treatment mean is known exactly: $\mu_1=12$. This case is handled directly; there is no need to define a nonresponder mean. In control, $q_0=0.6$ and $m_0=8$, so

\underline{\mu}_0=0.6(8)+0.4(0)=4.8, \qquad \overline{\mu}_0=0.6(8)+0.4(20)=12.8.

The treatment-effect interval is $[12-12.8,\ 12-4.8]=[-0.8,7.2]$. In outcome units, the average treatment effect could range from $-0.8$ to $7.2$. Complete response in the treatment arm removes uncertainty about $\mu_1$, but the missing control outcomes still leave uncertainty about the difference.

What the Bounds Do and Do Not Assume

These bounds allow response to be related to the potential outcome, even after conditioning on treatment. For example, people with worse unobserved outcomes may be less likely to respond. The calculation does not require response rates to be equal across arms. It also imposes no monotonicity restriction on whether a person would respond under treatment versus control; that restriction is central to Lee bounds, but is not assumed here.

The bounds do require a credible outcome support. If a stated upper limit is too low, or the lower limit too high, the resulting interval may exclude feasible outcomes and cease to be valid. More missingness generally widens the arm-specific bounds because a larger share of the mean can come from outcomes that are known only to lie within the support. A narrow support can make the bounds more informative, but should come from defensible knowledge about the outcome scale rather than convenience.

Random assignment is another essential condition for interpreting the observed arm-specific response rates and responder outcomes as information about the corresponding potential variables in the target population. In a nonrandomized setting, differences between assigned groups may also reflect treatment selection. The formulas alone do not remove that problem.

Common Mistakes

Treating the responder mean as the arm mean. The mean among observed outcomes describes responders. It equals the full-arm mean only if everyone responds or if additional assumptions justify that equality.
Subtracting the two lower bounds. The smallest treatment effect uses the treated lower bound minus the control upper bound. The largest uses the treated upper bound minus the control lower bound.
Using an undefined mean at a boundary. If $q_d=1$, there are no nonresponders and $\mu_d=m_d$. If $q_d=0$, there is no responder mean; the support alone bounds $\mu_d$.
Calling the interval a confidence interval. These are identification bounds based on population information and assumptions. Sampling uncertainty requires separate inferential methods.
Claiming the bounds prove a treatment effect is zero. If zero lies inside the interval, the assumptions and observed information do not determine the sign. They do not establish that the true effect equals zero.

Key takeaway: For each treatment arm, keep the observed responder contribution fixed and place the missing outcomes at the known support limits. Then subtract the largest feasible control mean for the lower treatment-effect bound, and the smallest feasible control mean for the upper bound. Handle full or zero response directly rather than invoking conditional means for empty groups.

Check Your Understanding

Use the definitions and bounds above to answer the following questions.

With outcomes in $[0,50]$, an arm has response rate $0.7$ and responder mean 20. What are the lower and upper bounds for that arm's population mean?
Why does the lower treatment-effect bound subtract the upper control-mean bound rather than the lower control-mean bound?
If every participant in an arm responds, what do the arm-mean bounds become, and why is no nonresponder conditional mean needed?
If no participant in an arm responds, what can be said about its population mean using only known support $[L,U]$?
What feature of the assumptions makes these bounds sharp, and what extra information could make them narrower?