Restrictions That Tighten Missing-Data Bounds

Why Restrictions Tighten Bounds

When outcomes are missing, the observed respondents do not by themselves determine the outcome distribution among nonrespondents. A useful alternative to choosing one untestable missing-outcome model is to state restrictions that rule out some possible values, then report the range of means still compatible with those restrictions. Two common restrictions are a known range for the outcome and a directional relationship between outcome values and response probability.

A range restriction says, for example, that every outcome lies between 0 and 100. Monotone response selection says that the chance of responding changes in one direction as the outcome changes. Each restriction rules out different complete-data possibilities. A range limits how extreme the unobserved outcomes can be; a selection direction limits how the missing outcomes compare with the observed ones. Neither assumption generally makes the missing mean point-identified.

Definition: Let $R=1$ indicate that an outcome is observed and $R=0$ that it is missing. Write $q=P(R=1)$ for the response rate, $m=E(Y\mid R=1)$ for the respondent mean, and $u=E(Y\mid R=0)$ for the missing-outcome mean. The population mean is $$ \mu=E(Y)=qm+(1-q)u. $$ A partial-identification bound is a lower and upper limit on a target quantity that holds for every complete-data distribution satisfying the observed-data information and stated assumptions.

For treatment-effect bounds, let $A\in\{0,1\}$ denote randomized assignment and $Y(a)$ the potential outcome under assignment $a$. Define $q_a=P(R=1\mid A=a)$, $m_a=E(Y\mid A=a,R=1)$, and $\mu_a=E\{Y(a)\}$. Under random assignment and consistency, the mean potential outcome in arm $a$ is the mean outcome for that arm before response selection. The average treatment effect is $\Delta=\mu_1-\mu_0$. The response assumptions below are made separately within each arm; one arm's selection direction does not automatically establish the other's.

Bounds From a Known Outcome Range

Suppose every outcome, observed or missing, is known to lie in $[L,U]$, where $L$ and $U$ are fixed outcome-scale limits. Then the missing mean must satisfy $L\leq u\leq U$. Substituting these limits into the mixture identity gives

qm+(1-q)L\leq \mu\leq qm+(1-q)U.

These are sharp bounds when the available information consists of the response rate, respondent mean, and support limits: the missing outcomes could all equal either endpoint. If the full respondent outcome distribution is available, additional restrictions or information may tighten the bounds. A claimed range must apply to every unit, including those whose outcomes are missing; a range seen only among respondents does not establish such a limit.

Worked Example: Outcome-Range Bounds for a Treatment Effect

Consider a hypothetical randomized study in which the outcome is known to lie between 0 and 100 points for every participant. Suppose the observed-data summaries are $q_1=0.80$, $m_1=70$ points, $q_0=0.75$, and $m_0=60$ points. Treat these as exact population summaries for illustration, not as estimates with sampling uncertainty.

For arm 1, the mean is bounded by $$ \mu_1\in[0.80(70)+0.20(0),\;0.80(70)+0.20(100)] =[56,76]\text{ points}. $$ For arm 0, $$ \mu_0\in[0.75(60)+0.25(0),\;0.75(60)+0.25(100)] =[45,70]\text{ points}. $$

The smallest treatment effect occurs with the smallest $\mu_1$ and largest $\mu_0$; the largest occurs with the largest $\mu_1$ and smallest $\mu_0$. Thus $$ \Delta\in[56-70,\;76-45]=[-14,31]\text{ points}. $$ The interval includes negative and positive values, so these assumptions and summaries do not determine the sign of the average treatment effect.

Monotone Response Selection

A directional selection assumption can narrow the range further. Define the response propensity $g(y)=P(R=1\mid Y=y)$. Positive monotone response selection means $g(y)$ is nondecreasing in $y$: higher outcomes do not have a lower response probability. Negative monotone response selection means $g(y)$ is nonincreasing: higher outcomes do not have a higher response probability. In a treatment study, define $g_a(y)=P(R=1\mid A=a,Y(a)=y)$ and state the direction separately for each arm.

This is a condition on response probabilities, not a claim that every person with a higher outcome must respond while every person with a lower outcome must not. Positive selection implies that the respondent mean is at least the missing-outcome mean; negative selection implies the reverse. Here is a short justification for the positive direction.

Express the respondent mean using the response propensity.
For $q=E\{g(Y)\}>0$, $E(Y\mid R=1)=E\{Yg(Y)\}/q$. Therefore its difference from the overall mean is $\operatorname{Cov}\{Y,g(Y)\}/q$.

Use monotonicity to determine the covariance sign.
For an independent copy $Y'$, $2\operatorname{Cov}\{Y,g(Y)\}=E[(Y-Y')\{g(Y)-g(Y')\}]$. If $g$ is nondecreasing, the two differences inside the product have the same sign, so the expectation is nonnegative.

Relate the overall mean to the missing mean.
The mixture identity gives $m-\mu=(1-q)(m-u)$. Since $m\geq\mu$, for $q<1$ it follows that $m\geq u$. If $g$ is nonincreasing, the covariance has the opposite sign and the inequality reverses.

Combining this implication with the outcome limits gives mean bounds. Under positive selection, $L\leq u\leq m$, so $\mu\in[qm+(1-q)L,\;m]$. Under negative selection, $m\leq u\leq U$, so $\mu\in[m,\;qm+(1-q)U]$. These bounds use the mean-order implication. A full respondent distribution together with monotone response propensity may impose further restrictions, so these mean bounds need not be sharp for all the observed-data information.

Worked Example: Positive Selection in Both Treatment Arms

Use the hypothetical summaries and the 0-to-100-point outcome range from the previous calculation. Now additionally assume, separately in both arms, that response probability is nondecreasing in the potential outcome. This is a substantive assumption; it does not follow from randomization.

In arm 1, the missing mean cannot exceed the respondent mean of 70 points. The mean bounds become $$ \mu_1\in[0.80(70)+0.20(0),\;70]=[56,70]\text{ points}. $$ In arm 0, the missing mean cannot exceed 60 points, giving $$ \mu_0\in[0.75(60)+0.25(0),\;60]=[45,60]\text{ points}. $$ Consequently, $$ \Delta\in[56-60,\;70-45]=[-4,25]\text{ points}. $$ The interval is narrower than the range-only interval $[-14,31]$, because positive selection rules out missing means above the corresponding respondent means. It still includes zero: the additional assumption narrows uncertainty but does not identify the treatment effect or determine its sign.

Applying a Restriction Only Where It Is Justified

Selection directions need not be the same in both arms. Applying positive selection to one arm and silently carrying it over to the other changes the identifying assumptions. If an arm has no defensible selection direction, retain its outcome-range bound. With arm-specific mean intervals $[\ell_a,u_a]$, the treatment-effect interval is $$ \Delta\in[\ell_1-u_0,\;u_1-\ell_0]. $$ The lower endpoint pairs the smallest treated mean with the largest control mean; the upper endpoint pairs the largest treated mean with the smallest control mean. The formula does not require the two arm means to attain their endpoints together.

Worked Example: A Directional Restriction in One Arm

In a hypothetical study, outcomes lie between 0 and 10 units. Suppose $q_1=0.60$, $m_1=8$ units, $q_0=0.80$, and $m_0=5$ units. Assume positive monotone response selection in arm 1, but make no directional selection assumption in arm 0.

For arm 1, positive selection and the range give $$ \mu_1\in[0.60(8)+0.40(0),\;8]=[4.8,8]\text{ units}. $$ For arm 0, use only the range: $$ \mu_0\in[0.80(5)+0.20(0),\;0.80(5)+0.20(10)] =[4,6]\text{ units}. $$ Thus $$ \Delta\in[4.8-6,\;8-4]=[-1.2,4]\text{ units}. $$ If positive selection were incorrectly imposed on arm 0 as well, its upper bound would be 5 rather than 6, and the calculated upper treatment-effect bound would be 4 units either way here only if paired with the same arm-1 upper limit; the control-arm lower endpoint still determines that upper bound. The lower treatment-effect endpoint, however, would change to $4.8-5=-0.2$ units. This illustrates why an unjustified arm-specific assumption can make the reported interval appear narrower.

Assumptions, Diagnostics, and Alternatives

A range restriction is credible only when the measurement scale or scientific definition truly bounds all outcomes. A positive-selection assumption rules out a lower average outcome among respondents than among nonrespondents within the specified population or arm; negative selection rules out the opposite ordering. The response indicator and observed outcomes do not reveal the missing outcomes, so the selection direction generally cannot be verified from the observed data alone. Substantive knowledge may motivate a direction, but it does not turn that direction into a testable fact.

Useful checks are therefore checks of the assumptions' meaning and scope, not proofs that they hold. Confirm that the proposed $L,U$ limits apply to missing as well as observed units, that the response rate and respondent mean refer to the same arm and target population, and that the selection direction has a defensible rationale. Compare results under no directional restriction, positive selection, and negative selection when the direction is uncertain. Large changes across these analyses show that conclusions depend materially on the untestable restriction.

The range-only approach is less restrictive than imposing a selection direction, but may produce wide bounds. A directional restriction can narrow them while retaining partial identification. A missing-at-random analysis or a parametric MNAR model can yield a point estimate under additional assumptions, but those approaches answer a different question: a single estimate depends on stronger modeling or missingness assumptions. Bound calculations do not make those assumptions for the analyst.

Common mistake: “Response probability rises with the outcome, so missing outcomes equal the respondent mean.” The assumption implies an ordering, $u\leq m$, not equality. Equality is possible, but is not forced.
Common mistake: using the observed respondent minimum and maximum as the known range for all outcomes. Those are sample or observed-data extrema, not guaranteed support limits for unobserved outcomes.
Common mistake: treating a narrower bound as proof that its assumption is true. A bound is conditional on the stated restrictions; observed-data fit alone cannot establish the missing-outcome selection direction.

Key takeaway: A known outcome range limits how extreme missing values can be. Monotone response selection additionally constrains the missing mean relative to the respondent mean. State each restriction explicitly, apply it only where justified, and interpret the resulting interval as conditional partial identification—not as a confidence interval or a point estimate.

Check Your Understanding

Answer each question, then compare with the solutions below.

Outcomes lie in $[0,20]$, the response rate is 0.70, and the respondent mean is 12. Find the range-only bound for the overall mean.
For the same summaries, suppose response probability is nondecreasing in the outcome. Find the mean bound implied by the range and this selection direction.
In a two-arm study, the mean intervals are $[3,7]$ in arm 1 and $[2,5]$ in arm 0. Find the treatment-effect interval.
Why does positive monotone response selection not establish that the missing mean equals the respondent mean?

Solutions. (1) The lower endpoint is $0.70(12)+0.30(0)=8.4$; the upper endpoint is $0.70(12)+0.30(20)=14.4$. Thus the bound is $[8.4,14.4]$ outcome units. (2) The missing mean is no greater than 12, so the bound is $[8.4,12]$ units. (3) Subtract the largest control mean for the lower endpoint and the smallest for the upper: $[3-5,\;7-2]=[-2,5]$ units. (4) The assumption entails only that the respondent mean is at least the missing mean. The two means may differ; equality is one possibility, not a consequence.