Interpreting Delta-Adjusted Outcome Distributions

Why Compare Distributions?

A delta value is an assumption about how outcomes for people with missing data differ from their reference predictions. A mean summarizes one aspect of that difference, but it cannot show how the assumption changes the spread, the tails, or the proportion meeting a clinical threshold. Two scenarios can have similar means and still imply different probabilities of a clinically important event.

The reference distribution is the predictive distribution for missing outcomes under the reference model, before applying the sensitivity adjustment. A delta-adjusted distribution is the distribution after applying the specified departure. Interpreting both makes the assumption visible: it shows not only how far a location summary moves, but also what happens to quantiles and decision-relevant probabilities.

Definition: Let \(Y_0\) denote a reference-model predictive outcome for a person with a missing outcome, with cumulative distribution function \(F_0(y)=P(Y_0\le y)\). Under a constant additive delta \(\delta\), define the adjusted outcome as \(Y_\delta=Y_0+\delta\). Its distribution is \(F_\delta(y)=F_0(y-\delta)\). The value of \(\delta\) has the same units as \(Y\); its sign and clinical meaning depend on the outcome scale.

This definition describes a location shift: the same delta is added to every reference predictive value. It preserves the shape and spread of the missing-pattern distribution, while changing its location. For a higher-is-better outcome, a negative delta represents less favorable outcomes than the reference predictions; for a lower-is-better outcome, the clinical interpretation of the sign may reverse. Always state the outcome scale and direction before interpreting a delta.

The comparison also depends on which distribution is being summarized. The missing-pattern predictive distribution concerns people whose outcomes are missing. The overall target-population distribution includes both observed and missing outcomes. If observed outcomes are left unchanged, the overall distribution is a mixture: the delta shifts only its missing-pattern component.

What Changes Under a Constant Shift?

Suppose the reference predictive distribution has a finite mean \(\mu_0\), and let \(Q_0(\tau)\) denote its \(\tau\)-quantile, defined as the smallest value \(y\) for which \(F_0(y)\ge \tau\). For the missing-pattern distribution, a constant additive delta changes the mean and quantiles by the same amount:

E(Y_\delta)=\mu_0+\delta,\qquad Q_\delta(\tau)=Q_0(\tau)+\delta.

The quantile identity follows because \(F_\delta(y)=F_0(y-\delta)\): the smallest \(y\) with \(F_0(y-\delta)\ge\tau\) is \(Q_0(\tau)+\delta\). The identity applies to discrete as well as continuous distributions when quantiles use the stated definition. The shift does not change the interquantile range or other spread summaries that are invariant to translation.

Threshold probabilities are also evaluated by shifting the threshold back to the reference scale. For a strict lower-tail event, the left limit matters:

P(Y_\delta<c)=F_0((c-\delta)-),\qquad P(Y_\delta\le c)=F_0(c-\delta).

Here \(F_0(a-)=P(Y_0<a)\) is the left limit of the reference cumulative distribution function at \(a\). The distinction is essential when the outcome is discrete and has probability mass at the threshold. For upper-tail events, \(P(Y_\delta>c)=1-F_0(c-\delta)\), while \(P(Y_\delta\ge c)=1-F_0((c-\delta)-)\).

Worked Example: A Lower-Is-Better Clinical Score

Suppose a symptom score is measured in points, with lower scores indicating fewer symptoms. For the missing-pattern reference predictions, assume \(Y_0\sim N(50,10^2)\). Consider \(\delta=-5\) points. This constant shift represents an assumption that missing outcomes average five points lower than their reference predictions. The normal reference model and constant additive shift are the assumptions for this calculation.

The reference mean is 50 points, so the adjusted mean is \(50+(-5)=45\) points. The reference median is also 50 points, and the adjusted median is \(50-5=45\) points. Both summaries describe the missing-pattern distribution, not the full target population.

For the probability that the score is below 40 points, the reference value is \(P(Y_0<40)=\Phi((40-50)/10)=\Phi(-1)=0.1587\). Because the normal distribution is continuous, strict and non-strict inequalities agree here. Under the delta assumption, \(P(Y_\delta<40)=P(Y_0<45)=\Phi((45-50)/10)=\Phi(-0.5)=0.3085\). Thus the assumed shift raises the predicted proportion below 40 by \(0.3085-0.1587=0.1498\), or about 15.0 percentage points.

For a score of at least 60 points, the reference probability is \(P(Y_0\ge60)=1-\Phi(1)=0.1587\). After adjustment it is \(P(Y_\delta\ge60)=P(Y_0\ge65)=1-\Phi(1.5)=0.0668\). This tail probability falls by about 9.19 percentage points. The two event probabilities answer different clinical questions; neither can be inferred from the change in the mean alone.

Overall Distributions and Their Summaries

Let \(r\) be the fraction of the target population with missing outcomes. Let \(G(y)\) be the cumulative distribution function of the observed outcomes, assumed unchanged across the sensitivity scenarios, and let \(F_0(y)\) describe reference predictions for missing outcomes. The overall reference distribution and its delta-adjusted version are

F_{\mathrm{ref}}(y)=(1-r)G(y)+rF_0(y),\qquad F_{\delta}(y)=(1-r)G(y)+rF_0(y-\delta).

This mixture representation assumes that \(r\) is fixed, the observed component is not altered, and the delta shifts the missing component by the same amount. It makes clear why overall summaries do not generally move as if the entire population had been shifted. If both component means are finite, the overall mean changes by \(r\delta\), not by \(\delta\):

\mu_{\delta}-\mu_{\mathrm{ref}} =\{(1-r)\mu_{\mathrm{obs}}+r(\mu_0+\delta)\} -\{(1-r)\mu_{\mathrm{obs}}+r\mu_0\} =r\delta.

No corresponding simple rule says that an overall quantile changes by \(r\delta\). It is the quantile of the mixture distribution, so its movement depends on where the observed and missing components place probability mass around that quantile. Overall event probabilities can be calculated by weighting the component probabilities, provided the event and component distributions are defined consistently.

Worked Example: A Pooled Median That Does Not Move

In a hypothetical target population, suppose 60% of outcomes are observed and are all 60 points. The remaining 40% are missing, and their reference predictive outcomes are all 50 points. Let \(\delta=-10\) points. These point-mass distributions and the fixed missing fraction are assumptions of this example.

Under the reference scenario, the overall distribution has probability 0.4 at 50 and probability 0.6 at 60. Its mean is \(0.4(50)+0.6(60)=20+36=56\) points. The cumulative probability at 50 is 0.4, so the 25th percentile is 50 points; the cumulative probability at 60 is 1, so the median is 60 points.

After adjustment, missing outcomes are all \(50-10=40\) points, while observed outcomes remain 60 points. The adjusted mean is \(0.4(40)+0.6(60)=16+36=52\) points, a decrease of \(56-52=4\) points, which equals \(r\delta=0.4(-10)=-4\). The 25th percentile is now 40 points, but the median remains 60 points: the cumulative probability at 40 is only 0.4, below 0.5, and it reaches 1 at 60.

The overall mean registers the shift in the missing component, while the median remains determined by the observed component at 60. Neither summary is erroneous. They reveal different features of this deliberately simple mixture.

Worked Example: A Clinically Relevant Responder Probability

Suppose a higher score is better and a responder is defined by \(Y\ge70\) points. In a hypothetical population, 30% of outcomes are missing. Among observed outcomes, assume half are responders. For missing outcomes, assume the reference predictive distribution is \(N(68,8^2)\), and consider \(\delta=-4\) points. The calculation assumes the observed responder proportion and missing fraction stay fixed.

Among missing outcomes under the reference model, the responder probability is \(P(Y_0\ge70)=1-\Phi((70-68)/8)=1-\Phi(0.25)=0.4013\). After the shift, it is \(P(Y_\delta\ge70)=P(Y_0\ge74)=1-\Phi((74-68)/8)=1-\Phi(0.75)=0.2266\). The missing-pattern responder probability therefore falls by \(0.4013-0.2266=0.1747\), or 17.5 percentage points.

For the whole population, weight the observed and missing probabilities. The reference responder proportion is \(0.70(0.50)+0.30(0.4013)=0.3500+0.1204=0.4704\). The adjusted responder proportion is \(0.70(0.50)+0.30(0.2266)=0.3500+0.0680=0.4180\). The overall decrease is about 0.0524, or 5.24 percentage points. It is smaller than the change among missing outcomes because the observed component is unchanged.

Interpreting, Checking, and Comparing Summaries

Each summary has a particular role. A mean gives the average location change; in the missing pattern it changes by \(\delta\), and in the overall mixture it changes by \(r\delta\) under the stated assumptions. Quantiles describe locations in the distribution and can highlight clinically important cut points or changes in the lower and upper portions. Threshold probabilities directly quantify how much probability the assumed distribution assigns to a defined event. Reporting more than one type of summary can therefore reveal consequences hidden by a single average.

Before interpreting a result, check that the model and calculation match the question:

Confirm the outcome units, the direction that is clinically favorable, and whether \(\delta\) applies only to missing outcomes.
State whether a number summarizes the missing-pattern distribution or the full target-population distribution.
Check the missing fraction and observed component used in any overall mixture calculation.
For threshold events, distinguish \(Y<c\), \(Y\le c\), \(Y>c\), and \(Y\ge c\). With discrete outcomes, inspect any probability mass at the boundary.
For bounded or otherwise restricted outcomes, verify that the adjusted distribution remains feasible; an additive shift may assign probability to impossible values.

A constant shift is not a general model of every plausible departure from the reference predictions. It preserves the missing-pattern spread and shape and moves all its quantiles equally. If clinical knowledge suggests a changed variance, a nonconstant shift across covariate groups, or a different change in one tail than another, a single constant delta may be inadequate. Similarly, averaging over covariates can make a simple marginal shift an imperfect description if the assumed departure differs across covariate strata.

The reference model also matters: a delta-adjusted distribution inherits its predictive shape from the reference distribution. A plausible shift cannot repair a poorly fitting reference model. Check whether the reference predictions are credible for the intended population, and whether the adjusted predictions are clinically plausible across the range of outcomes. In particular, examine tails and threshold probabilities, not just the mean. These checks assess the implications of the assumption; they do not identify the unobserved outcomes or establish which delta is correct.

Distributional summaries complement, rather than replace, other approaches. A mean-only comparison is compact but can obscure tail and threshold changes. Quantile comparisons describe distributional locations but do not directly answer a decision question such as the probability of response. Event probabilities answer such questions directly, but depend on the chosen threshold and may discard information about how far outcomes lie beyond it. A selection-model analysis represents missingness through a different model for the relationship between outcomes and missingness; delta-adjusted pattern-mixture summaries instead make the assumed departure in missing outcomes explicit. These are alternative sensitivity-model perspectives, not interchangeable summaries of the same assumption.

Key takeaway: Interpret each delta-adjusted summary against its reference value and identify the distribution it describes. Under a constant additive shift, missing-pattern means and quantiles translate by \(\delta\), but overall mixture quantiles need not; clinical event probabilities must be recalculated from the shifted distribution with the threshold inequality handled correctly.

Check Your Understanding

For each question, work from the stated assumptions. Solutions follow each problem.

For a missing-pattern reference distribution with mean 12 units and \(\delta=3\) units, what is the adjusted mean? If the missing fraction is 0.25 and the observed component is unchanged, by how much does the overall mean change?
Solution: The missing-pattern mean is \(12+3=15\) units. The overall mean changes by \(r\delta=0.25(3)=0.75\) units.
A missing-pattern reference distribution has 20% probability at 4 and 80% probability at 8. With \(\delta=-2\), calculate \(P(Y_\delta<6)\) and \(P(Y_\delta\le6)\).
Solution: The adjusted values are 2 with probability 0.20 and 6 with probability 0.80. Thus \(P(Y_\delta<6)=0.20\), while \(P(Y_\delta\le6)=1\). Equivalently, the strict event uses \(F_0((6-(-2))-)=F_0(8-)=0.20\); the non-strict event uses \(F_0(8)=1\).
Why can an overall median remain unchanged when the mean changes under delta adjustment?
Solution: The overall distribution is a mixture, with observed outcomes fixed and only the missing component shifted. The mean responds to the weighted change \(r\delta\), while the median depends on where the mixture’s cumulative probability reaches 0.5. That crossing may remain in the unchanged observed component.
For a continuous reference distribution, how would you calculate \(P(Y_\delta>c)\)? What changes if the outcome is discrete and the question instead asks for \(P(Y_\delta<c)\)?
Solution: For either a continuous or discrete distribution, \(P(Y_\delta>c)=1-F_0(c-\delta)\). For the strict lower-tail event, use \(P(Y_\delta<c)=F_0((c-\delta)-)\), not \(F_0(c-\delta)\), because a discrete distribution may have probability mass at the boundary.

Interpreting Delta-Adjusted Outcome Distributions

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