Pattern-Specific Delta Parameters

Why Give Different Patterns Different Deltas?

A single sensitivity parameter can be too restrictive when people with different missing-data patterns may have different departures from a missing-at-random (MAR) reference model. For example, people missing the outcome but observed on an auxiliary measure may differ plausibly from people missing both the outcome and that auxiliary measure. Assigning a separate delta to each pattern makes those assumptions visible. It does not make the missing outcomes identifiable from the observed data: the deltas remain sensitivity assumptions.

The key is to define what distinguishes the patterns before assigning parameters. A pattern is a configuration of which variables are observed and which are missing—not a clinical subgroup invented after looking at the outcomes. The interpretation of a delta then depends on the reference model, the outcome scale, and the particular pattern to which it applies.

Definition: Let $G$ denote the missingness pattern, with possible values $g$. For a target outcome $Y$, let $\mathcal{M}$ be the set of patterns in which $Y$ is missing. For each $g\in\mathcal{M}$, a pattern-specific additive delta $\delta_g$ specifies the departure of the target outcome in that pattern from its MAR reference conditional mean: $$ E(Y\mid W_g,G=g)=m_{0,g}(W_g)+\delta_g. $$ Here $W_g$ denotes the observed information used for that pattern, and $m_{0,g}(W_g)$ is the conditional mean under the chosen MAR reference model. The delta has the same units as $Y$. A positive delta shifts the conditional mean upward; a negative delta shifts it downward.

Patterns with observed $Y$ do not need a delta for that outcome: their $Y$ values are observed. A pattern may nevertheless be relevant to estimating the reference model. In applying the definition, the reference mean must be specified for the people whose outcome is missing, using the information available for their pattern. If the model conditions on a variable that is itself missing in a pattern, the analyst must specify how the reference prediction is formed there; silently treating unavailable information as observed is not justified.

Mean Shifts and Distributional Interpretation

The definition above specifies a conditional mean departure. It does not, by itself, determine the entire outcome distribution. One possible additional assumption is a location shift: within pattern $g$ and at given $W_g=w$, the distribution under the sensitivity model is the MAR reference distribution translated by $\delta_g$. If $f_{0,g}(y\mid w)$ is the reference density, this assumption is

f_{\delta,g}(y\mid w)=f_{0,g}(y-\delta_g\mid w).

This location-shift model keeps the reference distribution’s shape and spread while changing its location. It is stronger than a mean-only specification. With only the conditional mean restriction, different variances, tail behavior, or other features remain possible; those features cannot be inferred from $\delta_g$ alone.

The location-shift interpretation follows directly by a change of variable. If $Y_0$ has density $f_{0,g}(\,\cdot\mid w)$, define $Y_\delta=Y_0+\delta_g$. Its density at $y$ is the density of $Y_0$ at $y-\delta_g$, giving the expression above. Consequently,

E(Y_\delta\mid w,g)=E(Y_0\mid w,g)+\delta_g =m_{0,g}(w)+\delta_g, \qquad \operatorname{Var}(Y_\delta\mid w,g)=\operatorname{Var}(Y_0\mid w,g).

Thus translating the distribution implies the stated mean shift and preserves conditional variance. The reverse is not generally true: knowing a distribution’s mean does not establish that the rest of its distribution has been translated unchanged.

Worked Example: Interpreting Two Pattern-Specific Mean Shifts

Consider a hypothetical study with a continuous symptom score $Y$, measured in points, and an auxiliary score $W$. Pattern A has $Y$ missing but $W$ observed; pattern B has both $Y$ and $W$ missing. Suppose the specified MAR reference means for representative pattern members are 10 points in A and 8 points in B. The sensitivity assumptions are $\delta_A=-2$ points and $\delta_B=-5$ points.

For pattern A, the shifted conditional mean is $10+(-2)=8$ points. For pattern B, it is $8+(-5)=3$ points. These are departures from the respective pattern-specific reference means; the two deltas need not be equal because the assumptions concern different missingness patterns.

If the analyst additionally assumes a location shift and the reference distributions are $N(10,4)$ in A and $N(8,9)$ in B, the shifted distributions are $N(8,4)$ and $N(3,9)$, respectively. The second argument is the variance. The means change by the assigned deltas, while the variances remain 4 and 9 under this extra assumption. Without the location-shift assumption, those variances are not determined by the two mean deltas.

Worked Pattern-Specific Sensitivity Analysis

A pattern-specific analysis requires the pattern proportions, a reference model for the missing outcomes in each pattern, and a delta for each pattern being examined. The deltas should be stated on the outcome scale and tied explicitly to the corresponding reference distribution. The following calculation illustrates the bookkeeping; it is not a way to estimate the deltas from the observed data.

Worked Example: Combining Pattern-Specific Shifts

Suppose a hypothetical study has 100 participants and a continuous outcome in points. Sixty participants have the outcome observed, with mean 14 points. Of the remaining 40, 25 are in pattern A and 15 in pattern B. The MAR reference means for the missing outcomes are 10 points in A and 8 points in B. Consider $\delta_A=-2$ points and $\delta_B=-5$ points.

First shift each pattern’s reference mean. Pattern A has mean $10-2=8$ points, and pattern B has mean $8-5=3$ points. The weighted mean under these assumptions is $$ \frac{60(14)+25(8)+15(3)}{100} =\frac{840+200+45}{100} =10.85\text{ points}. $$

For comparison, the same pattern proportions with both deltas set to zero give $$ \frac{60(14)+25(10)+15(8)}{100} =\frac{840+250+120}{100} =12.10\text{ points}. $$

Under the specified reference model and deltas, the pattern-specific departures reduce the overall mean by $12.10-10.85=1.25$ points relative to the MAR-reference calculation. This is a sensitivity-analysis contrast, not evidence that the missing outcomes actually have those values.

Worked Example: Translating a Pattern’s Distribution

For a hypothetical missingness pattern C, suppose the MAR reference distribution for a continuous outcome is $Y\sim N(20,25)$, in points, and the assigned delta is $\delta_C=-3$ points. Assume a location shift for this pattern.

The shifted distribution is $N(20-3,25)=N(17,25)$. Its standard deviation remains $\sqrt{25}=5$ points. For example, the shifted probability of an outcome below 12 points is $$ P(Y<12\mid G=C) =\Phi\left(\frac{12-17}{5}\right) =\Phi(-1) \approx 0.1587. $$

Thus, under the stated reference distribution and location-shift assumption, about 15.87% of pattern C’s shifted outcomes fall below 12 points. This probability is conditional on pattern C and the assumptions just specified; it is not a claim that the observed data identify that proportion among the missing outcomes.

Assumptions Needed for Interpretation

A delta is interpretable only relative to a well-defined reference. For every pattern receiving a delta, the analyst should be able to state what information is conditioned on, how the MAR reference distribution or mean is obtained, and what the delta changes. The following assumptions and checks matter:

Pattern definition is substantively meaningful. The observed/missing configurations must be coded consistently. Combining patterns assumes the same delta is suitable for the combined configurations; separating patterns allows different assumptions but can create sparse groups.
The reference model is adequate. The MAR prediction must be relevant to the missing cases in each pattern. Its covariates, functional form, and handling of unavailable predictors need justification. A delta does not repair a poorly specified reference model.
There is support for prediction. If people in a missingness pattern have covariate values with little or no representation among cases with observed outcomes, reference predictions rely heavily on extrapolation. Pattern-specific deltas do not remove that limitation.
The scale and sign are explicit. A delta is measured in the units of the modeled outcome. The analyst must state whether higher values are favorable or unfavorable; “negative” alone does not convey clinical meaning.
Distributional assumptions are not hidden. A mean delta alone asserts a mean departure. Translating the whole distribution also assumes its other conditional features follow the reference shape and spread.
Each delta is an assumption, not an estimated fact. For a pattern in which $Y$ is never observed, its missing-outcome departure cannot be checked directly against those unobserved outcomes. Observed-data diagnostics can assess aspects of the reference model, but cannot establish the chosen delta.

Worked Example: A Sparse Pattern and Weak Reference Support

Suppose a hypothetical study has 200 participants. Pattern A contains 30 people with $Y$ missing and an auxiliary score $W$ observed. Pattern B contains only 4 people with both $Y$ and $W$ missing. The MAR reference model predicts pattern A outcomes using $W$, but provides a single unconditional mean for pattern B.

For A, the analyst can check whether the observed-$Y$ participants cover the range of $W$ found among the 30 pattern A participants. If the ranges overlap poorly, the conditional reference means depend on extrapolation. A delta for A expresses a further departure from those predictions; it does not make the extrapolation reliable.

For B, four participants provide little information about the pattern’s prevalence, and no $Y$ values are observed for that pattern. The reference mean and $\delta_B$ therefore require particularly cautious interpretation. Reporting a precise-looking shifted mean without describing this dependence would overstate what the data support. A practical response is to show results over a scientifically defensible range of $\delta_B$ values and report the small pattern size and reference-model limitation.

Comparisons, Diagnostics, and Limitations

A common delta across all patterns is simpler and may be appropriate when the same departure is defensible for every pattern. Pattern-specific deltas relax that restriction: they allow distinct departures, but require more assumptions and more choices to communicate. A zero delta in every pattern recovers the MAR reference analysis. Neither multiple deltas nor a more elaborate pattern definition makes the missing outcomes empirically identified.

Diagnostics should focus on observable parts of the analysis: confirm the pattern counts and definitions; examine the observed-data fit and covariate overlap used for each reference prediction; check that the delta units and signs match the outcome coding; and verify that shifted predictions are plausible on the outcome scale. If a location shift produces impossible values for a bounded outcome, that is a sign the chosen model or scale needs reconsideration. Also inspect the sensitivity results across a range of pattern-specific values rather than relying on one convenient assignment.

A central limitation is that pattern-specific parameters can multiply quickly. Small patterns can make the analysis difficult to summarize, and the observed data cannot select among competing delta values for outcomes entirely missing in a pattern. Results should therefore be presented as consequences of explicit assumptions, with each $\delta_g$ labeled by its pattern and its units. The earlier material on delta adjustment establishes the general sensitivity role of a delta; the additional discipline here is to keep each departure attached to the correct pattern-specific reference.

Common pitfall: “Pattern A and pattern B have different missingness configurations, so the data show that their deltas must differ.” This is incorrect: missingness patterns can be observed, but the departures of their unobserved outcomes from their MAR references generally are not. The correction is to treat $\delta_A$ and $\delta_B$ as separate, explicit sensitivity assumptions, explain why their values are plausible, and examine how conclusions change across plausible alternatives.

Check Your Understanding

Answer each question using the definitions and assumptions in this tutorial. Brief solutions follow.

A pattern’s MAR reference mean is 6 points and its assigned delta is $+1.5$ points. What is its shifted conditional mean, and what does the delta’s sign mean?
Under a location shift, a reference distribution is $N(12,16)$ and $\delta_g=-2$. Give the shifted distribution and its standard deviation.
Why does specifying a mean delta not, by itself, determine a pattern’s variance?
A target outcome is missing in patterns A and B. What additional interpretation is imposed by setting $\delta_A=\delta_B$?
What is one limitation of a reference prediction for a pattern whose covariate values have little overlap with those of participants with observed outcomes?

Solutions. (1) The shifted mean is $6+1.5=7.5$ points; the positive delta means an assumed upward departure from that pattern’s MAR reference mean. (2) The shifted distribution is $N(10,16)$, with standard deviation $\sqrt{16}=4$ points. (3) A mean restriction constrains only the mean; variance and other distributional features require additional assumptions or information. (4) It assumes the two patterns have the same additive departure on the outcome scale, despite having distinct missingness configurations. (5) The prediction may rely on extrapolation, making the reference distribution uncertain; adding a delta does not resolve the lack of support.