Delta Interactions for Clinically Distinct Subgroups

Why Let the Delta Differ by Subgroup?

A single delta shift assumes that, after conditioning on the variables in the reference model, the departure for missing outcomes is the same in every patient group. That may be too restrictive when groups have clinically distinct reasons for missingness. For example, missing outcomes among patients with severe disease could plausibly be lower than the reference model predicts, while the departure among patients with milder disease might be smaller.

A subgroup-specific delta allows these departures to differ. The key is to keep two ideas separate: the reference model may already predict different outcome levels for different subgroups, and the delta interaction describes whether the departure from that reference also differs. A group with a lower reference mean does not automatically have a more negative delta.

Definition: Let $Y$ be a continuous outcome, $R=1$ indicate that $Y$ is observed, and $R=0$ indicate that it is missing. Let $G$ be a prespecified clinical subgroup and $X$ denote other covariates. Write $m_{\text{ref}}(a,x,g)$ for the conditional mean supplied by the MAR reference model for treatment arm $a$, covariates $x$, and subgroup $g$. An additive subgroup-specific delta model for the missing outcomes is $$ E(Y\mid R=0,A=a,X=x,G=g) = m_{\text{ref}}(a,x,g)+\delta_g. $$ The delta $\delta_g$ is the assumed difference, in outcome units, between the missing-outcome conditional mean and the MAR reference conditional mean in subgroup $g$. A positive delta shifts that mean upward; a negative delta shifts it downward.

The reference model is not required to give every subgroup the same prediction. It may include subgroup indicators, treatment-by-subgroup terms, or covariate effects, as needed to describe the observed data. The delta interaction is an additional sensitivity assumption about missing outcomes, not a replacement for those reference-model terms.

Parameterizing the Interaction

With two subgroups, choose one as the reference subgroup. For example, if $G=0$ denotes the reference group and $G=1$ the other group, write

\delta_g=\delta_0+\gamma g.

Then the departure in the reference subgroup is $\delta_0$, and the departure in the other subgroup is $\delta_0+\gamma$. The interaction coefficient $\gamma$ is the difference between those departures:

\gamma=\delta_1-\delta_0.

For more than two subgroups, one can use one delta for each group, or use a reference delta and one interaction coefficient for each nonreference group. These are alternative parameterizations of the same set of subgroup-specific shifts. The values are generally sensitivity parameters: observed outcomes alone do not identify how the unobserved outcomes depart from the reference model.

Worked Example: Interpreting a Delta Interaction

Suppose the prespecified groups are lower-risk ($G=0$) and higher-risk ($G=1$). On a hypothetical symptom scale measured in points, the reference model predicts missing-outcome means of 50 points and 62 points, respectively, at the covariate values under consideration. Consider $\delta_0=-2$ and $\gamma=1.5$.

The lower-risk departure is $\delta_0=-2$ points. The higher-risk departure is $$ \delta_1=\delta_0+\gamma=-2+1.5=-0.5\text{ points}. $$

Thus the adjusted missing-outcome means are $50-2=48$ points for the lower-risk group and $62-0.5=61.5$ points for the higher-risk group. The interaction says that the higher-risk group’s departure is 1.5 points higher (less negative) than the lower-risk group’s departure. It does not say that the higher-risk group’s outcome mean is 1.5 points higher: the reference means already differ by 12 points.

What the Interaction Changes

A short derivation makes the interpretation precise. For two subgroups, let their reference means at the same specified covariates and treatment arm be $m_{\text{ref},0}$ and $m_{\text{ref},1}$. Their adjusted missing-outcome mean difference is

\begin{aligned} &(m_{\text{ref},1}+\delta_1)-(m_{\text{ref},0}+\delta_0)\\ &=(m_{\text{ref},1}-m_{\text{ref},0})+(\delta_1-\delta_0)\\ &=(m_{\text{ref},1}-m_{\text{ref},0})+\gamma. \end{aligned}

The subgroup difference in the reference predictions remains present; the delta interaction adds the difference in assumed departures. If $\gamma=0$, the departures are equal, even if the reference predictions differ substantially. If $\gamma\ne0$, the sensitivity scenario changes the subgroup contrast among missing outcomes by $\gamma$ points on this scale.

Worked Example: Common Delta Versus an Interaction

Suppose the MAR reference model predicts missing-outcome means of 40 points in subgroup A and 60 points in subgroup B. Compare a common delta of $-4$ points with a scenario using $\delta_A=-2$ and $\delta_B=-8$.

Common delta. The adjusted means are $40-4=36$ points and $60-4=56$ points. Their difference is $56-36=20$ points, the same as the reference difference.

Subgroup-specific deltas. The adjusted means are $40-2=38$ points and $60-8=52$ points. Their difference is $52-38=14$ points. The departure difference is $\delta_B-\delta_A=-8-(-2)=-6$ points, so the subgroup contrast is shifted downward by 6 points relative to the reference contrast: $20+(-6)=14$.

These calculations describe hypothetical sensitivity scenarios. They do not establish that subgroup B truly has a 6-point more negative departure; that difference is an assumption to examine.

Using Subgroup Deltas in an Overall Summary

When the target is an overall mean, a group’s delta affects the contribution of its missing outcomes. The calculation must respect the numbers of observed and missing outcomes in each subgroup. The following example uses simple subgroup-specific reference predictions; in a fitted analysis, predictions may also vary with $A$ and $X$.

Worked Example: Computing an Adjusted Overall Mean

In hypothetical data, subgroup L has 40 observed outcomes with mean 50 points and 10 missing outcomes. Its reference-model mean for the missing outcomes is 49 points, and its assumed delta is $-2$ points. Subgroup H has 30 observed outcomes with mean 60 points and 20 missing outcomes. Its missing-outcome reference mean is 58 points, and its assumed delta is $-5$ points. Assume these counts and means refer to the same target population being summarized.

The adjusted missing-outcome means are $49-2=47$ points in subgroup L and $58-5=53$ points in subgroup H. Combining observed outcomes with their adjusted missing-outcome predictions gives $$ \frac{40(50)+10(47)+30(60)+20(53)}{40+10+30+20} = \frac{2000+470+1800+1060}{100} = 53.3\text{ points}. $$

For comparison, using the reference predictions without a delta gives $$ \frac{40(50)+10(49)+30(60)+20(58)}{100} = \frac{5450}{100} = 54.5\text{ points}. $$

Under these specified assumptions, the subgroup-specific shifts lower the overall mean by $53.3-54.5=-1.2$ points. This weighted result depends on the subgroup counts as well as the delta values: a shift in a subgroup with more missing outcomes can have a larger effect on the overall summary.

Assumptions and Practical Checks

A subgroup-specific delta analysis is interpretable only when its reference model and sensitivity assumptions are stated clearly. In particular:

Prespecify meaningful groups. Define the clinical subgroup using information available for the analysis, and state its categories. Choosing groups after inspecting which ones produce a desired result undermines the interpretation.
Specify the reference model. The model must give credible conditional predictions for the missing outcomes under its MAR reference assumption. It should represent relevant observed-data relationships, including subgroup differences where warranted.
Choose a scale and direction. Each $\delta_g$ is measured on the outcome scale used in the model. The sign convention here is that a positive value means higher missing outcomes than the reference prediction.
State the within-group restriction. The model assumes a constant additive shift within each subgroup, conditional on the variables in the reference model. It does not allow the shift to vary further with $X$ unless that variation is explicitly included.
Check data support. Review the observed and missing counts in each subgroup and whether observed patients cover the covariate ranges where predictions are required. Very sparse observed data can make the reference predictions unstable.
Separate estimation from sensitivity assumptions. The observed data can inform the reference-model estimates, but they do not by themselves determine the delta interaction for unobserved outcomes. Report conclusions conditional on the delta values considered.

A practical diagnostic is to tabulate observed and missing counts by subgroup and inspect the reference-model predictions across the covariate ranges used. Look for empty or nearly empty cells, extreme predictions, and groups for which the fitted reference model relies on extrapolation. Also compare the implications of a common delta with clinically plausible subgroup-specific values. These checks can reveal fragile modeling or an overly influential scenario, but they cannot verify the assumed departures for outcomes that were never observed.

How This Differs From Other Subgroup Effects

Quantity	What it describes	What it does not mean
Subgroup term in the reference model	Differences in reference predictions across subgroups, conditional on modeled variables	A difference in missing-outcome departures
Delta for subgroup $g$	The assumed shift from the reference prediction for missing outcomes in that subgroup	A subgroup mean difference by itself
Delta interaction $\gamma$	The difference in departures between a subgroup and the reference subgroup	A treatment effect or evidence that the departure difference is identified by observed outcomes

A common delta is the simpler alternative when equal departures across subgroups are clinically defensible. Subgroup-specific deltas are useful when that equality is doubtful, but introduce additional sensitivity parameters and therefore more choices to report. Estimating a separate departure as though it were an ordinary observed-data regression coefficient is not a substitute: the missing outcomes needed to identify that departure are unavailable.

Common Mistakes and Limitations

Confusing a subgroup main effect with a delta interaction. A lower reference prediction in one group is a modeled subgroup difference. It does not imply a more negative delta.
Calling $\gamma$ the subgroup outcome difference. It is the difference in departures. The adjusted missing-outcome contrast also includes the reference-model contrast.
Assuming observed data select the right delta. The data can assess the reference model against observed outcomes, but cannot directly test the shift for unobserved outcomes.
Ignoring subgroup-specific uncertainty or instability. Few observed outcomes in a group can make its reference predictions unreliable, while additional delta parameters can make conclusions depend strongly on the scenario chosen.

Finally, an additive shift is meaningful only on the selected outcome scale and under the stated within-group restriction. A scenario that produces implausible predictions should be reconsidered, not presented as supported merely because the arithmetic is possible. A careful report identifies the groups, reference model, delta values, units, and the conclusions that change across the scenarios considered.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A two-group model uses $\delta_0=-3$ and interaction $\gamma=2$. What is the delta in group 1, and how does its departure compare with group 0?
Reference missing-outcome means are 45 and 55 points in two groups. Their deltas are both $-4$ points. What are the adjusted means, and what happens to their difference?
In a model with group 1 as the reference, explain why a group indicator in $m_{\text{ref}}$ is not the same as a delta interaction.
A subgroup has very few observed outcomes but many missing outcomes. Name one concern and one conclusion that observed data cannot establish.

Solutions. (1) $\delta_1=\delta_0+\gamma=-3+2=-1$ point. Group 1’s departure is 2 points higher, or less negative, than group 0’s. (2) The adjusted means are $45-4=41$ and $55-4=51$ points. Their difference remains 10 points because the common shift cancels. (3) The group indicator describes differences in the reference predictions; the delta interaction describes differences in assumed shifts from those predictions for missing outcomes. They affect different parts of the model. (4) The reference prediction may be unstable or require extrapolation because observed support is sparse. The observed data cannot establish the true mean departure among that subgroup’s missing outcomes.