Subgroup-Specific Delta Adjustments

Why Allow Different Departures by Subgroup?

A single delta adjustment assumes that, after conditioning on the variables in an imputation model, the departure of missing outcomes from their MAR reference is the same for everyone to whom that adjustment applies. That assumption may be implausible when a scientifically defined subgroup has a distinct reason for missing outcomes. For example, participants with greater baseline symptom burden might have missing follow-up outcomes that are systematically worse than their MAR predictions, while the corresponding departure in a lower-burden subgroup could be smaller or even point in the other direction.

A subgroup-specific adjustment makes those departures explicit. It does not identify which departure is true from the observed data alone. Instead, it defines a set of plausible assumptions, applies each to the missing outcomes in the relevant subgroup, and examines how the subgroup estimates and contrasts respond. As in the earlier material on delta adjustments, the MAR imputation model is the reference; the delta parameters describe departures from that reference for missing outcomes.

The subgroup should be defined in a way that is meaningful for the analysis and available when the adjustment is assigned. A baseline characteristic such as a prespecified age category can often serve this purpose. If subgroup membership itself is missing, it is not generally valid to assign a fixed subgroup adjustment using an unknown group label. The uncertainty in membership must instead be handled in a coherent imputation strategy. Subgroups should not be selected after inspecting which definition produces the most favorable result.

Specification and Assumptions

For a continuous outcome on its original scale, an additive adjustment is \(Y^{\mathrm{mis}}_{\delta}=Y^{\mathrm{mis}}_{\mathrm{MAR}}+\delta_g\) for a missing outcome in subgroup \(g\). Here \(\delta_g\) is measured in the outcome's units. For a binary outcome, a log-odds adjustment can be written as \(\operatorname{logit}(p_{\delta,g})=\operatorname{logit}(p_{\mathrm{MAR}})+\delta_g\), where \(p_{\mathrm{MAR}}\) is the reference probability for a missing outcome given the observed information. Thus, a delta of zero recovers the MAR reference, and a negative log-odds delta lowers the probability of the event.

These adjustments require more than correct arithmetic. The MAR reference model must be a defensible model for the observed data, including relevant predictors and the subgroup structure. The chosen scale must make sense for the outcome, and the resulting distributions must be plausible and respect the outcome's support. The subgroup definition and the rationale for its delta should be specified before interpreting the sensitivity results. A delta does not correct a poor MAR model, and a different delta in each subgroup does not by itself guarantee that the assumed departures are realistic.

For each fixed vector \((\delta_1,\ldots,\delta_K)\), apply the appropriate adjustment to missing outcomes in each subgroup in every imputed data set, then calculate the analysis estimate in each completed data set. Re-estimate subgroup estimates and their contrasts under that scenario. If \(Q_m\) is an estimate from imputation \(m\), and \(U_m\) its estimated variance, the usual multiple-imputation summaries are \(\bar Q=M^{-1}\sum_{m=1}^M Q_m\), \(\bar U=M^{-1}\sum_{m=1}^M U_m\), \(B=(M-1)^{-1}\sum_{m=1}^M(Q_m-\bar Q)^2\), and \(T=\bar U+(1+M^{-1})B\). The sensitivity parameter is held fixed within a scenario; uncertainty across imputations reflects imputation and analysis uncertainty under that scenario, not uncertainty about which delta is correct.

How Subgroup Adjustments Change Estimates

A useful justification is available for an additive adjustment when the target is a subgroup mean. Let subgroup \(g\) contain \(N_g\) individuals, of whom \(n_{\mathrm{mis},g}\) have missing outcomes. If every missing outcome is shifted by the same \(\delta_g\), the sum of the completed outcomes increases by \(n_{\mathrm{mis},g}\delta_g\), while the observed outcomes remain unchanged. Dividing by the subgroup size gives

The formula shows why the effect depends both on the delta and on the fraction missing in the subgroup. For two subgroups \(A\) and \(B\), the adjusted contrast \(B-A\) changes by \(\{n_{\mathrm{mis},B}/N_B\}\delta_B-\{n_{\mathrm{mis},A}/N_A\}\delta_A\). This is a point-estimate identity for the stated additive setup. With stochastic imputations, analysis-model estimates, nonlinear estimands, or other adjustment scales, calculate the estimand in each completed data set rather than assuming this simple identity applies.

Contrasts, Diagnostics, and Alternatives

Separate subgroup estimates are often reported because an overall average can hide different sensitivity patterns. The contrast must be computed from estimates produced under the same delta scenario. A useful sensitivity analysis includes the MAR reference vector, with all deltas set to zero, and a set of scientifically defensible alternatives. For continuous outcomes, the preceding identity helps anticipate how the contrast changes. For binary outcomes, odds shifts alter probabilities nonlinearly, so recompute the subgroup proportions or model-based estimands at each delta setting.

Before interpreting results, check the subgroup-specific missingness counts and fractions, the distribution of observed outcomes and predictors, and whether the MAR model has adequate information within each subgroup. Sparse data can make subgroup-specific reference predictions unstable. Inspect the shifted distributions, especially for bounded outcomes or unusual tail behavior, and verify that group assignments are applied consistently in every imputation. If the subgroup-specific deltas are so large that shifted values become implausible, that is evidence to reconsider the range or the scale, not a reason to overlook the problem.

A shared delta is simpler and may be appropriate when the same departure is scientifically plausible across subgroups. Pattern-specific adjustments instead distinguish missingness patterns; a subgroup adjustment distinguishes defined population groups, and those classifications need not coincide. Subgroup-specific deltas are more flexible but require more assumptions and can yield poorly supported estimates in small subgroups. They are not a substitute for a richer, well-specified MAR model or for reporting uncertainty.

A second common error is to compare subgroup estimates calculated under different delta settings as if they formed one coherent contrast. For a scenario-based comparison, specify the full delta vector and apply it consistently. Report the subgroup definitions, delta scale and values, the MAR reference, and the resulting estimates and contrasts. This makes clear which conclusions are supported across assumptions and which depend on a particular departure.