Why Use Arm-Specific Delta Adjustments?
A cardiovascular prevention trial may measure each participant’s estimated event risk at follow-up, but some measurements will be missing. If the probability of a missing measurement depends on the unobserved risk itself, an analysis relying only on observed data may not capture the risk among participants who were not measured. A delta sensitivity analysis asks how conclusions change when those missing measurements are assumed to differ from their predictions under a missing-at-random (MAR) model.
The aim is not to estimate the delta values from the observed trial data. Delta values encode assumptions about unobserved outcomes, so the analysis compares a set of clinically interpretable assumptions. Because missingness can differ between randomized arms, the departures can be specified separately by arm. The resulting range of treatment effects describes sensitivity to those choices; it is not a confidence interval for an unknown delta.
This additive definition is on the risk scale: a delta of \(1\) means one percentage point, not a one-percent relative increase. The conditional distribution of the MAR imputations is otherwise retained in this simple location-shift model. The outcome, however, must remain within its possible range. If risk is recorded from 0 to 100%, proposed shifts should not create impossible values. Truncating values at a boundary changes the effective average shift, so it must be handled explicitly rather than treated as an unchanged additive adjustment.
How the Assumptions Enter the Estimate
Suppose there are \(n_a\) randomized participants in arm \(a\), of whom \(m_a\) have missing follow-up measurements. Write \(q_a=m_a/n_a\) for that arm’s missing fraction. Let \(\bar{Y}_{a,obs}\) be the observed mean and \(\bar{Y}_{a,mis}^{MAR}\) the mean predicted for missing participants under the MAR model. The arm mean under a delta scenario is
This expression weights each group’s mean by its share of the randomized arm. Only the missing fraction receives the delta shift. Define the treatment effect as the treatment-arm mean minus the control-arm mean, so a negative risk difference favors treatment. Subtracting the two arm means gives
where \(RD^{MAR}\) is the risk difference under the MAR imputations. This formula assumes that the same specified delta applies to missing participants within an arm, while their MAR-predicted values and other modeled features can vary across participants. It is a useful check on the direction and size of a proposed shift.
For this analysis to be meaningful, the risk outcome must be defined for the randomized participants being compared, and the treatment contrast must be specified in advance. The MAR model should account for relevant observed information, including predictors of the outcome and missingness, and should be fitted in a way that respects the randomized arms and appropriate outcome scale. The delta then represents a departure from that model for missing outcomes, not a correction to observed outcomes. Randomization supports a treatment-arm comparison, but it does not identify the outcomes that were not measured or determine the delta values.
Worked Analyses
Worked Example: Calculating an Arm-Specific Risk Difference
Hypothetical data. A trial randomizes 100 participants to treatment and 100 to control. In the treatment arm, 80 risk measurements are observed, with mean \(4.0\%\); MAR imputations for the 20 missing measurements have mean \(4.2\%\). In control, 90 measurements are observed, with mean \(4.5\%\); MAR imputations for the 10 missing measurements have mean \(4.6\%\). Assume, for this calculation, that the MAR model is the chosen reference and that no delta shift is applied.
The missing fractions are \(q_T=20/100=0.20\) and \(q_C=10/100=0.10\). The corresponding MAR arm means are
Thus \(RD^{MAR}=4.04-4.51=-0.47\) percentage points. Under these assumptions, estimated risk is 0.47 percentage points lower in the treatment arm. This is a model-based comparison using all randomized participants, with the missing outcomes supplied by MAR imputations; it is not a direct comparison of observed means alone.
Worked Example: Comparing Arm-Specific Delta Scenarios
Use the hypothetical trial above. Suppose the sensitivity analysis considers positive deltas, meaning missing participants are assumed to have higher risk than their MAR predictions. For each scenario, calculate \(RD^{(\delta)}=-0.47+0.20\delta_T-0.10\delta_C\). Deltas and effects are in percentage points.
| \(\delta_T\) | \(\delta_C\) | Adjustment to \(RD^{MAR}\) | \(RD^{(\delta)}\) |
|---|---|---|---|
| 0 | 0 | \(0.20(0)-0.10(0)=0\) | \(-0.47\) |
| 1 | 0 | \(0.20(1)-0.10(0)=0.20\) | \(-0.27\) |
| 0 | 1 | \(0.20(0)-0.10(1)=-0.10\) | \(-0.57\) |
| 2 | 1 | \(0.20(2)-0.10(1)=0.30\) | \(-0.17\) |
| 2.5 | 0 | \(0.20(2.5)-0.10(0)=0.50\) | \(0.03\) |
For example, under \(\delta_T=1\) and \(\delta_C=0\), the treatment-arm mean increases by \(0.20(1)=0.20\) percentage points, while the control mean is unchanged. The estimated contrast is therefore \(-0.47+0.20=-0.27\) percentage points. Under \(\delta_T=0,\delta_C=1\), the control-arm mean increases by \(0.10(1)=0.10\) percentage points, making the treatment-minus-control contrast more negative: \(-0.47-0.10=-0.57\).
The displayed values form a sensitivity range for these particular assumptions, not a range of effects guaranteed to contain the truth. In particular, the \(0.03\) result says that this chosen scenario gives a treatment-minus-control risk difference of 0.03 percentage points. It does not establish that the treatments have equal effects or that the assumptions in that scenario are correct.
Worked Example: Combining Multiple-Imputation Uncertainty
For one fixed delta scenario, suppose five completed-data analyses give hypothetical risk differences \(Q_m=-0.30,-0.26,-0.28,-0.24,-0.26\) percentage points. Suppose each has within-imputation variance \(U_m=0.0400\) percentage-points squared. The mean estimate is
The deviations from \(-0.268\) are \(-0.032,0.008,-0.012,0.028,0.008\). Their squared deviations sum to \(0.00208\), so the between-imputation variance is \(B=0.00208/(5-1)=0.000520\). The average within-imputation variance is \(\bar{U}=0.0400\). Rubin’s total variance is
Using a large-sample normal approximation, a 95% interval is \(-0.268\pm1.96(0.2016)\), or approximately \((-0.663,0.127)\) percentage points. Under this delta scenario, the interval includes zero. It describes uncertainty in the estimated contrast conditional on the specified sensitivity assumptions; it does not account for uncertainty about which delta values are appropriate. In an application, use the appropriate multiple-imputation degrees of freedom when a normal approximation is not justified.
Choosing and Checking a Sensitivity Range
A delta grid should be chosen to make the assumptions interpretable and clinically discussable. For example, investigators might ask whether missing participants could plausibly have risk one or two percentage points higher than the MAR prediction in either arm. The scale and range should reflect the outcome, follow-up time, and clinical context. A grid that is too narrow can make conclusions appear robust simply because it excludes meaningful departures; an implausibly wide grid can emphasize scenarios that are not credible.
Before interpreting the resulting range, check the analysis and its assumptions:
- Check missingness by arm. Report the number and fraction of missing measurements in each arm. These fractions help explain how strongly each delta changes the contrast.
- Check the MAR predictions. Compare imputed and observed risk distributions by arm and across important observed covariates. Predictions far outside the observed support may indicate poor model fit or extrapolation.
- Check outcome bounds. Confirm that shifted imputed risks remain possible. If a bounded-scale transformation or truncation is needed, report it and verify the actual shifted means.
- Check implementation. Confirm that deltas are applied only to missing outcomes, in the correct arm, and in percentage-point units. Verify calculated contrasts against \(RD^{MAR}+q_T\delta_T-q_C\delta_C\) when the additive assumptions apply.
- Check imputation stability. Review convergence and Monte Carlo variability. Too few imputations can make reported uncertainty unstable even if the point estimate appears consistent.
The delta is an assumption, not a parameter identified by the observed data. A sensitivity analysis should therefore show the selected scenarios and their rationale, rather than selecting a preferred delta because it produces a desired result. For any one scenario, a confidence interval reflects sampling and imputation uncertainty under that assumption. Differences between scenario estimates reflect sensitivity to assumptions, not random sampling variation alone.
How This Approach Differs From Alternatives
Under a MAR analysis, the imputation model uses observed information and makes no additional delta shift, so \(\delta_T=\delta_C=0\). Arm-specific delta sensitivity analysis keeps that MAR-based structure as a reference and asks how the result changes when missing outcomes depart from it. Reference-based imputation instead borrows a specified reference-arm outcome distribution for some missing outcomes; it is a different assumption, not a synonym for an arm-specific delta shift. Selection models describe missingness conditional on outcomes, whereas the delta approach directly specifies a departure for the missing-outcome distribution. These approaches can address related concerns, but their assumptions and interpretations differ.
A limitation of a single delta per arm is that it imposes a simple average departure. The true departure could vary with baseline risk, time, or the reason a measurement is missing. A constant additive shift may also be inappropriate near the bounds of a risk scale. These limitations do not make delta analysis unhelpful; they mean that the stated assumptions, outcome scale, and range of scenarios must remain visible when interpreting the results.
Common Pitfall
Check Your Understanding
Answer each question using the treatment-minus-control risk difference convention. Solutions follow the questions.
- An analysis has \(RD^{MAR}=-0.60\) percentage points, \(q_T=0.25\), and \(q_C=0.10\). Under \(\delta_T=1.2\) and \(\delta_C=0.5\), what is the adjusted risk difference?
- If both arms have missing fraction \(0.20\) and both deltas are \(1.5\) percentage points, how much does the risk difference change?
- Does observing a treatment effect under \(\delta_T=\delta_C=0\) establish that missing outcomes are MAR? Explain.
- What should an analyst check if adding a positive delta produces imputed risks above 100%?
Solutions. (1) The adjustment is \(0.25(1.2)-0.10(0.5)=0.30-0.05=0.25\) percentage points, so \(RD^{(\delta)}=-0.60+0.25=-0.35\) percentage points. (2) The adjustment is \(0.20(1.5)-0.20(1.5)=0\), so the contrast is unchanged even though each arm mean increases. (3) No. The zero-delta analysis uses the MAR imputation model as its reference assumption; observed data alone do not establish that the missing outcomes follow that model. (4) The analyst should verify the outcome scale and the proposed delta range, then use a defensible bounded-scale approach if needed. Simple truncation changes the effective mean shift and must be reported and checked.