Apply Delta Adjustments to Imputed Outcomes

Why Apply a Delta Adjustment?

A baseline multiple-imputation analysis may fill in missing outcomes under a missing-at-random (MAR) assumption. A delta adjustment changes selected imputed values to represent a different, prespecified assumption about outcomes after dropout. It is a practical pattern-mixture sensitivity analysis: the data used for the baseline analysis are retained, while specified post-dropout imputations are shifted on the outcome scale.

The shift is not learned from the observed data. Its value expresses a sensitivity assumption, so it should be specified before comparing results and interpreted in the outcome’s units. For example, if higher symptom scores are worse, a positive delta means imputed post-dropout symptoms are assumed to be worse than their baseline-imputed values. The sign has the opposite practical meaning when lower scores are worse.

Definition: Let $Y_{it}^{(m,0)}$ be the outcome for participant $i$ at assessment time $t$ in completed dataset $m$ under the baseline imputation analysis. Let $a_i$ denote the participant’s randomized arm. Let $P_{it}=1$ indicate that the outcome is missing at a scheduled time after the participant’s dropout, according to the prespecified dropout definition; otherwise $P_{it}=0$. An additive arm- and time-specific delta adjustment produces $$ Y_{it}^{(m,\delta)} = \begin{cases} Y_{it}^{(m,0)}+\delta_{a_i,t}, & P_{it}=1,\\ Y_{it}^{(m,0)}, & P_{it}=0. \end{cases} $$ Here $\delta_{a,t}$ is a fixed shift in outcome units for arm $a$ at time $t$. The rule applies only to designated post-dropout missing outcomes; it leaves observed outcomes and all other completed values unchanged.

Specify the Adjustment Before Applying It

An implementation needs a clear rule for which cells receive a shift. Define the dropout event and identify the scheduled times after that event. Do not infer eligibility merely from a value being missing: a missed visit before dropout, for example, is not automatically a post-dropout outcome. Keep the participant’s arm and the assessment time attached to each record so that the correct $\delta_{a,t}$ is selected.

The baseline MAR imputation model and its completed datasets provide the starting point. Apply the same prespecified delta table to every imputation. Do not redraw the imputations, alter the observed outcomes, or apply an additional random shift unless the sensitivity specification explicitly calls for that different procedure. After adjustment, fit the same substantive analysis model and target the same estimand as in the baseline analysis. The adjusted and baseline estimates can then be compared; inferential summaries must account for the multiple imputations using the course’s standard pooling procedure.

The adjustment changes the imputed outcome values; it does not by itself refit the imputation model under a new likelihood. With a fixed delta, the shifted values represent a chosen departure from the baseline imputation assumptions. The baseline imputation model still needs to be appropriate for its role, and the delta values need substantive justification. A delta analysis cannot establish that its chosen assumption is true.

Worked Example: Apply Two Shifts and Preserve Other Values

Suppose a hypothetical trial measures a symptom score at weeks 4, 8, and 12; higher scores indicate worse symptoms. The trial has a control arm and an active-treatment arm. A participant in the active arm drops out after week 4. In one baseline completed dataset, the participant’s observed week-4 score is 18, and the imputed week-8 and week-12 scores are 20 and 19. The prespecified active-arm shifts are $+2$ points at week 8 and $+3$ points at week 12.

The week-8 value is post-dropout and missing, so it is adjusted as $20+2=22$. The week-12 value is also post-dropout and missing, so it becomes $19+3=22$. Week 4 is observed, so its value remains 18. The completed record is therefore 18, 22, 22. The interpretation is that, under this sensitivity specification, the imputed post-dropout symptom scores are assumed to be 2 and 3 points worse than their baseline-imputed values at the respective visits.

If the participant had instead missed week 4 while still under follow-up and then dropped out before week 8, the week-4 imputation would remain unchanged under this rule. Its missingness alone would not make it eligible for the post-dropout adjustment.

What the Shift Does to a Summary

For a simple mean, the effect of a constant shift can be derived directly. In a group of $n$ values, suppose exactly $k$ are eligible for a common delta $\delta$, and the other values remain unchanged. The adjusted mean is the original mean plus the total added shift divided by $n$:

\bar{Y}^{(\delta)} = \frac{\sum_{i=1}^{n}Y_i^{(0)}+k\delta}{n} = \bar{Y}^{(0)}+\frac{k}{n}\delta.

This shows why the arm-specific contrast does not generally change by the full value of delta: only eligible records are shifted, and their share of the analysis group matters. For a model-based estimate, the effect can be more complex because the model is refit to the adjusted dataset. In either case, calculate the analysis estimate from the completed data rather than assuming that delta can simply be added to the final treatment effect.

Worked Example: Compare a Baseline and Adjusted Endpoint Contrast

Consider one hypothetical completed dataset with four participants per arm at a final visit. All four control outcomes are observed: 8, 9, 7, and 8 points. In the active arm, two observed outcomes are 6 and 7 points, and two post-dropout outcomes are imputed as 7 and 8 points. Higher scores are worse. Use the simple endpoint difference, active mean minus control mean, and apply an active-arm delta of $+2$ points to each eligible imputation. The control outcomes are all observed, so none is adjusted.

At baseline, the control mean is $(8+9+7+8)/4=32/4=8$. The active mean is $(6+7+7+8)/4=28/4=7$, so the baseline contrast is $7-8=-1$ point.

After adjustment, the two eligible active values are $7+2=9$ and $8+2=10$. The active mean becomes $(6+7+9+10)/4=32/4=8$; the control mean remains 8. The adjusted contrast is $8-8=0$ points. In this dataset, the $+2$-point shift moves the contrast by $+1$ point, not $+2$, because only two of the four active-arm outcomes were adjusted. This arithmetic illustrates a point estimate in one completed dataset, not a pooled multiple-imputation inference.

Implementation and Verification

A transparent workflow separates the baseline completed datasets from their adjusted copies. Save the baseline values, the delta table, and the rule that identifies post-dropout cells. Apply the rule consistently, then run the same analysis on each adjusted dataset. Keeping the baseline data intact makes it possible to reproduce and audit both analyses.

Define the eligible cells.
Using the recorded arm, visit time, dropout status, and missingness pattern, identify only scheduled post-dropout outcomes that were imputed.

Check the delta specification.
Confirm that each required arm-time combination has one prespecified value, on the outcome scale and in the correct direction. Record the units and rationale.

Adjust each completed dataset.
For every eligible cell in every imputation, add the corresponding fixed $\delta_{a,t}$. Leave observed values and other imputed values unchanged.

Verify and analyze.
Compare missingness and observed values before and after adjustment, check the changed cells against the delta table, then fit the same substantive model used for the baseline analysis.

Useful automated checks include confirming that observed values are identical before and after adjustment; that the missingness pattern and row count are unchanged; and that every changed value was eligible and changed by exactly its assigned delta. A run with every delta set to zero should reproduce the baseline completed datasets and their analysis results, apart from any nondeterminism in the analysis software. It is also useful to summarize adjusted outcome ranges by arm and visit and check for impossible values.

Worked Example: Check Arm- and Time-Specific Cell Counts

In a hypothetical study, the delta table for symptom scores is $+1$ point for active-arm week 8 and $+2$ points for active-arm week 12; control-arm deltas are 0 at both visits. Across a completed dataset, there are three eligible active-arm week-8 cells, two eligible active-arm week-12 cells, and four eligible control-arm cells at each visit. The expected total shift is

3(1)+2(2)+4(0)+4(0)=7\text{ points}.

If an audit finds that the sum of all adjusted-minus-baseline differences is 9 points, the implementation does not match the specification. Possible causes include applying the week-12 shift at week 8, adjusting an ineligible cell, or applying a control-arm delta incorrectly. Checking cell counts and differences by arm and time can locate the error; the overall sum alone cannot identify its cause.

Diagnostics, Alternatives, and Limits

A useful comparison holds the imputation draws, analysis model, and estimand fixed while changing only the specified delta values. Compare the adjusted analysis with the baseline MAR analysis, and report how the estimates change across scientifically meaningful delta specifications. That comparison isolates sensitivity to the chosen outcome shifts more clearly than changing the imputation model and delta rule simultaneously.

Delta adjustment is one pattern-mixture approach to departures from MAR. A selection-model sensitivity analysis instead describes missingness conditional on outcome information and varies assumptions in that model. These approaches express sensitivity in different parameters, so their values are not directly interchangeable. The choice should follow the scientific question and the outcome process; neither approach makes assumptions identifiable from observed data alone.

Additive adjustment is most straightforward for continuous outcomes on a meaningful scale. For bounded outcomes, a shifted value may fall outside the allowable range. Silently truncating such values changes the specified adjustment and can alter its implications; inspect the range and choose a model or sensitivity parameterization suited to the outcome. For binary outcomes, adding a numeric delta to a 0/1 value does not produce a valid binary outcome. A shift on a link or probability scale requires a different, explicitly defined procedure. Likewise, if the scientific assumption concerns a change in variability or distribution shape rather than a location shift, a constant delta does not express it.

A common mistake is to apply delta to every missing value, on the grounds that every missing value is imputed. That is not the rule defined here: a value is adjusted only if it is missing after the prespecified dropout event. Another mistake is to say that a $+2$ shift necessarily increases the treatment contrast by 2 points. As the mean derivation shows, the change depends on how many records are shifted, and a model-based contrast must be recalculated. Finally, a delta is a sensitivity assumption, not a correction known to recover the unobserved outcomes.

Key takeaway: Apply a fixed delta only to the specified post-dropout imputations, on a clearly defined outcome scale. Preserve observed values, rerun the same analysis, and verify the adjusted data before interpreting how the estimate differs from the baseline MAR analysis.

Check Your Understanding

Answer the questions, then compare your reasoning with the solutions.

A participant has an observed week-4 score, an imputed missed week-6 score before dropout, and an imputed week-8 score after dropout. Which values receive the post-dropout delta?
In an arm with 10 endpoint outcomes, 4 eligible imputed values receive a common $+1.5$-point delta. By how much does that arm’s simple mean change?
Why should observed values be checked after the adjustment, and what should a zero-delta run reproduce?
Why is adding a delta directly to a binary 0/1 imputation generally inappropriate?

Solutions. 1. Only the week-8 value receives the delta. The observed week-4 score is preserved, and the week-6 imputation is not post-dropout under the stated rule. 2. The mean changes by $(4/10)(1.5)=0.6$ points. 3. Observed values must remain unchanged because the procedure modifies designated imputations only. A zero-delta run should reproduce the baseline completed datasets and analysis results, subject to any software nondeterminism. 4. Adding a nonzero value to 0 or 1 can produce a value outside the binary outcome’s support; a binary-outcome sensitivity adjustment needs a suitable scale and a separately defined procedure.