Additive Delta Adjustments for Continuous Outcomes

Why Add a Shift to MAR-Imputed Outcomes?

An imputation model under a missing-at-random (MAR) assumption predicts missing outcomes from observed information. But the unobserved outcomes could, even after accounting for that information, tend to be systematically higher or lower than the MAR predictions. An additive delta adjustment represents one such departure: it shifts missing continuous outcomes by a chosen constant while leaving observed outcomes unchanged.

The delta value is a sensitivity parameter, not a quantity estimated from the observed data. Its purpose is to ask how a conclusion would change under specified assumptions about the missing outcomes. The value zero recovers the MAR reference. Positive and negative values represent shifts upward and downward, respectively, in the outcome’s units.

Definition: For a continuous outcome \(Y\), let \(Y^{\mathrm{MAR}}_{ij}\) be the MAR-imputed value for a missing outcome of individual \(i\) in imputed data set \(j\). An additive delta adjustment replaces it with \(Y^{\delta}_{ij}=Y^{\mathrm{MAR}}_{ij}+\delta\). Observed outcomes are not changed. The same \(\delta\) is applied to every missing outcome in this tutorial.

The adjustment is made to the MAR-imputed values in each completed data set, and the chosen treatment-effect analysis is then applied to each adjusted data set. This keeps the reference analysis and the sensitivity analysis aligned: the comparison changes the assumption about missing outcomes, not the observed data or the definition of the treatment effect.

How the Shift Changes a Difference in Means

Consider a two-arm study with a continuous outcome, where the estimated treatment effect is the treatment-group mean minus the control-group mean. Let \(n_T\) and \(n_C\) be the numbers assigned to treatment and control, and let \(r_T\) and \(r_C\) be the numbers with missing outcomes in those groups. In a completed data set, shifting all missing outcomes in both groups by the same \(\delta\) changes the group means by \(\delta r_T/n_T\) and \(\delta r_C/n_C\), respectively.

\begin{aligned} \bar{Y}^{\delta}_T &= \bar{Y}^{\mathrm{MAR}}_T+\delta\frac{r_T}{n_T},\\ \bar{Y}^{\delta}_C &= \bar{Y}^{\mathrm{MAR}}_C+\delta\frac{r_C}{n_C},\\ \hat{\theta}^{\delta} &=\bar{Y}^{\delta}_T-\bar{Y}^{\delta}_C\\ &=\hat{\theta}^{\mathrm{MAR}}+ \delta\left(\frac{r_T}{n_T}-\frac{r_C}{n_C}\right). \end{aligned}

This is an exact identity for the unadjusted difference in completed-data means. It explains why a common shift need not cancel from a treatment contrast. It cancels when the missing-outcome proportions are equal; otherwise the arm with the larger missing proportion receives a larger change in its mean. The sign of the resulting effect change also depends on the sign of \(\delta\).

Key result: For a treatment-minus-control difference in means, a common additive shift changes the estimate by the delta value multiplied by the difference in missing-outcome proportions. If the treatment and control missing proportions are equal, the point estimate is unchanged, although other features of the completed data can still change.

The identity relies on using the difference in group means as the estimator and applying the same shift to every missing outcome in both arms. For a different estimator, such as a covariate-adjusted regression coefficient, the change is not generally given by this missing-proportion formula; the specified analysis must be refitted to each adjusted data set.

Worked Example: A Negative Shift With More Missing Outcomes in Treatment

Suppose a hypothetical trial measures a continuous symptom score in points, with higher scores indicating worse symptoms. In one MAR-completed data set, the treatment group has \(n_T=40\), \(r_T=10\), and mean \(12.4\) points. The control group has \(n_C=50\), \(r_C=5\), and mean \(10.0\) points. Apply \(\delta=-2\) points to missing outcomes.

The MAR treatment effect is \(12.4-10.0=2.4\) points. The missing proportions are \(10/40=0.25\) in treatment and \(5/50=0.10\) in control. Therefore the change in the contrast is \( -2(0.25-0.10)=-0.30\) points, and the adjusted effect is \(2.4-0.30=2.1\) points.

Directly, the treatment mean falls by \(2(10/40)=0.50\) points, to \(11.9\). The control mean falls by \(2(5/50)=0.20\) points, to \(9.8\). Their difference is \(11.9-9.8=2.1\) points, agreeing with the formula. Under this selected shift, the estimated treatment-minus-control contrast is 2.1 points rather than the MAR reference of 2.4 points.

Applying Selected Values Across Multiple Imputations

With \(m\) imputed data sets, apply the same selected delta to each data set’s missing outcomes, calculate the treatment-effect estimate in every adjusted data set, and pool those estimates using the course’s usual multiple-imputation analysis. The point estimate is the average of the \(m\) completed-data estimates. The within-imputation variances and between-imputation variation must be calculated from the adjusted analyses; they should not simply be assumed to remain the same as under MAR.

For an unadjusted difference in means, if the treatment and control missing proportions are fixed across the \(m\) completed data sets, the shift in the point estimate is the same in every data set. Thus it also shifts the pooled point estimate by the amount in the identity above. More generally, calculate the analysis in every adjusted data set and pool the results rather than relying on that shortcut.

Choose a reference.
Specify the MAR-imputed data sets and the treatment-effect estimator that define the reference analysis.

Set a delta scenario.
Choose a shift in outcome units, with its sign and magnitude tied to a defensible sensitivity question.

Adjust and analyze.
Add delta to missing outcomes in every completed data set, leave observed outcomes unchanged, and refit the chosen analysis.

Pool and compare.
Pool the adjusted estimates and their variances, then compare the result with the MAR reference and other selected delta scenarios.

Worked Example: A Positive Shift With More Missing Outcomes in Control

In a hypothetical study of a continuous function score measured in points, a MAR-completed data set has treatment mean \(6.2\) and control mean \(4.7\), so the treatment-minus-control estimate is \(1.5\) points. There are \(n_T=30\) participants in treatment, with \(r_T=3\) missing outcomes, and \(n_C=20\) in control, with \(r_C=8\) missing outcomes. Consider \(\delta=3\) points.

The missing proportions are \(3/30=0.10\) for treatment and \(8/20=0.40\) for control. The change in the treatment effect is \(3(0.10-0.40)=-0.90\) points. The adjusted estimate is therefore \(1.5-0.90=0.60\) points.

The treatment mean increases by \(3(3/30)=0.30\) points, to \(6.5\). The control mean increases by \(3(8/20)=1.20\) points, to \(5.9\). The adjusted difference is \(6.5-5.9=0.60\) points. Although the same positive shift was applied in both arms, the control mean rose more because a larger proportion of its outcomes was missing.

Worked Example: Pooling Three Adjusted Estimates

Suppose, hypothetically, that three MAR-completed data sets give treatment-effect estimates of \(1.8\), \(2.0\), and \(2.2\) points. In each, 20% of treatment outcomes and 10% of control outcomes are missing. A common shift of \(\delta=-5\) points changes each treatment-effect estimate by \(-5(0.20-0.10)=-0.50\) points. The adjusted estimates are \(1.3\), \(1.5\), and \(1.7\) points.

Their pooled point estimate is \((1.3+1.5+1.7)/3=4.5/3=1.5\) points. The MAR estimates average \((1.8+2.0+2.2)/3=6.0/3=2.0\) points, so the pooled estimate has moved down by \(0.50\) points, as predicted.

For clarity, suppose the within-imputation variance recalculated for each adjusted analysis is \(U_j=0.64\) points squared. The between-imputation variance of the adjusted estimates is \(B=[(1.3-1.5)^2+(1.5-1.5)^2+(1.7-1.5)^2]/(3-1)=0.04\) points squared. The pooled variance is \(T=\bar{U}+(1+1/m)B=0.64+(1+1/3)(0.04)=0.6933\) points squared, giving a pooled standard error of \(\sqrt{0.6933}=0.8326\) points. This calculation uses the stipulated adjusted within-imputation variances; in an actual analysis, they must be obtained by refitting the analysis to the adjusted data sets.

Assumptions, Diagnostics, and Limits

An additive delta analysis is meaningful only when its assumptions and interpretation are made explicit. It uses a MAR analysis as a reference and represents a departure from that reference by a constant shift in the outcome scale. It leaves observed outcomes unchanged, applies the chosen shift to the missing outcomes, and holds the analysis model and target contrast fixed across scenarios. The MAR imputation model and analysis must themselves be suitable for the intended estimand; a delta adjustment does not repair a poorly specified reference model.

The constant-shift assumption is substantive: it treats the departure from MAR as the same number of outcome units for every missing case. That may be a useful scenario, but it is not established by the observed data. The delta should therefore be reported with its units, direction, and rationale. A range of selected values can show how sensitive the conclusion is, but cannot prove that any particular value is correct.

Useful checks include confirming that observed values were not changed, verifying the shift on a few missing values, and checking the arithmetic against the missing proportions. Inspect adjusted outcome distributions and group summaries for implausible values or an unrealistic concentration of shifted observations. Report estimates across the selected delta values, not only a scenario that supports a preferred conclusion. If the analysis uses a regression coefficient rather than a difference in means, compare the refitted coefficients directly; do not substitute the simple missing-proportion formula.

The MAR reference (\(\delta=0\)) is the natural comparison, not a competing empirical finding. A complete-case analysis is another possible analysis, but it discards cases with missing outcomes and generally targets the same treatment effect only under additional assumptions; it does not encode the same explicit shift scenario. Additive adjustments also have limits: a single constant cannot represent departures that differ systematically across types of missing cases, and it may produce values outside a scientifically valid outcome range. Those concerns call for an explicitly justified alternative sensitivity model rather than silently clipping values.

Common Pitfall

Common mistake: “I added \(-2\) points to missing outcomes in both arms, so the treatment effect must decrease by 2 points.” This confuses a shift to individual outcomes with a shift to the treatment contrast. For a difference in means, the contrast changes by \(\delta\) times the difference between the two missing proportions. In the first example the change was \(-2(0.25-0.10)=-0.30\) points, not \(-2\) points. With equal missing proportions, the common shift would not change that point estimate at all.

Check Your Understanding

Use the treatment-minus-control convention. Assume each question concerns an unadjusted difference in completed-data means unless stated otherwise.

A study has 25% missing outcomes in treatment and 15% in control. If the MAR estimate is 4 points and \(\delta=-3\) points, what is the adjusted estimate?
In a hypothetical study, 6 of 30 treatment outcomes and 4 of 20 control outcomes are missing. What does a common additive shift do to the treatment-effect point estimate?
A MAR estimate is 2.5 points. The missing proportions are 0.10 in treatment and 0.35 in control. Find the estimate when \(\delta=2\) points.
Why should within-imputation variances be recalculated after adjusting outcomes?

Solutions. (1) The missing-proportion difference is \(0.25-0.15=0.10\). The change is \(-3(0.10)=-0.30\) points, so the adjusted estimate is \(4-0.30=3.70\) points. (2) Both missing proportions equal \(0.20\), so the change is \(\delta(0.20-0.20)=0\); the point estimate is unchanged for any common delta. (3) The change is \(2(0.10-0.35)=-0.50\) points, giving \(2.5-0.50=2.00\) points. (4) Shifting only some observations can change the spread of outcomes within a completed data set and hence the estimated variance of the treatment effect. The adjusted analysis, including its variance, must be recalculated in each data set before pooling.