Treatment-Arm-Specific Delta Adjustments

Why Separate Treatment-Arm Adjustments Matter

In a randomized or observational comparison, missing outcomes in the treatment and control arms need not be missing for the same reasons, and the unobserved outcomes need not depart from a missing-at-random (MAR) reference in the same way. A single shared shift can conceal that possibility. Treatment-arm-specific delta adjustments make the departures explicit: one parameter describes the assumed shift among missing treatment outcomes, and another describes the shift among missing control outcomes.

The key point is that the treatment-effect estimate does not generally change by the difference between the two deltas alone. Each delta is weighted by the proportion of outcomes missing in its arm. Thus, the same pair of assumptions can have different effects when arm-specific missingness rates differ.

Definition: Let $A=1$ denote the treatment arm and $A=0$ the control arm. Let $Y$ be a continuous outcome, and let $R=1$ indicate that $Y$ is observed. A treatment-arm-specific additive delta adjustment replaces each MAR-reference imputation for a missing outcome in arm $a$ by $$ Y_{i,\mathrm{mis}}^{(\delta)}=Y_{i,\mathrm{mis}}^{(\mathrm{MAR})}+\delta_a,\qquad a\in\{0,1\}. $$ The parameter $\delta_a$ is measured in the same units as $Y$. A positive value shifts missing outcomes in arm $a$ upward relative to the MAR reference; a negative value shifts them downward. The values $\delta_0$ and $\delta_1$ are sensitivity assumptions, not quantities identified from the observed outcomes alone.

How the Adjustments Change a Mean Difference

For a simple marginal comparison, define $n_a$ as the number assigned to arm $a$, $m_a$ as its number of missing outcomes, and $q_a=m_a/n_a$ as its missing fraction. In one completed data set, let $\bar{Y}_{a,\mathrm{MAR}}$ be the arm mean after MAR-reference imputations. The treatment effect is defined here as the treatment mean minus the control mean: $\hat{\tau}_{\mathrm{MAR}}=\bar{Y}_{1,\mathrm{MAR}}-\bar{Y}_{0,\mathrm{MAR}}$.

Adding $\delta_a$ to each of the $m_a$ imputed values adds $m_a\delta_a$ to the sum of outcomes in that arm, while leaving the observed outcomes unchanged. Dividing this change in the sum by $n_a$ gives the change in the arm mean:

\bar{Y}_{a,\delta} =\bar{Y}_{a,\mathrm{MAR}}+\frac{m_a}{n_a}\delta_a =\bar{Y}_{a,\mathrm{MAR}}+q_a\delta_a.

Subtracting the adjusted control mean from the adjusted treatment mean gives the central result.

Result: For a treatment-minus-control difference in marginal means, the arm-specific delta adjustment changes the MAR-reference estimate by $q_1\delta_1-q_0\delta_0$:

\hat{\tau}_{\delta} =\hat{\tau}_{\mathrm{MAR}}+q_1\delta_1-q_0\delta_0.

The signs follow from the definition of the contrast. Raising missing treatment outcomes increases the treatment-minus-control effect, while raising missing control outcomes decreases it. When the missing fractions are equal, a shared shift in both arms cancels from the contrast. When they differ, even a common delta can alter the effect.

Worked Example: Computing the Arm-Weighted Adjustment

Suppose a hypothetical study has 100 people in each arm. In the control arm, 80 outcomes are observed with mean 52, and the 20 missing outcomes have MAR-reference imputed mean 50. In the treatment arm, 70 outcomes are observed with mean 58, and the 30 missing outcomes have MAR-reference imputed mean 56. Assume $\delta_0=-3$ and $\delta_1=-1$, in outcome units.

Control MAR mean. The completed-data control mean under the MAR reference is $(80\times52+20\times50)/100=(4160+1000)/100=51.60$. The adjusted missing-outcome mean is $50-3=47$, so the adjusted control mean is $(80\times52+20\times47)/100=(4160+940)/100=51.00$.

Treatment MAR mean. The completed-data treatment mean under the MAR reference is $(70\times58+30\times56)/100=(4060+1680)/100=57.40$. The adjusted missing-outcome mean is $56-1=55$, so the adjusted treatment mean is $(70\times58+30\times55)/100=(4060+1650)/100=57.10$.

The MAR-reference treatment effect is $57.40-51.60=5.80$ outcome units. The adjusted effect is $57.10-51.00=6.10$ outcome units. The formula gives the same result: $q_1=0.30$, $q_0=0.20$, so the change is $0.30(-1)-0.20(-3)=-0.30+0.60=0.30$ outcome units. The treatment-minus-control estimate therefore increases from 5.80 to 6.10 outcome units under these assumptions.

Assumptions and Interpretation

A delta analysis does not recover the unobserved outcomes or prove which delta values are correct. It asks how the analysis would change under explicitly chosen departures from the MAR reference. The reference imputation model must first be specified and fit appropriately, including relevant predictors and the treatment-arm structure needed for the analysis. The deltas then describe departures from that reference for missing outcomes.

Meaningful additive scale. The outcome must support an additive shift that is interpretable in its units. A shift of $-2$ means two outcome units lower, not a two-percent decrease.
Arm assignment is known. The treatment or control arm for each person must be available so that the appropriate delta is applied to each missing outcome.
Within-arm shift is the stated assumption. The basic adjustment applies the same additive shift to missing outcomes within an arm. It does not assert that every person's true unobserved outcome is exactly the imputed value plus that constant; it defines a sensitivity scenario for the imputed distribution.
Other aspects of the imputation model remain specified. The delta changes the assumed missing-outcome distribution relative to the MAR reference; it does not automatically fix misspecification in the reference model or unmeasured predictors.
Values remain plausible. Shifted imputations should be checked against the outcome's possible range and substantive meaning. If an additive shift creates impossible values, the simple adjustment is not appropriate without a defensible alternative construction.

The adjustment formula is exact for a difference in arm means when each missing value receives the same additive shift in its arm. In multiple imputation, the shifts are applied within each imputed data set, and the specified treatment-effect analysis is then run on each completed data set. The completed-data estimates and their uncertainty are pooled using the usual multiple-imputation procedure. A fixed pair of deltas represents a fixed sensitivity scenario; uncertainty about which deltas are plausible is not automatically included in the pooled standard error or interval.

Worked Example: Why a Shared Delta May Not Cancel

Suppose a hypothetical analysis has a MAR-reference treatment-minus-control estimate of 4.0 points. The missing fractions are $q_1=0.30$ in the treatment arm and $q_0=0.10$ in the control arm. Investigators choose a common shift of $-2$ points for missing outcomes in both arms, so $\delta_1=\delta_0=-2$.

The treatment-arm contribution is $0.30(-2)=-0.60$ points, and the control-arm contribution is $0.10(-2)=-0.20$ points. Because the control mean is subtracted, the effect adjustment is $q_1\delta_1-q_0\delta_0=-0.60-(-0.20)=-0.40$ points. The adjusted effect is therefore $4.0-0.40=3.60$ points.

The shared shift does not cancel because the missing fractions differ. If instead both fractions were 0.20, the adjustment would be $0.20(-2)-0.20(-2)=0$, and the treatment-effect estimate would remain 4.0 points. Equal shifts cancel from this contrast only when their weights are equal.

Comparing Sensitivity Scenarios

It is useful to distinguish three scenarios. Under the MAR reference, both deltas are zero. Under a common-delta scenario, the same shift is imposed in each arm, but the estimate can still move if missingness rates differ. Under treatment-arm-specific deltas, the two shifts can differ, allowing the missing outcomes to depart from the MAR reference in different directions or by different amounts.

The following hypothetical calculation shows how the weighted contrast depends on both parameters. Suppose the MAR-reference estimate is 5.0 points, $q_1=0.25$, and $q_0=0.15$.

$\delta_1$ (treatment)	$\delta_0$ (control)	Adjustment $0.25\delta_1-0.15\delta_0$	Adjusted estimate
0	0	0.00 points	5.00 points
-2	-2	-0.20 points	4.80 points
-2	-1	-0.35 points	4.65 points
-1	-3	0.20 points	5.20 points

For example, in the second row the adjustment is $0.25(-2)-0.15(-2)=-0.50+0.30=-0.20$ points. In the final row it is $0.25(-1)-0.15(-3)=-0.25+0.45=0.20$ points. A more negative delta in the control arm can increase the treatment-minus-control contrast because it lowers the control mean, which is subtracted.

Diagnostics, Failure Modes, and Alternatives

A sensitivity scenario is informative only if its implementation and implications are checked. Before interpreting adjusted estimates, verify that deltas were applied only to missing outcomes and to the intended arm. Compare the MAR-reference and adjusted imputed distributions by arm, and inspect their ranges, means, and quantiles. Confirm that the implied shift has the intended direction and units. For continuous outcomes, also check whether a constant shift produces implausible values or an implausible distribution shape. A location shift moves the distribution but does not represent every possible way that missing outcomes could differ.

The compact formula is a useful check for an unadjusted difference in means. If results do not agree with $q_1\delta_1-q_0\delta_0$, check the arm coding, the sign of the contrast, the missing fractions, and whether the stated deltas were actually added to the missing values. For a covariate-adjusted regression coefficient, this simple formula need not hold: changing imputed outcomes can affect the fitted model in ways that depend on the covariates and model specification. In that case, rerun the planned analysis in each adjusted imputed data set rather than using the marginal-mean formula as a substitute.

A common alternative is to retain the MAR reference with $\delta_0=\delta_1=0$. This is a reference scenario, not evidence that MAR is true. Another is to use a common delta; it is simpler, but may not reflect distinct departures by arm and may not leave the contrast unchanged if missingness rates differ. Treatment-arm-specific deltas are more flexible, but they introduce more assumptions that must be justified and reported. Neither approach can determine the right delta from observed data alone.

Common mistake: “The two arms receive the same delta, so the treatment-effect estimate cannot change.” This is true only if the missing fractions are equal for the mean difference considered. In general, the change is $q_1\delta_1-q_0\delta_0$; with a common delta $\delta$, it becomes $(q_1-q_0)\delta$, which is not zero when $q_1\ne q_0$.

Worked Example: Reading the Direction of the Contrast Change

Consider a hypothetical continuous symptom score for which higher values indicate worse symptoms. The MAR-reference estimate, treatment minus control, is $-3.0$ points. Suppose 20% of treatment outcomes and 40% of control outcomes are missing, and investigators set $\delta_1=-1$ point and $\delta_0=-2$ points.

The change in the effect is $0.20(-1)-0.40(-2)=-0.20+0.80=0.60$ points. Thus the adjusted estimate is $-3.0+0.60=-2.40$ points. Both arm-specific assumptions shift missing outcomes toward lower symptom scores, but the larger missing fraction and larger downward shift in the control arm raise the treatment-minus-control contrast. The adjusted estimate still indicates a lower mean score in the treatment arm, by 2.40 points in this scenario; it does not establish a causal effect without the study design and analysis assumptions required for causal interpretation.

Check Your Understanding

Answer each question using the treatment-minus-control convention. Solutions are included below.

The MAR-reference estimate is 6 points, $q_1=0.10$, $q_0=0.30$, $\delta_1=-2$, and $\delta_0=-1$. What is the adjusted estimate?
Under what condition does a common delta leave a marginal mean difference unchanged? State the algebraic reason.
If $\delta_1=2$ outcome units and $\delta_0=0$, what is the direction of the treatment-effect change when $q_1>0$?
Why should an analyst rerun a covariate-adjusted regression in each adjusted imputed data set rather than assume the marginal formula gives its coefficient change?

Solutions. (1) The adjustment is $0.10(-2)-0.30(-1)=-0.20+0.30=0.10$ points. The adjusted estimate is $6+0.10=6.10$ points. (2) With $\delta_1=\delta_0=\delta$, the change is $q_1\delta-q_0\delta=(q_1-q_0)\delta$. It is zero when the missing fractions are equal, or when the common delta is zero. (3) The change is $q_1(2)-q_0(0)=2q_1>0$, so the treatment-minus-control estimate increases by $2q_1$ outcome units. (4) A regression coefficient depends on covariates and the fitted model as well as arm means. Changing imputed outcomes can alter the regression fit, so the coefficient change is not generally just the weighted difference of arm-specific deltas. Refit the specified model in each adjusted completed data set and pool its estimates.