Time-Specific Delta Adjustments in Longitudinal Data

Why the Timing of a Delta Matters

For a repeated outcome, a single delta applied to every missing visit may not represent the concern that motivates a sensitivity analysis. For example, a departure from the missing-at-random (MAR) imputation model might be thought small at an early follow-up but larger at a later one. A time-specific delta makes that pattern explicit: the shift applied to a missing outcome depends on its visit.

This choice can change a longitudinal contrast even when the same direction of shift is used throughout. The result depends on which visits are missing, how often they are missing in each group, and which visits enter the estimand. A delta is therefore not just a technical setting: it encodes a particular assumption about unobserved outcomes over time.

Definition: Let \(Y_{it}\) be the outcome for individual \(i\) at visit \(t\), and let \(Y^{MAR}_{it}\) be a draw for a missing outcome from the specified imputation model under MAR. An additive time-specific delta adjustment replaces that draw by \(Y^{MAR}_{it}+\delta_t\). If the assumed departure can differ by treatment arm \(a\), write \(\delta_{a,t}\). The shift is applied to missing outcomes only; observed outcomes are left unchanged. A delta of zero reproduces the MAR draw, while a negative delta represents an assumption that the missing outcome tends to be lower than the corresponding MAR prediction, in the outcome's units.

Model, Assumptions, and Estimand

A time-specific delta analysis starts with an imputation model that produces plausible draws under MAR. For each missing \(Y_{it}\), that model should condition on the observed information specified for the analysis, which may include treatment, baseline variables, observed outcomes at other visits, and other relevant predictors. Its repeated-outcome structure should represent the dependence across visits well enough for the intended analysis. The delta adjustment changes the assumption about missing outcomes; it does not repair a misspecified imputation model.

The sensitivity scenario also requires an additive shift to be meaningful on the outcome scale, and each shifted value should remain in a valid range. A fixed shift in outcome units may be unsuitable when the outcome is strictly positive with a strongly multiplicative scale, or when it creates impossible values. In that case, a different scale or a constrained approach is needed. The chosen \(\delta_{a,t}\) values are sensitivity parameters: they are specified for analysis, not estimated from the observed data.

For clarity, consider the treatment-versus-control difference in change from baseline to a follow-up visit \(T\):

D = (\mu_{1T}-\mu_{10})-(\mu_{0T}-\mu_{00}),

where \(\mu_{at}\) denotes the mean outcome for arm \(a\) at visit \(t\), with \(a=1\) for treatment and \(a=0\) for control. This estimand compares the mean change in the two arms. Other longitudinal estimands, such as a contrast between adjacent visits or a model-based trajectory contrast, can also be examined, but the delta effect must be calculated for the particular estimand being reported.

How Missingness and Delta Combine

For a simple completed-data arm mean at visit \(t\), let \(q_{at}\) be the fraction of individuals in arm \(a\) whose outcome is missing at that visit. If every missing outcome in that arm and visit is shifted by the same fixed \(\delta_{a,t}\), the mean changes by \(q_{at}\delta_{a,t}\). This follows because each of the \(n_a\) outcomes contributes equally to the mean, and \(n_a q_{at}\) of them receive the shift:

\frac{1}{n_a}\left(n_a q_{at}\delta_{a,t}\right) = q_{at}\delta_{a,t}.

Therefore, for the difference in change from baseline to \(T\), the exact change in the completed-data contrast of arm means is

\Delta D = (q_{1T}\delta_{1,T}-q_{10}\delta_{1,0}) - (q_{0T}\delta_{0,T}-q_{00}\delta_{0,0}).

Here \(\Delta D\) means the delta-adjusted contrast minus the MAR-based contrast. If baseline is fully observed, \(q_{10}=q_{00}=0\), so only the follow-up shifts contribute. If missingness occurs at baseline too, its contribution must be included with the opposite sign because baseline is subtracted in a change.

This weighted formula is exact for the specified arm means and contrast. It is not automatically an exact formula for a regression coefficient from a covariate-adjusted or otherwise model-based analysis: the fitted coefficient can depend on the full data structure and model. In that setting, apply the chosen shifts to each imputed data set, refit the prespecified analysis, and pool the resulting estimates using the standard multiple-imputation procedure.

Worked Example: A Common Shift at One Follow-Up

Suppose a hypothetical study has 100 people in each arm, and baseline outcomes are complete. At the final visit, 20% of treatment-arm outcomes and 10% of control-arm outcomes are missing. The MAR-based completed-data means imply a treatment-minus-control difference in change of \(2.00\) points. Consider a common time-specific shift of \(-2.00\) points at this final visit in both arms. Assume the additive shift is plausible on this outcome scale and does not create invalid values.

The visit-specific missing fractions are \(q_{1T}=0.20\) and \(q_{0T}=0.10\). The adjusted contrast change is

\Delta D = (0.20)(-2.00)-(0.10)(-2.00) = -0.400-(-0.200) = -0.200\text{ points}.

Thus the delta-adjusted difference in change is \(2.00-0.200=1.80\) points. Although both arms receive the same shift, the contrast moves because a larger fraction of treatment outcomes is missing at that visit. The calculation concerns the shift in the contrast, not a claim that the missing outcomes are known to be 2 points lower.

Time Patterns Can Change the Longitudinal Contrast

A constant delta over time and a time-varying delta describe different assumptions. For a contrast comparing changes between two follow-up visits \(t_1\) and \(t_2\), define

D_{t_2-t_1} = (\mu_{1,t_2}-\mu_{1,t_1})-(\mu_{0,t_2}-\mu_{0,t_1}).

For arm means, the delta-induced change in this contrast is

\Delta D_{t_2-t_1} = (q_{1,t_2}\delta_{1,t_2}-q_{1,t_1}\delta_{1,t_1}) -(q_{0,t_2}\delta_{0,t_2}-q_{0,t_1}\delta_{0,t_1}).

The signs follow the contrast: a shift at the later visit adds to the within-arm change, whereas a shift at the earlier visit is subtracted. This is why a delta pattern should be specified by visit before calculating its effect on a longitudinal contrast.

Worked Example: Constant Versus Increasingly Negative Shifts

Suppose a hypothetical two-arm study measures an outcome at visits 1 and 2. The MAR-based difference in change from visit 1 to visit 2 is \(1.20\) points. Missing fractions are \(q_{1,1}=0.10\), \(q_{1,2}=0.40\), \(q_{0,1}=0.10\), and \(q_{0,2}=0.20\). Assume each shift is additive and plausible, and that the analysis uses the completed-data arm means.

First consider a constant shift of \(-1.50\) points at both visits in both arms. Then

\Delta D_{2-1} = [(0.40)(-1.50)-(0.10)(-1.50)] -[(0.20)(-1.50)-(0.10)(-1.50)] = -0.450-(-0.150) = -0.300\text{ points}.

The adjusted contrast is \(1.20-0.300=0.900\) points. Now consider a time-varying pattern: \(\delta_1=-0.50\) points at visit 1 and \(\delta_2=-2.00\) points at visit 2, again in both arms. The result is

\Delta D_{2-1} = [(0.40)(-2.00)-(0.10)(-0.50)] -[(0.20)(-2.00)-(0.10)(-0.50)] = (-0.750)-(-0.350) = -0.400\text{ points}.

Under this increasing-negative pattern, the adjusted contrast is \(1.20-0.400=0.800\) points. The two scenarios have the same visit-specific missing fractions but differ by \(0.100\) points in their adjusted contrasts because they assign different shifts to the earlier and later visits.

Worked Example: Arm-Specific Shifts at Two Visits

Suppose a hypothetical study compares treatment and control at a final visit, with baseline fully observed. The final-visit missing fractions are 0.30 in treatment and 0.10 in control. The MAR-based difference in change is \(3.00\) points. Consider a scenario in which missing treatment outcomes at the final visit are shifted by \(-2.00\) points, while missing control outcomes are shifted by \(-1.00\) point. Assume these arm- and visit-specific additive shifts are the intended sensitivity scenario and remain within the outcome's valid range.

The change in the treatment mean is \((0.30)(-2.00)=-0.600\) points. The change in the control mean is \((0.10)(-1.00)=-0.100\) points. Therefore,

\Delta D=(-0.600)-(-0.100)=-0.500\text{ points}, \qquad D_{\text{adjusted}}=3.00-0.500=2.50\text{ points}.

The treatment-minus-control difference in change is \(2.50\) points under this specified scenario. Both shifts lower their respective arm means, but their unequal weighted effects move the contrast downward by \(0.500\) points.

Implementation, Diagnostics, and Limitations

For multiple imputation, use a clearly specified visit-by-visit delta pattern and apply it to the corresponding missing values in every imputed data set. Keep observed outcomes unchanged. Then fit the same prespecified longitudinal analysis to each completed data set and pool the estimates and uncertainty in the usual way. The delta scenario should be held fixed across imputations so that variation among the imputed data sets reflects the imputation procedure rather than random changes in the sensitivity assumption.

Useful checks include confirming that each shifted value belongs to the intended visit and arm, verifying that no observed values were shifted, and comparing the number of missing values and mean shift at every visit with the specified pattern. Plotting imputed outcome distributions by visit and arm under each scenario can reveal implausible ranges or abrupt patterns. Also check whether the specified shift is sensible in outcome units and whether the resulting longitudinal trajectories are coherent. A delta may be individually plausible at each visit yet create an implausible trajectory when applied across visits.

A common mistake is to reason, “The same delta is used in both arms, so the treatment contrast cannot change.” That is true only under conditions such as equal missing fractions at the relevant visits for the arm-mean contrast. In general, the shift in an arm mean is \(q_{at}\delta_{a,t}\), not simply \(\delta_{a,t}\). Another mistake is to treat the formula for arm means as an exact adjustment to every longitudinal model coefficient. The formula gives the exact arithmetic for the stated contrast of means; model-based analyses should be rerun on the adjusted imputed data.

Time-specific delta analysis does not identify the true distribution of missing outcomes from observed data alone. Results depend on the chosen delta pattern, the MAR imputation model, the outcome scale, and the analysis model. It is a structured sensitivity analysis, not a test that establishes which missing-data assumption is correct. Report the pattern in interpretable units, describe which missing visits receive each shift, and distinguish scenario-specific estimates from estimates supported solely by the observed data.

Key takeaway: In longitudinal data, specify a delta for each relevant visit before evaluating a contrast. The effect on an arm mean is weighted by the fraction missing at that visit, and the effect on a longitudinal contrast follows the signs and weights in that contrast.

Check Your Understanding

Try each calculation, then compare it with the solution that follows.

At a final visit, 25% of treatment outcomes and 15% of control outcomes are missing. Baseline is complete. If both groups receive a delta of \(-1.20\) points, by how much does the treatment-minus-control difference in change move?
For a visit-2-minus-visit-1 contrast, treatment missing fractions are 0.20 and 0.50 at visits 1 and 2; control fractions are 0.10 and 0.30. A common delta is \(-1.00\) point at both visits. Calculate the change in the contrast.
Why does a common delta across arms not necessarily leave an arm-mean treatment contrast unchanged?
Name two checks that can identify a problem with a time-specific delta scenario.

Solutions. (1) \(\Delta D=(0.25)(-1.20)-(0.15)(-1.20)=-0.300-(-0.180)=-0.120\) points, so the contrast decreases by \(0.120\) points. (2) The treatment change is \((0.50)(-1.00)-(0.20)(-1.00)=-0.300\) points; the control change is \((0.30)(-1.00)-(0.10)(-1.00)=-0.200\) points. Thus \(\Delta D=-0.300-(-0.200)=-0.100\) points. (3) Each arm mean changes by its missing fraction times the delta. Different missing fractions therefore give different mean shifts, even with a common delta. (4) Examples include confirming that shifts were applied only to missing outcomes at the specified visit and arm, checking that shifted values remain in the valid range, and examining whether the resulting trajectories are plausible.