Worked Example: Longitudinal Delta Adjustment

Why Longitudinal Delta Adjustment Needs a Trajectory

For a single missing outcome, a delta adjustment shifts a reference-model prediction by a specified amount. In a longitudinal study, however, a missing outcome at one visit can also become part of the history used to predict later outcomes. Applying a shift at each visit therefore does not generally produce a trajectory that is just the reference trajectory plus the same fixed amount. Earlier shifts can be carried forward through the sequential model.

This tutorial works through that propagation for monotone dropout: once a participant drops out, all later outcomes are missing. The calculations concern hypothetical continuous symptom scores measured at baseline and at weeks 4, 8, and 12. Higher scores indicate worse symptoms. The numerical trajectories are conditional means, useful for understanding the adjustment; actual multiple imputations also include random variation.

Because higher symptom scores are worse in the examples, a positive \(\delta_t\) represents a sensitivity assumption that missing outcomes are worse than their reference-model predictions. The sign and size of a delta are substantive choices, not quantities identified by the observed outcomes alone. The earlier material on calibrating delta values and anchoring deltas to clinical thresholds motivates how such choices can be made.

Assumptions Behind the Sequential Calculation

The arithmetic below relies on a clearly specified reference model and a clearly specified sensitivity assumption. In practice, both are essential: an adjustment cannot repair a poorly fitting reference model, and a delta value does not by itself describe every feature of a missing-outcome distribution.

• Reference model adequacy. The conditional models should represent the observed longitudinal data reasonably well, including important predictors and dependence on prior outcomes. The earlier material on checking the reference model and diagnosing implausible predictions addresses this prerequisite.
• Sequential conditioning. At each visit, the model conditions on the history available through the preceding visit. For a participant who has dropped out, this history includes earlier imputed outcomes as well as observed outcomes.
• Specified delta scale and pattern. Each \(\delta_t\) is expressed in the outcome's units and applies to the missing outcomes to which it was assigned. It does not alter outcomes already observed.
• Distributional choice. A mean shift alone does not specify the full imputation distribution. A common implementation retains the reference model's residual distribution and variance while shifting its location. That is an additional assumption; another distributional change would need to be defined explicitly.
• Clinical feasibility. Adjusted predictions and draws should be compatible with the outcome's possible range and with plausible within-person change. For a bounded scale, an unconstrained normal model can generate impossible values even when its mean is plausible.

The examples use specified conditional means to make the propagation transparent. They do not claim that a particular delta is correct, nor that a mean trajectory alone captures the distribution of outcomes. Sensitivity analysis compares conclusions over defensible assumptions; it does not identify the unobserved outcomes.

Dropout Timing Determines Which Visits Are Adjusted

The delta is applied only to missing outcomes. A participant who remains observed through week 8 has no week-8 delta adjustment, even if another participant with the same treatment assignment dropped out earlier. The later conditional prediction can nevertheless differ between these participants because their observed histories differ. If delta values vary by dropout pattern, that pattern must be specified along with the visit.

From Adjusted Means to Imputed Values

In multiple imputation, the adjusted conditional mean is generally not substituted as the missing outcome. Instead, a random draw is made from a specified adjusted predictive distribution. For example, if the reference conditional distribution is normal with standard deviation \(\sigma_t\), and the chosen assumption shifts its mean but keeps its standard deviation, then a missing value is drawn from \(N(m_t(H_{t-1})+\delta_t,\sigma_t^2)\). That draw becomes part of the history used at the next visit. Repeating this sequentially preserves both model-based uncertainty and the dependence induced by the longitudinal model.

Diagnostics, Alternatives, and Limitations

A trajectory should be checked as a sequence, not only one visit at a time. Useful checks include plotting observed and imputed outcome profiles by visit and dropout pattern, comparing adjusted means and quantiles with clinical thresholds, and checking the frequency of values outside the scale's bounds. For a clinical scale, also inspect changes between adjacent visits: plausible individual-level values can still form an implausible trajectory if they imply abrupt, sustained deterioration or improvement unsupported by the scenario.

Look especially for these warning signs:

• Adjusted means, imputed draws, or clinically important quantiles exceed the outcome's possible range.
• The same delta is applied to observed visits, or a visit is shifted despite being observed for that participant.
• Later predictions are calculated from reference-only earlier values when the intended sequential procedure requires adjusted imputed history.
• A delta schedule gives a pattern that conflicts with the stated clinical direction or produces implausible change across visits.
• Predictions depend heavily on poorly supported histories, such as extreme imputed values in a sparse part of the data.

Longitudinal delta adjustment differs from applying an independent fixed shift to every missing value: sequential conditioning lets earlier adjusted values affect later conditional means. It also differs from leaving the reference model unchanged, which corresponds to no delta shift under the specified location-shift setup. A different sensitivity approach could alter variance, distributional shape, or the evolution of change over time; those are different assumptions and should not be silently combined with a mean shift.

The method is limited by the reference model, the chosen delta schedule, and the assumed distribution for imputation. If dropout patterns are sparse, a detailed pattern-specific schedule may be weakly supported and unstable. If outcomes are bounded, shifting a normal mean does not ensure valid draws; a model respecting the bounds or a carefully justified constrained procedure may be needed. Finally, a coherent adjusted trajectory is not evidence that it is the true trajectory. It shows the implications of a stated sensitivity assumption.