Delta Adjustment for Intermittent Missingness

Why Intermittent Missingness Needs Special Care

A missing visit is intermittent when an outcome is unobserved at one time but observed again at a later time. For example, a participant may have outcomes recorded at visits 1 and 3, but not at visit 2. This pattern differs from dropout: the later measurement remains available and contains information about the unobserved visit. An imputation for visit 2 should therefore respect the observed outcome at visit 3 rather than treating the sequence as though follow-up ended at visit 1.

Delta adjustment represents a sensitivity assumption about the distribution of outcomes that are missing, beyond the reference model used to predict them. In the intermittent setting, the reference distribution should generally condition on all relevant observed information, including observed later outcomes. The delta then modifies that conditional distribution without replacing or changing those observed values.

The direction of the shift must be stated in context. If larger values indicate worse health, a positive delta means the missing outcomes are assumed to be worse, on average, than the reference model predicts. A delta is a sensitivity parameter, not a quantity identified by the observed data alone. Its value should be justified by substantive knowledge or explored across plausible values.

Condition on the Observed Future

For a participant with \(Y_1\) observed, \(Y_2\) missing, and \(Y_3\) observed, a bridge-style reference model uses \(f_0(Y_2\mid Y_1,Y_3,X,R)\). The future value \(Y_3\) is part of the participant’s observed data. It can inform the likely value at the missing visit through the longitudinal association between visits. A delta adjustment changes the assumed distribution of \(Y_2\) conditional on those observations; it does not impute a new \(Y_3\).

The future observation does not necessarily impose a deterministic rule such as \(Y_2\) having to lie between \(Y_1\) and \(Y_3\). Such a rule would require a substantive constraint, such as a known monotone trajectory. In ordinary longitudinal data, outcomes can rise and fall. The later value instead constrains the imputation probabilistically through the conditional model: values of \(Y_2\) that fit poorly with the observed endpoints receive less probability under the reference distribution.

Why the Conditional Construction Is Coherent

A useful justification comes from factorizing the joint distribution into an observed-data distribution and a conditional distribution for the missing components. For simplicity, condition throughout on \(X\) and \(R\). Let \(f_0(Y_O)\) be the reference distribution of the observed outcomes and \(f_0(Y_M\mid Y_O)\) the reference conditional distribution for the missing outcomes. Define the sensitivity model by keeping the observed-data distribution fixed and replacing the conditional distribution:

This is a proper joint distribution whenever \(f_\delta(Y_M\mid Y_O)\) is a normalized conditional distribution for each \(Y_O\). Indeed, integrating over the missing values gives \(\int f_\delta(Y_O,Y_M)\,dY_M=f_0(Y_O)\). Integrating over the observed values then gives 1. Consequently, the construction changes the assumed distribution of missing values while retaining the reference distribution for the observed values.

For an unbounded continuous outcome, an additive shift can be written as \(f_\delta(y_M\mid y_O)=f_0(y_M-\delta\mid y_O)\). A change of variable \(u=y_M-\delta\), with \(dy_M=du\), shows that the shifted density integrates to 1. For several missing visits, the corresponding vector shift is \(f_{\boldsymbol{\delta}}(\mathbf y_M\mid\mathbf y_O)=f_0(\mathbf y_M-\boldsymbol{\delta}\mid\mathbf y_O)\), provided the shifted distribution is well-defined on the outcome’s support. The joint conditional distribution matters: shifting components separately while ignoring their dependence can produce a different and potentially incoherent longitudinal model.

Several Intermittently Missing Visits

When multiple visits are missing but later and earlier visits are observed, specify the missing outcomes jointly conditional on the observed outcomes. This makes the dependence among missing visits explicit. For example, if visits 2 and 4 are missing while visits 1, 3, and 5 are observed, use a reference distribution for \((Y_2,Y_4)\) conditional on \((Y_1,Y_3,Y_5)\), then define a sensitivity shift for each missing visit. The observed values at visits 1, 3, and 5 remain fixed.

A vector of deltas allows the sensitivity assumptions to differ by visit. They may also depend on covariates or the missingness pattern if that is scientifically justified. Such flexibility does not remove the need for a coherent joint conditional model: the adjusted distributions should retain reasonable dependence across missing visits and should be compatible with any constraints on the outcome.

Assumptions, Diagnostics, and Alternatives

Delta adjustment does not make an inadequate reference model adequate. The reference conditional distribution must represent the relevant observed history, future outcomes, covariates, and missingness pattern, with sufficient overlap to estimate the needed conditional relationships. A common reference is based on a missing-at-random assumption: conditional on observed data included in the model, the missingness does not further depend on the unobserved outcome. The delta then expresses a chosen departure from that reference assumption. The observed data generally cannot determine whether the delta is correct.

Check the reference model before interpreting adjusted imputations. Useful diagnostics include comparing predicted and observed outcome distributions at visits with data, checking conditional residual patterns, and examining whether simulated trajectories reproduce the dependence among observed visits. After applying delta, inspect adjusted distributions and complete trajectories for implausible values, abrupt changes, or contradictions with known clinical constraints. If the outcome has bounded support, a simple additive shift of an unbounded model may assign probability outside the valid range; use a distribution and adjustment that respect the support. A numerical delta that pushes many imputations into impossible values is a model warning, not a reason to ignore those values.

A history-only imputation model predicts a missing visit from information available before that visit. It can be appropriate when the future outcome is unavailable at the time of prediction or when the analysis specifically requires that conditioning set. For retrospective imputation of an intermittently missing outcome, however, an observed later outcome is usually relevant information and should not be omitted without reason. Last observation carried forward is not a substitute for a conditional model: it fixes the missing value to a prior observed value and does not represent uncertainty or the information in a later observed visit.

A selection model is another standard framework for sensitivity analysis, but it parameterizes the relationship between outcome values and their observation probabilities rather than directly specifying the missing-outcome distribution conditional on observed outcomes. The choice of framework does not make the sensitivity assumption empirically identified. Delta adjustment is useful when it is easier to state a clinically meaningful departure in outcome units, but its results depend on both the reference model and the chosen delta values.

Common Pitfall

A second error is to shift an entire participant trajectory, including observed visits. Delta adjustment here concerns the distribution of missing outcomes. Changing observed values would alter the data rather than express a sensitivity assumption about unobserved outcomes.