Common Errors in Delta Parameterization

Why Delta Parameterization Errors Matter

A delta adjustment is meaningful only when its direction, scale, and target are clearly specified. A minus sign may represent worse outcomes or better outcomes depending on the outcome’s direction; a numerical shift in outcome units is not the same as a shift in log odds; and an adjustment for missing outcomes must not alter outcomes that were observed. These distinctions are easy to lose when a sensitivity analysis is implemented across several imputation steps.

This tutorial uses one explicit convention throughout. First, a reference model predicts the distribution of an outcome for a person with missing data, conditional on the information included in that model. Delta then shifts that reference distribution for missing outcomes only. Under this convention, positive delta means shifting the missing-outcome distribution upward on the stated scale, and negative delta means shifting it downward. Whether “upward” is clinically better depends on how the outcome is defined.

This definition does not make delta an observed-data quantity. The reference model is informed by observed data and its assumptions; delta represents an additional sensitivity assumption about outcomes that are missing. Unless a model explicitly assigns a probability distribution to delta and estimates it, a chosen delta value is fixed for that sensitivity analysis.

Trace the Sign Before Doing the Arithmetic

For an outcome where higher values are better, a negative delta says that missing outcomes are assumed to be worse than their reference-model predictions. For an outcome where lower values are better, the same negative delta says the missing outcomes are assumed to be better. The sign has no clinical interpretation without the outcome’s direction and the stated definition of the shift.

A useful check is to write the defining equation before entering a delta value into software. If the intended assumption is that missing outcomes are worse on a higher-is-better scale, the adjusted expectation should be less than the reference expectation. This simple inequality catches sign reversals before they affect a treatment comparison.

The Scale Is Part of the Parameterization

A delta value has units, even when those units are not immediately visible. For a continuous outcome, an additive delta may be measured in points, kilograms, or another outcome unit. For a binary outcome, a shift can instead be defined on the probability scale or on a link scale such as the logit. Those are different parameterizations: the same numerical value does not have the same meaning across scales.

For example, a logit-scale delta adds to the reference-model log odds. If the reference probability is \(p_{\mathrm{ref}}\), then the adjusted probability is $$ p_{\delta}=\frac{\exp\{\operatorname{logit}(p_{\mathrm{ref}})+\delta\}}{1+\exp\{\operatorname{logit}(p_{\mathrm{ref}})+\delta\}}. $$ Equivalently, delta multiplies the reference odds by \(\exp(\delta)\). It does not add directly to the probability.

For binary data, a probability-scale shift must also keep adjusted probabilities between 0 and 1 for the cases to which it is applied. A constant logit-scale shift keeps probabilities within that interval automatically, although it may still imply implausible probabilities. For bounded continuous outcomes, an additive shift can also produce values outside the possible range; feasibility must be checked rather than assumed.

Apply the Adjustment to Missing Outcomes Only

The reference prediction and its delta adjustment describe the distribution assigned to outcomes that were not observed. They do not revise an observed measurement. In longitudinal settings, this means an observed visit remains observed at its recorded value; delta affects a missing visit’s imputed distribution, conditional on the information the model uses. A later missing visit can be adjusted without changing an earlier observed visit.

The consequence for an overall mean follows directly from averaging observed and missing outcomes. Write \(q=P(R=0)\) for the missing fraction, \(\mu_{\mathrm{obs}}=E(Y\mid R=1)\), and \(\mu_{\mathrm{ref,mis}}=E_{\mathrm{ref}}(Y\mid R=0)\). Under an additive delta applied to missing outcomes, $$ E(Y;\delta)=(1-q)\mu_{\mathrm{obs}}+q(\mu_{\mathrm{ref,mis}}+\delta) =E(Y;0)+q\delta. $$ The observed component has no delta term. For a treatment-minus-control contrast, the corresponding shift is \(q_T\delta_T-q_C\delta_C\). This identity is also a useful implementation check: shifting all outcomes rather than missing outcomes would not produce the stated adjustment.

A Fixed Sensitivity Value Is Not an Estimated Parameter

In a fixed-delta sensitivity analysis, the analyst chooses delta (or a set of values) to represent assumptions not identified by the observed outcomes. The imputation model may estimate regression coefficients, residual variation, or other quantities from observed data, but those estimates do not turn the chosen delta into an estimated parameter. At each fixed delta, uncertainty in the reference model and the completed-data analysis can still be represented according to the analysis method.

In multiple imputation, the same fixed delta is used for each imputation in a given sensitivity analysis. Imputed values can vary across imputations because the reference-model parameters and imputation draws vary. That variation is not evidence that delta itself has been estimated. If several delta values are examined, they define separate sensitivity analyses; they should not be averaged together as though they were repeated estimates of one unknown delta unless a separate probabilistic model for delta has been specified.

Derivation and Practical Checks

The overall-mean identity above follows from conditioning on whether the outcome is observed. Because \(R\) takes the values 0 and 1, the law of total expectation gives

With \(P(R=0)=q\), \(P(R=1)=1-q\), and the definition \(E(Y\mid R=0;\delta)=\mu_{\mathrm{ref,mis}}+\delta\), substitution yields

This derivation assumes the additive shift is applied to the missing-outcome expectation, the observed-outcome distribution is left unchanged, and \(q\) is the relevant missing fraction. For covariate-specific shifts, the analogous adjustment is averaged over the covariate distribution among missing cases; it need not equal \(q\) times one constant unless delta is constant.

Before trusting a result, check the implementation in a small, inspectable set of records. Confirm that each shifted value belongs to a missing outcome, that its scale matches the stated delta, and that its direction agrees with the clinical assumption. Compare the adjusted and reference predictions for a few cases. Also check for impossible values, unexpected changes to observed records, and discrepancies between the claimed and implemented delta.

How This Choice Relates to Alternatives

A zero delta on the stated scale reproduces the reference-model assumption for missing outcomes. A nonzero fixed delta departs from that reference assumption in a transparent, specified way. Other delta values can be considered to assess sensitivity, but each value must retain the same definition and scale for comparisons to be interpretable.

A delta adjustment is one way to represent assumptions about missing outcomes. Selection models instead represent missingness through a model for the probability of observation conditional on outcomes and other information. These approaches parameterize different parts of the joint distribution and their parameters should not be treated as interchangeable. Whichever approach is used, the assumptions about unobserved outcomes are not diagnosed solely from observed data; the purpose of sensitivity analysis is to examine conclusions under explicit alternatives.