Worked Example: Continuous Outcome Delta Imputation

From a Fitted Reference Model to Delta-Adjusted Imputations

A continuous-outcome delta analysis turns a fitted model for observed outcomes into a transparent set of imputations for missing outcomes. The calculation has two parts: predict from the reference model, then add a prespecified adjustment. Keeping those parts separate makes it easier to verify the arithmetic and interpret the resulting summaries.

Consider a clinical endpoint measured at a follow-up visit. Let \(Y\) denote the endpoint, \(X\) the observed covariates used in the imputation model, and \(R\) indicate whether the endpoint is observed: \(R=1\) means observed and \(R=0\) means missing. A fitted reference model supplies a conditional distribution for \(Y\) given \(X\), estimated from participants with observed outcomes. The sensitivity parameter \(\delta\) shifts that reference distribution for participants with missing outcomes. Its units are the same as those of \(Y\).

For example, if higher scores indicate better health, a negative delta represents worse outcomes among participants with missing data than the reference model predicts for participants with the same covariates. This is a sensitivity assumption, not a quantity estimated from the observed outcomes alone. The earlier material on interpreting delta-adjusted outcome distributions provides the context for choosing and interpreting delta values; here the emphasis is on calculating and summarizing the imputations.

Model, Assumptions, and the Shift Calculation

For clarity, suppose the reference model is a normal linear regression,

The fitted model replaces \(m(x)\), \(\beta_0\), \(\beta_1\), and \(\sigma\) with estimates. For a missing outcome, a conditional-mean imputation is \(\hat m(x)+\delta\). A stochastic imputation also includes a residual draw:

The formula makes an important distinction. The delta-adjusted mean prediction is \(\hat m(x)+\delta\); an individual stochastic imputation generally differs from it because it includes residual variation. Adding the same fixed delta moves every conditional mean by the same amount. Under the stated normal location-shift model, it leaves the residual variance unchanged.

These calculations rely on assumptions that should be made explicit:

• The reference model adequately describes the conditional outcome distribution among participants with observed outcomes over the covariate values of participants with missing outcomes. Its functional form and residual distribution are reasonable for this use.
• The delta-shift assumption is appropriate for the sensitivity scenario: conditional on the modeled covariates, the missing-outcome distribution has the reference shape and spread, shifted by \(\delta\). A constant delta does not allow the shift to vary with covariates.
• The chosen delta is expressed on the same scale and in the same units as the outcome, and the resulting predictions are clinically and mathematically plausible.
• When point estimates of the reference-model parameters are used, the calculation treats them as fixed. It does not, by itself, propagate uncertainty in the fitted model.

The normal model is a working assumption for the examples, not a requirement for every continuous outcome. With another suitable reference distribution, an additive delta still shifts its location, but the resulting distribution must be checked on the actual outcome scale. For bounded outcomes, for instance, a shifted distribution can place mass outside the feasible range.

The shift in a pattern-specific mean follows directly from the definition. For participants with missing outcomes, average the reference predictions and then add the common delta:

If \(n_1\) outcomes are observed with mean \(\bar Y_{R=1}\) and \(n_0\) are missing, the corresponding completed-data marginal mean based on these pattern means is

Thus, when the observed outcomes and fitted reference predictions are held fixed, changing delta by one outcome unit changes the overall mean by \(n_0/(n_1+n_0)\) outcome units. The missing pattern receives the full shift; the overall sample receives a shift weighted by the fraction missing.

Summarizing the Missing Pattern and Its Distribution

A pattern-specific mean is only one summary. When the fitted model supplies a predictive distribution, the missing-pattern distribution can be described as a mixture over the participants’ covariate values. Under the normal location-shift model, participant \(i\)'s predictive distribution is \(N\bigl(\hat m(X_i)+\delta,\hat\sigma^2\bigr)\). Averaging those distributions gives the missing-pattern predictive distribution. Its mean is the average of the shifted means; its variance includes both residual variation and variation in predicted means across participants.

For a finite set of \(n_0\) missing participants, treating their covariates as fixed, the variance of that equally weighted predictive mixture is

The delta does not appear in the second term because adding the same number to every predicted mean does not change their spread. Nor does it change \(\hat\sigma^2\) under this model. This distributional summary describes predictive variation conditional on the fitted parameters; it is not a confidence interval and does not account for uncertainty in estimating those parameters.

How the Overall Mean Changes Across Delta Values

The same predictions can be reused to examine several fixed delta scenarios. This is useful for checking that pattern summaries and the overall mean change in the expected direction and by the expected amount. It is a calculation under each specified scenario, not an estimate of which delta is correct.

Checks, Alternatives, and Limitations

Before interpreting a calculation, check the fitted reference model over the covariate range of the missing participants, the direction and units of delta, and the plausibility of every adjusted prediction. These checks do not establish that the missing outcomes truly follow the shifted distribution; they can reveal when the proposed scenario or its reference model is unsuitable. Model diagnostics and implausible predictions are important because a perfectly executed shift can still be applied to poor predictions.

A frequent mistake is to add delta to the overall mean, or to the observed outcomes, instead of to the missing-pattern predictions. If \(\delta=-3\) points and 40% of outcomes are missing, the missing-pattern mean shifts by \(-3\) points, whereas the overall mean shifts by \(0.40(-3)=-1.2\) points. Only the missing pattern receives the full shift. Another mistake is to treat \(\hat m(x)+\delta\) as a stochastic individual draw; it is the conditional mean, and stochastic imputation also requires residual variation.

Delta adjustment is not the only way to handle missing outcomes. Complete-case analysis excludes the missing pattern and can target a different population summary. Conditional-mean imputation gives missing participants predictions without residual variation. Stochastic multiple imputation adds predictive variation and can propagate reference-model uncertainty, while the delta specifies the sensitivity shift applied to the missing pattern. A selection-model analysis instead specifies missingness conditional on the outcome; it is a different modeling route, not a reason to interpret delta as an observed-data estimate.

The additive model is limited when the departure from the reference distribution varies across covariates, when a shift also changes variance or shape, or when a constant shift violates the outcome’s feasible range. A fixed fitted model and delta also omit uncertainty in the fitted coefficients and residual scale unless that uncertainty is explicitly propagated. Accordingly, report the reference model, the delta and its units, the pattern-specific summaries, and whether summaries reflect conditional means, stochastic imputations, or predictive distributions.