Additive Delta Shifts for Continuous Outcomes

Why Shift a Reference Distribution?

For a continuous outcome with missing values, an analysis needs a model for outcomes that were not observed. A common starting point is a MAR reference distribution: conditional on the recorded covariates and observed history, missing outcomes are assigned the distribution specified by the reference model. This is a useful benchmark, but it does not establish that the benchmark is correct for missing outcomes. An additive delta adjustment makes a chosen departure from that reference explicit.

This tutorial focuses on one specific construction: shift the conditional mean of a continuous missing outcome by a fixed amount, while retaining the reference model’s residual distribution. The adjustment applies to missing outcomes only. The observed outcome distribution is not altered.

Here \(\delta\) is expressed in the original outcome units. For example, if \(Y\) is a symptom score measured in points, \(\delta=2\) shifts the conditional mean for missing outcomes upward by 2 points. This definition uses the convention that a positive \(\delta\) adds to the reference mean; the direction is part of the specification and should be stated when reporting results.

The reference distribution is a model-based benchmark, not a directly observed distribution of missing outcomes. In particular, the data do not generally identify \(\delta\). It is chosen to examine how conclusions change under departures from the MAR reference assumption.

What the Additive Shift Does

The defining operation is a translation of the reference outcome. If \(Y_0\mid X=x\) has reference density \(f_0(\cdot\mid x)\), then adding \(\delta\) gives the shifted density \(f_\delta(y\mid x)=f_0(y-\delta\mid x)\). The same translation gives the shifted cumulative distribution function. These relations hold whether or not the reference residuals are normally distributed.

If the reference model is written as \(Y_0=m_0(X)+\varepsilon\), the additive adjustment can be represented as \(Y_\delta=m_0(X)+\delta+\varepsilon\). The same residual \(\varepsilon\) is retained. In particular, whenever the relevant moments exist,

The variance result follows because adding a constant does not change deviations from the mean: \(Y_\delta-E(Y_\delta\mid x)=Y_0-E(Y_0\mid x)\). Consequently, the conditional variance and the shape of the residual distribution are retained; only location changes. If the reference model has covariate-dependent residual distributions, the adjustment retains each such conditional distribution rather than imposing a single common variance.

For this construction to have the stated interpretation, the reference distribution must be defined for the covariate and history values at which outcomes are missing, and the same additive shift must be meaningful over the range to which it is applied. The reference model itself must be adequate for its intended conditional prediction. The adjustment does not repair a misspecified reference model; it adds a specified location departure to it.

Applying the Shift to Imputations and Summaries

In a completed-data analysis, the shift is applied to the missing outcomes generated from the reference distribution. If a reference-model draw for a missing record is \(y_0^{(j)}\), the corresponding shifted draw is \(y_\delta^{(j)}=y_0^{(j)}+\delta\). The draw’s residual variation is retained; replacing all draws by a conditional mean would instead remove that variation.

For a sample with \(n\) people, let \(n_1\) outcomes be observed and \(n_0\) be missing, so \(n=n_1+n_0\). If the reference predictions for missing outcomes are \(m_0(X_i)\), the reference completed-data mean is the sum of observed outcomes and predicted missing means divided by \(n\). Under a common additive shift, each missing prediction gains \(\delta\), while observed outcomes do not change. Therefore the completed-data mean changes by \(n_0\delta/n\), not by \(\delta\) unless every outcome is missing.

This identity is for the completed-data mean when the missing outcomes are represented by their conditional means. With stochastic imputations, the same shift adds \(n_0\delta/n\) to each completed-data mean for a fixed set of reference draws. The imputation uncertainty from those draws remains present.

Consequences for Comparisons Between Groups

A common shift does not necessarily cancel from a contrast between groups. Suppose group \(g\) has missing fraction \(q_g\), and its reference completed-data mean is \(\bar{Y}_{g,0}\). When every missing outcome in both groups receives the same \(\delta\), the group mean becomes \(\bar{Y}_{g,0}+q_g\delta\). For the contrast of group A minus group B, the adjusted contrast is therefore the reference contrast plus \((q_A-q_B)\delta\). If missing fractions differ, a common shift changes the contrast.

Assumptions, Diagnostics, and Alternatives

An additive delta analysis has both a reference-model component and a sensitivity component. The reference model supplies the conditional distributions for missing outcomes; the analyst supplies \(\delta\) to represent a departure from those distributions. A useful analysis states both clearly rather than presenting the shifted result as if it were identified by observed data.

• Check the reference predictions. Examine whether covariate profiles among missing cases are represented in the data used to fit the reference model. Predictions far outside the observed covariate range may rely heavily on extrapolation.
• Check the residual assumption. Compare the reference model’s residual spread and shape with observed residuals where outcomes are available. The additive shift retains those features, so a poor residual model remains poor after shifting.
• Check scale plausibility. A location shift preserves the distribution’s shape and spread. Consider whether a translated distribution remains plausible for the outcome’s possible range and for all relevant covariate profiles.
• Check the analysis over specified values. Report results across a justified range of \(\delta\) values, including the reference value \(\delta=0\). A single chosen shift can hide how sensitive the conclusion is to the unobserved-outcome assumption.

The additive construction is appropriate when a constant change in the conditional mean, in outcome units, is a meaningful sensitivity departure and retaining the reference residual distribution is defensible. It may be unsuitable when the effect of a departure should vary with the covariates or when a constant translation produces implausible values. In such cases, the model’s assumptions need to be changed deliberately rather than ignored.

The main comparison is with the unshifted MAR reference analysis, which corresponds to \(\delta=0\). Reference-based imputation instead specifies missing outcomes through a reference arm or reference trajectory, as covered in the earlier material on reference-based imputation; it is not generally equivalent to adding a constant to a MAR conditional mean. More broadly, selection models parameterize missingness conditional on outcomes, whereas this pattern-mixture sensitivity construction directly specifies the missing-outcome distribution. These approaches encode different assumptions and should not be treated as interchangeable.

Common Mistake: Shifting More Than the Missing Outcomes

A frequent error is to add \(\delta\) to every outcome, including those observed, or to add the full \(\delta\) to the overall completed-data mean. The first action changes the observed data and no longer represents a departure confined to missing outcomes. The second ignores the proportion missing. For a common shift, the overall mean changes by \(q\delta\), where \(q=n_0/n\), as shown by the completed-data identity above. A second error is to shift the mean and also inflate the residual variance without specifying a different model: that does not retain the reference residual distribution.