Deriving the Marginal Mean Under Additive Delta Adjustment

Why Pattern Proportions Matter

An additive delta describes a shift in the assumed mean outcome for a missing-data pattern relative to a reference model. But a shift in one pattern does not, by itself, determine the shift in the population mean. Its marginal effect also depends on how common that pattern is. A large delta for a rare pattern can have less impact on the overall mean than a modest delta for a common pattern.

The calculation follows a basic principle: partition the target population into its missingness patterns, find the adjusted mean within each pattern, and average those means using the patterns’ proportions in the target population. The weights must include people whose outcome is missing; using only observed outcomes changes the target quantity.

Definition and Marginal-Mean Formula

The delta is measured in the same units as \(Y\). For example, if \(Y\) is a score in points, a delta of \(2\) points means that the assumed mean among people with missing outcomes is 2 points higher than the reference-model mean for that pattern. A negative delta represents a lower assumed mean. In either case, \(\delta_r\) is a sensitivity parameter, not something identified by the observed outcomes alone.

For clarity, the formula treats \(\delta_r\) as a constant shift for the outcome mean within pattern \(r\). If the outcome is observed in a pattern, that pattern contributes its actual mean and receives no shift. If the outcome is missing, \(\mu_r^{\mathrm{ref}}\) is the mean predicted for that pattern under the reference model, and the adjusted mean is \(\mu_r^{\mathrm{ref}}+\delta_r\). Thus the formula does not replace observed outcomes with model predictions.

In a sample of \(n\) people, let \(n_r\) be the number in pattern \(r\), so that the sample pattern proportion is \(\hat{\pi}_r=n_r/n\). Replacing population quantities with estimates gives the plug-in marginal mean:

Here \(\hat{\mu}_r^{\mathrm{ref}}\) is the observed pattern mean when the outcome is observed, or the reference-model predicted mean when it is missing. The proportions must sum to one. If the target population is defined by a study cohort, use pattern counts from that cohort, not proportions calculated only among people with observed outcomes.

Derivation and Sensitivity to Delta

The marginal mean is the expectation of \(Y\) over the entire target population. Partitioning that expectation according to the mutually exclusive patterns gives the law of total expectation:

Under the specified additive adjustment, the conditional mean within pattern \(r\) is \(\mu_r^{\mathrm{ref}}+\delta_r\). Substituting this into the partitioned expectation yields

This derivation requires no special weighting result beyond averaging conditional means over a complete partition. The substantive assumption is the pattern-specific mean specification: the reference mean is appropriate as a starting point, and the chosen delta represents the intended departure from it for that pattern.

The formula also makes the role of pattern proportions explicit. If only \(\delta_s\) changes, then

A one-unit increase in \(\delta_s\) therefore increases the marginal mean by \(\pi_s\) units, with the other inputs held fixed. If every missing-outcome pattern shares a common delta \(\delta\), while observed-outcome patterns have delta zero, the formula becomes

where \(\mathcal{M}\) is the set of patterns in which the outcome is missing. The multiplier of the common delta is the proportion of the target population with a missing outcome. This result is a useful check: a common additive shift cannot change the population mean by more than the shift itself when the missing-outcome proportion is at most one.

Several Missingness Patterns

A single missing-versus-observed split can be too coarse when distinct patterns have different reference means or different delta assumptions. In that setting, retain each pattern as a separate term. Patterns should be defined so that every person belongs to exactly one, and the outcome’s missingness status in each pattern should be clear.

Applying the Formula to a Contrast

The same calculation can be made separately in each treatment arm before forming a contrast. This matters because arms can have different pattern proportions. Even if the same delta is used in both arms, its contribution to each arm mean depends on that arm’s missing-outcome proportion.

Assumptions, Checks, and Failure Modes

The arithmetic is simple, but its interpretation depends on the setup being coherent. Before using the formula, check the following:

• The patterns form a partition. They must be mutually exclusive and exhaustive for the target population. Their proportions are nonnegative and sum to one.
• The target population is explicit. Pattern proportions must refer to the population whose mean is sought. Excluding people with missing outcomes and then renormalizing the remaining weights targets a different group.
• Each reference mean is defined for its pattern. Observed-pattern means can be calculated from observed outcomes. Missing-pattern reference means require a reference model that supplies predictions for those outcomes.
• The delta matches the outcome scale and pattern. A delta in points cannot be added to an outcome measured in a different unit. A delta intended for one pattern should not be applied to all patterns without stating that assumption.
• The mean exists and the proposed adjustment is meaningful. For a bounded outcome, a constant shift might imply impossible values or an infeasible mean. The mean formula alone does not guarantee that the adjusted outcome distribution is scientifically plausible.

A practical arithmetic check is to calculate the reference-model marginal mean first, then add each pattern’s weighted delta contribution. The result should agree with direct weighting of the adjusted pattern means. If not, likely causes include omitting a pattern, using percentages that do not sum to one, applying a shift to an observed pattern, or multiplying a delta by the wrong proportion.

One common mistake is to say, “The missing-pattern delta is 2 points, so the overall mean increases by 2 points.” That is wrong unless every person belongs to an adjusted pattern. With a common delta, the population-mean change is the delta multiplied by the missing-outcome proportion. For example, if 30% of outcomes are missing and the common delta is 2 points, the marginal mean changes by \(0.30(2)=0.60\) points, not 2 points.

How This Calculation Differs From Alternatives

The observed-case mean averages only people whose outcomes are observed. It can be a useful descriptive summary of those people, but it does not incorporate the missing-pattern means and generally is not the target-population mean. The delta-adjusted marginal mean instead gives every pattern its target-population weight.

The reference-model marginal mean is the same weighted calculation with all deltas set to zero. It represents the reference assumptions for missing outcomes. A delta-adjusted mean departs from that reference by the sum of the pattern proportions times their respective deltas. Thus the formula makes the sensitivity assumption’s effect visible rather than concealing it in a single overall result.

These comparisons do not establish which delta is correct. Observed data can inform pattern proportions and observed means, and the reference model supplies its predictions, but the delta represents an assumption about outcomes that are not observed. The resulting marginal mean should therefore be reported with the delta values and pattern weights that produced it. A point calculation also does not, by itself, account for uncertainty in estimated pattern proportions or reference means.