Delta Adjustment as a Pattern-Mixture Sensitivity Parameter

Why Parameterize a Departure?

A pattern-mixture analysis describes outcomes separately for groups defined by their missingness patterns. For people whose outcome is missing, the data alone do not reveal the distribution of that outcome. An analysis must therefore specify, directly or indirectly, what that distribution would be. A delta sensitivity parameter gives a structured way to describe departures from a chosen reference distribution, rather than treating that reference as the only possibility.

The reference distribution is a starting assumption, not an observed fact about people with missing outcomes. Delta labels or indexes alternative distributions relative to it. The parameter can concern a whole distribution, not just its mean: an alternative may differ in location, spread, shape, or other features. No universal numerical scale or sign convention follows from the word “delta”; these depend on how the family of alternatives is defined.

The notation allows \(\delta\) to be scalar for a simple sensitivity analysis, but the family must still be explicit: one needs to know which distributions its values select. In particular, \(\delta\) is not automatically an additive change in outcome units, nor must it move every feature of a distribution in the same way. Those are possible choices of parameterization, not part of the general definition.

A valid \(Q_\delta\) must be a probability distribution on plausible outcome values, conditional on the stated information and pattern. It should also be clear whether the same family applies across patterns and covariate histories. The sensitivity analysis then compares conclusions under specified values of \(\delta\), including the reference value, rather than pretending that the missing outcomes have been observed.

How Delta Enters a Pattern-Mixture Summary

For clarity, consider a single outcome \(Y\) and two patterns: observed outcome and missing outcome. Let \(R=1\) indicate that \(Y\) is observed and \(R=0\) that it is missing. The pattern probabilities are \(P(R=1)\) and \(P(R=0)\). The observed-pattern distribution can be summarized from observed data; the missing-pattern distribution is supplied by the reference assumption and its delta alternatives.

The overall mean under a specified \(\delta\) follows from conditioning on the pattern. Provided the outcome has a finite mean in each pattern, the law of total expectation gives:

This identity is a mixture calculation, not an extra missing-data assumption. The sensitivity assumption enters through \(E_\delta(Y\mid R=0)\), which is the mean implied by \(Q_\delta\). The pattern probability and the observed-pattern mean also need to be estimated or specified appropriately; the formula does not make those quantities known.

With several missingness patterns, the same reasoning sums across them: each pattern-specific mean is weighted by the probability of that pattern. With covariates or observed histories, the conditional distribution is indexed by that information and the relevant quantities must be averaged over its distribution. A delta specification therefore needs to say what information conditions the reference and alternative distributions, not merely give a number.

Assumptions and Interpretation

A delta analysis is meaningful only when its ingredients are stated. At minimum, specify the outcome and missingness patterns, the information \(H\) conditioning the missing-outcome distribution, the reference distribution \(Q_0\), and the family \(Q_\delta\). State which values of \(\delta\) are examined and what each means in that family. If pattern probabilities or the observed-pattern distribution are estimated, the analysis also relies on those estimates being suitable for the target population and estimand.

The reference distribution must be a defensible benchmark for the target pattern and conditioning information. A departure family should preserve the outcome’s support and produce coherent distributions for relevant histories. For example, probabilities must remain between 0 and 1 and sum to 1; a distribution for a bounded outcome must not assign probability outside its allowable range. These checks apply to every scenario, not only the reference.

Crucially, observed outcomes generally do not identify the distribution of outcomes that are missing. Delta is therefore a sensitivity parameter, not an ordinary model coefficient that the observed data can estimate without further assumptions. Data can inform the reference model and observed portions of the analysis, but selecting among plausible departures requires substantive judgment or external information. Reporting a single delta value as if the data established it overstates what is learned.

Comparing Scenarios and Diagnosing Problems

The purpose of varying \(\delta\) is to show how conclusions depend on unobserved-outcome assumptions. A useful report gives the reference result alongside alternative scenarios and explains the substantive meaning of each. Where an estimand is a contrast, compute it under each scenario using the same target population and clearly stated assumptions. Do not describe one scenario as empirically confirmed merely because its result is more desirable or statistically significant.

Several checks help identify a specification that may be going wrong. Confirm that every proposed distribution is normalized and respects the outcome’s support. Compare its implied summaries and shape with observed outcomes and relevant external knowledge, while recognizing that observed outcomes cannot verify the missing-outcome distribution. Examine whether conditional distributions vary implausibly across histories or whether small changes in delta create abrupt, unexplained changes in the estimand. Such patterns can indicate a poorly chosen parameterization, unstable estimation, or a scenario that lacks substantive credibility.

Delta sensitivity analysis is related to, but distinct from, other approaches. Reference-based imputation specifies a particular reference rule for missing outcomes; a delta family can be used to examine departures from a chosen reference. A selection model instead describes how the probability of being observed depends on outcome information. These approaches encode missingness assumptions differently, and results need not agree. Delta analysis does not remove the need to justify a reference or guarantee robustness; it makes selected departures explicit so their consequences can be assessed.

Common Pitfalls and Limits

• Assuming delta has a universal meaning. The correction is to define the reference point and the family \(Q_\delta\) before interpreting a value. A positive value might indicate a particular departure in one parameterization and something different in another.
• Changing only a summary while leaving the distribution undefined. A mean alone does not specify tails, spread, or support. If the analysis requires draws or distributional probabilities, define a full valid distribution rather than only a target mean.
• Treating delta as estimated from the observed outcomes. Its role is to index assumptions about missing outcomes that are not identified by those outcomes. Separate data-estimated quantities from sensitivity choices in both analysis and reporting.
• Ignoring conditioning and pattern composition. The missing-outcome distribution can depend on observed history, and pattern weights affect the overall result. State both clearly and use the appropriate weights.
• Calling a scenario “conservative” without specifying the target. A departure that lowers one group’s mean may not be conservative for every contrast or decision. Describe its consequences for the estimand actually being reported.