Defining the Delta Sensitivity Parameter

Why Define Delta Precisely?

A missing-at-random (MAR) analysis provides a reference for imputing unobserved values, conditional on the variables included in the imputation model. A sensitivity analysis asks how conclusions might change if the unobserved values systematically differ from that reference. The parameter called delta describes this departure. It is not meaningful to report a delta value without also stating the outcome, the reference, the direction of subtraction, and the scale.

This tutorial defines delta for a conditional location, such as a conditional mean. That definition makes its sign and units explicit. Delta can be defined differently for other outcome models and scales, but those definitions must likewise identify the quantity being shifted and the reference against which it is compared.

Under MAR, and assuming the conditional distribution is correctly specified for the relevant variables, the reference mean is the observed-outcome conditional mean: \(m_0(x)=E(Y\mid X=x,R=1)\). In that case MAR corresponds to \(\delta(x)=0\). A nonzero delta represents a departure from this reference, not a quantity directly learned from the observed outcomes. The direction convention matters: here delta is missing minus MAR reference. Positive delta means a higher conditional mean among cases with missing outcomes; negative delta means a lower one.

Assumptions Behind the Definition

A delta value only has a clear interpretation after the reference and comparison have been specified. In the conditional-mean definition above, \(X\) must be stated: comparing groups with different values of \(X\) would not isolate a departure at the same covariate values. The MAR reference also depends on the chosen imputation model. If that model is misspecified, a nonzero delta could reflect model error as well as a departure from MAR.

The definition does not assume that missingness is actually MAR. Rather, MAR supplies a reference model from which the sensitivity departure is described. The missing outcomes cannot, in general, reveal their own conditional mean: they are unobserved. Consequently, the observed data alone do not identify delta. Analysts must assess plausible values using subject-matter knowledge, external information, or a deliberately stated range of scenarios. A single selected value is an assumption, not an estimate made identifiable by the data.

A conditional mean is not always the right feature to describe. A delta defined for a mean says nothing by itself about how the entire distribution differs: the spread, shape, or tail behavior could also change. The scale must be appropriate to the outcome, and any proposed departure must respect its possible values. These points are reasons to state the definition, not reasons to interpret every delta as a universal shift of every individual value.

Reading the Sign and Magnitude

Suppose the outcome is a score measured in points, and the MAR reference conditional mean for a given set of covariates is 70 points. If delta is \(+4\) points, the stipulated conditional mean for missing outcomes is 74 points. If delta is \(-4\) points, that mean is 66 points. The sign answers which direction the missing-outcome mean departs from the reference; the magnitude answers how large that conditional difference is on the stated scale.

A value such as \(+0.4\) is ambiguous until its scale is named. It could mean 0.4 outcome units, 0.4 standard deviations, or 0.4 units on a transformed scale. Those are different assumptions and generally imply different departures on the original outcome scale. Always report delta together with its scale and the direction convention.

Justification: The Difference Is the Parameter

The definition is simple, but writing it algebraically prevents a frequent sign error and makes clear what a conditional-mean delta encodes. Starting from the missing-minus-reference definition, $$ \delta(x)=E(Y\mid X=x,R=0)-m_0(x), $$ add \(m_0(x)\) to both sides: $$ E(Y\mid X=x,R=0)=m_0(x)+\delta(x). $$

This identity justifies the interpretation: if delta is positive, the missing-outcome mean exceeds the reference mean by that amount; if it is negative, the missing-outcome mean is lower by its absolute magnitude. The identity concerns conditional means only. It does not imply that individual missing values can be recovered by adding the same number to particular observed values, nor does it determine the conditional variance or distribution.

What Delta Does—and Does Not—Tell You

Delta makes an untestable departure explicit. It does not test MAR, identify the missing values, or show that any chosen scenario is plausible. Observed data can help estimate the MAR reference and can reveal whether the imputation model is inadequate for observed cases. They cannot, without additional assumptions or information, verify the conditional mean of outcomes that are missing.

The main alternatives differ in what they assume. A MAR-only analysis sets the departure to zero and reports results under that reference. A delta sensitivity analysis states departures from the reference and assesses how conclusions depend on them. Other missing-data models can parameterize the relationship between missingness and unobserved outcomes in different ways, but changing the model does not remove the need for assumptions about unseen values. Complete-case analysis, which discards records with missing outcomes, does not supply a delta and is not automatically protected from bias.

Delta is also limited by its chosen summary and scale. A mean difference may be inadequate if the scientific concern is a change in the probability of an extreme value, or if a constant mean difference is implausible across covariate profiles. A value that implies an impossible outcome mean signals a poor scale choice or an implausible scenario. And even a mathematically coherent delta cannot compensate for an unsuitable MAR reference model.

Diagnostics and Common Pitfalls

Because delta is not estimable from the missing outcomes, diagnostics focus on clarity and coherence rather than on testing its value. Before interpreting a scenario, check that the reference conditions on the intended variables, that the subtraction order is explicit, and that the delta units match the stated scale. Check whether the implied values are possible and scientifically credible. Separately assess the imputation model using observed data; a plausible delta does not validate a poorly fitting reference.

• Sign reversal: Calling a positive missing-minus-reference delta a decrease reverses the definition. Write the subtraction explicitly before interpreting the sign.
• Unspecified scale: Reporting “delta is 0.5” gives no usable interpretation. Name the outcome scale and units, including any standardization or transformation.
• Overclaiming individual effects: A conditional mean departure does not mean each missing value differs by delta. It specifies a difference in group-level conditional means.
• Treating delta as observed-data evidence: The missing outcomes do not identify their own departure. Present delta as a sensitivity assumption, not as a data-estimated fact.
• Using delta to conceal model failure: A departure parameter does not correct misspecified relationships in the MAR reference. Diagnose that reference separately.

The introductory material on missing-data assumptions provides the MAR context. The key point here is narrower: delta is meaningful only as a defined, scale-specific contrast with that reference. A careful statement should name the missing group, the conditioning variables, the reference quantity, the subtraction direction, and the units.