Applying Delta Adjustments After Predictive Mean Matching

What Changes When a Delta Is Applied After Predictive Mean Matching?

Predictive mean matching (PMM) fills a missing outcome by selecting an observed outcome from a donor whose predicted mean is close to the recipient’s predicted mean. This lets the imputation retain values that actually occurred among observed cases, rather than substituting a model-predicted mean. But when PMM is used as a reference for a sensitivity analysis, the missing outcomes may be posited to differ systematically from the corresponding MAR imputation. A delta adjustment expresses that departure by changing the donor value after it has been drawn.

The order matters. The PMM model and matching step determine which donor is selected. A post-draw delta then changes the selected outcome; it does not revisit the match. Consequently, an adjusted value need not be an outcome value observed among donors. This is not automatically an error: it is the intended consequence of positing a departure from the MAR reference. It does require checking that the shifted values remain plausible for the outcome being imputed.

Definition: Let $Y_i^{\text{PMM}}$ be the donor outcome drawn for a missing value on case $i$, before sensitivity adjustment. An additive post-PMM adjustment is $Y_i^{\text{adj}}=Y_i^{\text{PMM}}+\delta$, where $\delta$ is measured in the outcome’s units. A multiplicative adjustment is $Y_i^{\text{adj}}=cY_i^{\text{PMM}}$, where $c>0$ is a dimensionless factor. The adjusted draw represents the sensitivity assumption; the unadjusted PMM draw is the MAR-reference draw.

The adjustment’s interpretation depends on its scale and direction. For an additive adjustment, a positive $\delta$ raises every affected draw by $\delta$ units, and a negative $\delta$ lowers it. For a multiplicative adjustment, $c=0.8$ reduces each positive draw by 20%, while $c=1.2$ increases it by 20%. If a sensitivity parameter is defined on a log scale, its corresponding multiplicative factor is $c=\exp(\delta)$; the scale convention must be stated rather than inferred from the word “delta.”

These are assumptions about the distribution of outcomes among cases with missing values, relative to the MAR reference. The observed data do not identify the appropriate delta merely by fitting the PMM model. For an adjustment to be interpretable, the reference imputation should be suitable for the analysis, the chosen donor pool should be defensible, and the direction and size of the adjustment should make scientific sense. The adjustment also does not repair poor matching or an inadequate imputation model.

Worked Adjustments and Their Consequences

Worked Example: An Additive Shift in Outcome Units

Suppose a hypothetical study imputes a continuous biomarker measured in mg/L. For four missing cases, PMM draws donor outcomes of 14, 17, 20, and 17 mg/L. Consider a sensitivity assumption that these missing values are 4 mg/L lower than the MAR reference, so $\delta=-4$ mg/L.

The additive rule is $Y_i^{\text{adj}}=Y_i^{\text{PMM}}+\delta$. Applying it gives $14+(-4)=10$, $17+(-4)=13$, $20+(-4)=16$, and $17+(-4)=13$ mg/L. The mean of the original draws is $(14+17+20+17)/4=17$ mg/L. The adjusted mean is $(10+13+16+13)/4=13$ mg/L, exactly 4 mg/L lower.

Suppose the eligible donor outcomes in this hypothetical imputation range from 12 to 22 mg/L. The adjusted values 10 and 13 mg/L show that some draws are below the observed donor range; indeed, none of these adjusted draws is required to equal a donor outcome. That is different from violating a scientific boundary: all four adjusted values remain positive. The donor range is an empirical feature of the available donors, not necessarily the full set of scientifically valid values.

Worked Example: A Multiplicative Adjustment for Positive Outcomes

Suppose PMM imputes a positive hypothetical outcome measured in dollars, with donor draws of $25, $40, $55, and $40. A sensitivity assumption posits values 20% lower than the MAR reference, so $c=0.80$.

The rule is $Y_i^{\text{adj}}=cY_i^{\text{PMM}}$. The adjusted values are $0.80(25)=20$, $0.80(40)=32$, $0.80(55)=44$, and $0.80(40)=32$ dollars. Their mean is $(20+32+44+32)/4=32$ dollars. The unadjusted mean is $(25+40+55+40)/4=40$ dollars, and $0.80(40)=32$ dollars, as expected.

The factor preserves positivity because all donor values and $c$ are positive. It does not preserve the exact donor values: for example, $32 may not occur among the available donors. A multiplicative shift can be useful when a proportional difference is meaningful, but it can create especially large absolute changes for large donor values. Its plausibility should therefore be checked across the full range of imputed draws, not only at their mean.

What the Adjustment Does to a Distribution

For a fixed additive $\delta$, adding the same constant to every donor draw translates the distribution without changing its spread. If the donor draws have mean $\mu_D$ and variance $\sigma_D^2$, the adjusted draws have mean $\mu_D+\delta$ and variance $\sigma_D^2$. For a positive fixed factor $c$, the adjusted mean is $c\mu_D$, the variance is $c^2\sigma_D^2$, and the rank ordering of the draws is preserved. These statements describe the transformation of the draws for a given imputation; they do not account for uncertainty about which delta is scientifically appropriate.

The additive result follows directly. If the donor draws are $Y_1,\ldots,Y_n$, their adjusted mean is

\bar{Y}_{\text{adj}} =\frac{1}{n}\sum_{i=1}^{n}(Y_i+\delta) =\frac{1}{n}\sum_{i=1}^{n}Y_i+\frac{1}{n}(n\delta) =\bar{Y}+\delta.

Subtracting the adjusted mean from each adjusted draw gives $(Y_i+\delta)-(\bar{Y}+\delta)=Y_i-\bar{Y}$. Thus the deviations, and therefore the variance computed from those deviations, are unchanged. For a multiplicative adjustment, the deviations become $c(Y_i-\bar{Y})$, so their squared values are multiplied by $c^2$. This explains why the variance is multiplied by $c^2$, while the standard deviation is multiplied by $c$.

Worked Example: The Match Is Unchanged by a Post-Draw Shift

Consider a hypothetical PMM step for one missing case with predicted mean 32. Three observed candidate donors have predicted means 29, 31, and 34, and observed outcomes 28, 34, and 39 units, respectively. With nearest-neighbour matching, their absolute prediction distances are $|32-29|=3$, $|32-31|=1$, and $|32-34|=2$. The two closest candidates are therefore the donors with predicted means 31 and 34.

Suppose random donor selection chooses the donor whose observed outcome is 39 units. That is the PMM draw; now consider the post-draw additive adjustment $\delta=5$ units. The adjusted imputation is $39+5=44$ units. The selected donor remains the same because the shift was applied after matching. In particular, the adjustment did not change the distances 3, 1, and 2, or cause the donor with outcome 34 to be selected instead.

If the eligible donor outcomes are 28, 34, and 39 units, then 44 is outside the exact donor support and above its observed range of 28 to 39. This makes clear what is being assumed: the missing-case value may be higher than the observed donor values under the sensitivity scenario. Whether 44 is plausible depends on the outcome’s scientifically valid range and context, not on donor membership alone.

Assessing Donor Support and Plausibility

“Support” can refer to more than one thing. The exact donor support is the set of observed outcome values eligible to be drawn. The donor range is the minimum-to-maximum interval in that pool. A shifted draw may leave the exact set, or even fall outside the range, while still being scientifically valid. Conversely, a value inside the donor range may still be invalid for a particular case or inconsistent with known outcome constraints. Check both the empirical donor distribution and the outcome’s valid range; do not treat them as interchangeable.

A practical assessment should be made for each proposed delta and across imputations. Review the adjusted values’ minimum and maximum, distribution, and frequency of values at or beyond meaningful boundaries. Compare them with the MAR-reference draws and, when relevant, observed outcomes in comparable covariate regions. Check whether shifts produce implausible concentration, extreme tails, or impossible values. If different cases have different donor pools, assess the adjusted values against the relevant pools rather than only against the overall observed range.

For a constant additive shift, a range $[L,U]$ becomes $[L+\delta,U+\delta]$; for a positive multiplier, it becomes $[cL,cU]$. These simple checks help anticipate boundary problems. If the outcome has a strict lower bound of zero, for example, an additive negative delta could create negative values even when every donor value is positive. A multiplicative factor $c>0$ preserves positivity for positive donor values, but it does not guarantee that the adjusted values respect an upper limit or any more complex constraint.

Clipping shifted values to a boundary changes the assumed distribution: several distinct adjusted outcomes may collapse to the same boundary value. Redrawing until a value is valid also changes the distribution and can alter its mean. Such corrections should not be treated as neutral repairs. If a proposed adjustment regularly creates invalid values, reconsider the scale, the size of the shift, or the plausibility of the scenario. Boundary handling should be explicit and consistent with the scientific estimand.

Comparison, Diagnostics, and Limitations

A post-PMM adjustment has a useful separation of roles: PMM supplies a MAR-reference donor draw, and delta changes that draw according to a stated sensitivity scenario. It differs from changing a predictor model, modifying predicted means before donor matching, or changing donor-selection rules. Those alternatives can change which donors are eligible or selected, whereas a post-draw adjustment leaves the match intact. A direct parametric adjustment may instead transform model-based draws and need not retain observed donor outcomes as its reference. These approaches answer different modeling questions; none is automatically preferable for every outcome.

Approach	What is changed?	Key implication
Post-PMM additive delta	Selected donor outcome is shifted by a fixed number of outcome units.	Donor selection is unchanged; values may leave donor support or valid bounds.
Post-PMM multiplicative delta	Selected donor outcome is multiplied by a positive factor.	Proportional change is imposed; absolute change depends on donor value.
Change to prediction or matching stage	Predicted means, candidate donors, or selection can change.	The selected donor distribution can change, not just the values after selection.

A useful diagnostic is to compare both stages rather than inspecting adjusted values alone: record the donor selected, its observed outcome, and the final adjusted value. If the donor identities differ across runs that were intended to use the same random draw and matching setup, investigate the implementation; the delta itself should not cause that difference when it is applied only after selection. Also check that the same adjustment is applied consistently to the intended missing values and that the scale is not confused—for example, an additive shift in dollars is not equivalent to a percentage reduction.

PMM’s donor-based reference remains dependent on the matching model, the availability of suitable donors, and the donor-selection procedure. A delta does not fix poor overlap in predicted means, a sparse donor pool, or a model that omits important predictors. Large shifts can place imputed values far from observed outcomes and make conclusions depend strongly on extrapolations beyond the donor data. Finally, a single fixed shift may be an inadequate description if the departure from MAR varies across cases; defining more complex patterns of adjustment requires additional assumptions.

Common Mistake

Common mistake: “PMM always keeps imputations within the observed donor range, so applying a delta after PMM must still produce donor-supported values.” The first claim applies to an unadjusted donor draw, not a shifted value. For example, adding 5 to a donor outcome of 39 produces 44, which is not among donors with outcomes 28, 34, and 39. The correction is to distinguish the donor selected from the final imputed value, then assess the adjusted value against both the donor distribution and the outcome’s scientifically valid range.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

PMM draws 8, 11, and 14 units. What are the adjusted values under an additive delta of $-2$ units, and what happens to their mean?
A positive outcome has PMM draws 10 and 30. What are the adjusted draws for $c=1.25$? Does this factor preserve their ordering?
Explain why applying an additive delta after donor selection does not change which donor was selected.
The donor values range from 5 to 18, and an additive adjustment produces a value of 3. Is the value necessarily invalid? What should be checked?

Solutions. (1) The values are $8-2=6$, $11-2=9$, and $14-2=12$ units. The original mean is $11$ units and the adjusted mean is $9$ units, a decrease of 2. (2) The values are $1.25(10)=12.5$ and $1.25(30)=37.5$; because $c$ is positive, their ordering is preserved. (3) Donor selection used the predictions and distances before the outcome was shifted; a later adjustment changes the selected value, not the match. (4) Not necessarily. The value is below the observed donor range, but validity depends on the outcome’s scientific constraints and context; check those constraints and whether the assumed shift is plausible.