From a Reference Imputation to a Delta-Adjusted Imputation
Multiple imputation represents uncertainty about missing outcomes by creating several completed data sets. In a sensitivity analysis, the reference imputation model supplies a starting distribution for those outcomes. A prespecified delta then encodes a chosen departure from that reference. The essential operation is to adjust the missing outcomes within each imputation, while leaving observed outcomes unchanged.
Let \(Y_i\) denote the outcome for individual \(i\), let \(R_i=1\) indicate that \(Y_i\) is observed, and let \(R_i=0\) indicate that it is missing. Let \(X_i\) denote the covariates used in the imputation model. For imputation \(m\), the reference procedure generates a draw for each missing outcome from its predictive distribution given the available data and covariates. Write this draw as \(Y_{i,\mathrm{mis}}^{(m,0)}\), where 0 labels the reference, unadjusted imputation.
The reference imputation procedure must be appropriate for the outcome and covariates, and its predictive draws must represent the intended reference distribution. The adjustment also requires a clearly defined delta scale and direction. A positive delta does not have a universal clinical meaning: its interpretation depends on the outcome scale and, for a binary outcome, on which event is coded as 1.
For each imputation, the workflow is therefore: generate reference predictive draws for missing outcomes, apply the specified transformation to those draws or their predictive distributions, and retain the observed outcomes as recorded. The same prespecified delta is used across imputations for a fixed sensitivity scenario. The imputation-specific draws can differ because their reference predictions differ.
Continuous Outcomes: Shift Each Missing Draw
For an additive delta on the outcome scale, the adjusted draw is the reference draw plus \(\delta\):
Here \(\delta\) has the same units as \(Y\). If the reference predictive distribution has cumulative distribution function \(F_0\), this transformation produces the shifted distribution \(F_\delta(y)=F_0(y-\delta)\). Thus its mean increases by \(\delta\), while its variance is unchanged. This is a location shift, not a change in residual variability or distributional shape.
Worked Example: Shifting a Continuous Imputation Draw
Suppose a hypothetical study records a biomarker in mg/dL. Two observed values are 100 and 110 mg/dL, and a third outcome is missing. The reference imputation procedure produces three draws for that missing outcome: 120, 130, and 140 mg/dL. The prespecified sensitivity value is \(\delta=+5\) mg/dL.
Apply the shift separately to each reference draw:
The three completed data sets are therefore \((100,110,125)\), \((100,110,135)\), and \((100,110,145)\), all in mg/dL. Their means are \(335/3=111.7\), \(345/3=115.0\), and \(355/3=118.3\) mg/dL, respectively. The observed values remain 100 and 110 mg/dL in every data set. The positive shift increases each completed-data mean by \(5/3=1.667\) mg/dL, not by 5 mg/dL, because only one of the three outcomes is missing.
The example also shows why the delta belongs on each missing draw: each imputation retains its own reference draw and uncertainty. Adding the delta to the observed outcomes as well would change the target data, not merely the assumptions for missing outcomes.
Binary Outcomes: Shift the Log Odds Before Drawing
For a binary outcome, let \(Y_i=1\) denote the event of interest and let \(p_{i}^{(m,0)}\) be its reference-model probability in imputation \(m\). A delta on the log-odds scale changes that probability according to
The adjusted probability is between 0 and 1 for any finite \(\delta\). The multiplier \(\exp(\delta)\) is the factor by which the event odds are multiplied, conditional on the reference probability. After calculating the adjusted probability, draw the missing binary outcome from a Bernoulli distribution with that probability. In other words, the adjustment changes the probability distribution from which the binary outcome is drawn; it does not add a real-valued quantity to an already drawn 0 or 1.
Worked Example: A Binary Odds Shift
Suppose a hypothetical reference model gives a missing patient’s probability of an event as \(p^{(m,0)}=0.20\). Take \(\delta=\log(2)\), so the event odds are doubled. The reference odds are \(0.20/(1-0.20)=0.25\). The adjusted odds are \(2(0.25)=0.50\), giving
Now suppose the random number used to generate the Bernoulli outcome is \(U=0.30\), with the outcome set to 1 when \(U<p^{(m,\delta)}\). Since \(0.30<0.3333\), this imputation draw is \(Y_{\mathrm{mis}}^{(m,\delta)}=1\). Under the reference probability alone, the same \(U\) would yield 0 because \(0.30\) is not less than 0.20. The adjusted probability is 0.3333, not a guarantee that this particular missing outcome equals 1.
The calculation assumes event coding is fixed as stated. If the event were instead coded as 0, the sign and interpretation of the delta would need to be defined for that coding; one should not switch event definitions without also changing the parameterization.
Applying the Same Delta Across Imputations
A fixed sensitivity scenario uses the same delta value in every imputation, but does not force the imputed outcomes to be identical. The reference draws vary across imputations, and the delta transformation is applied to each one. For a continuous outcome, a single missing value in each imputation might have draws 43, 47, and 45. With \(\delta=+4\), the adjusted draws are 47, 51, and 49. For a binary outcome, each imputation’s reference probability is shifted on the log-odds scale before its missing outcome is drawn.
Worked Example: Adjusting Several Missing Outcomes in Each Imputation
Consider a hypothetical continuous outcome measured in score points. Two observed scores are 40 and 50, and two scores are missing. Three reference imputations produce the following missing-score pairs:
Imputation 1: \((43,55)\); imputation 2: \((47,52)\); imputation 3: \((45,58)\). Use the same prespecified shift, \(\delta=+4\) points, for every missing score.
The adjusted pairs are \((47,59)\), \((51,56)\), and \((49,62)\). The observed scores remain 40 and 50. The completed-data means are
For comparison, the three reference completed-data means are \(47.00\), \(47.25\), and \(48.25\). In each imputation, two of four scores were missing, so the shift raises that imputation’s overall mean by \(2(4)/4=2\) points. This agrees with the displayed calculations. It also illustrates that the delta shifts the missing values by 4 points each; it does not shift the overall completed-data mean by 4 points.
Why the Order of Operations Matters
The transformation must match the outcome scale and be applied at the appropriate stage. For a continuous additive shift, adding \(\delta\) to each reference missing-value draw gives the intended location-shift distribution. For a binary outcome, the log-odds transformation must be applied to the predictive probability before the Bernoulli draw. Adding \(\delta\) to a 0/1 draw would create values that are not binary outcomes. Adding \(\delta\) directly to a probability is also a different model: it is an additive probability shift, can produce values outside the interval from 0 to 1, and does not represent a constant odds multiplier.
Likewise, adjusting only a final average after imputation is generally not equivalent to adjusting the missing draws. The number and pattern of missing outcomes determine how a shift in missing values affects a statistic, and many analyses depend on more than the overall mean. Applying the transformation within each completed data set preserves the intended outcome-level adjustment for subsequent analysis.
Assumptions, Diagnostics, and Alternatives
Delta adjustment does not repair a poor reference imputation model. The model should be suitable for the outcome type and include relevant predictors; its predictive behavior should be checked before interpreting an adjusted analysis. The delta must also be specified on a meaningful scale, with its sign and target population clear. A fixed delta is a sensitivity assumption, not a value identified by the observed outcomes. Different plausible delta values therefore represent different assumptions about the missing outcomes.
After adjustment, inspect the resulting imputations in context. For continuous outcomes, check whether shifted values violate natural bounds, create implausible tails, or yield patterns inconsistent with the intended location shift. For binary outcomes, check that the event coding is correct and examine whether adjusted probabilities are nearly 0 or 1 for many missing cases. Extreme probabilities can make the result highly sensitive to small changes in predictions or delta. Also compare the adjusted and reference imputations to confirm that observed values have not changed and that the intended shift was applied to every missing value.
A delta shift is not the only possible sensitivity specification. The unadjusted reference imputation corresponds to \(\delta=0\). An additive shift in continuous outcomes changes location while preserving the reference spread; a model that also changes residual variability or distributional shape expresses a different assumption. For binary outcomes, a log-odds shift encodes a constant odds multiplier, whereas an additive probability shift encodes a different relationship and may require restrictions to keep probabilities valid. These alternatives should be chosen for their scientific interpretation, not because one is automatically more correct.
Finally, a delta-adjusted analysis remains conditional on the reference model and the chosen sensitivity value. It does not reveal the unobserved outcomes, and agreement with observed-data checks cannot establish that the missing outcomes follow the adjusted distribution. The value of the procedure is that it makes a departure from the reference explicit and allows its consequences to be analyzed consistently across completed data sets.
Common Mistakes
- Shifting observed outcomes. This changes recorded data as well as the missing-data assumption. Apply delta only to outcomes that were imputed.
- Adding a binary delta to a 0/1 draw. A binary delta is applied to the log odds of the event probability before drawing the outcome.
- Calling \(\exp(\delta)\) a probability change. For a binary log-odds shift, it is an odds multiplier. The probability change depends on the starting probability.
- Assuming the completed-data mean shifts by exactly delta. Only missing outcomes are shifted. The change in an overall mean depends on the fraction missing and on which values are missing.
- Interpreting a plausible adjusted data set as evidence for that delta. Plausibility checks can reveal failures, but they do not identify the unobserved outcomes or validate the sensitivity assumption.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A continuous outcome has three missing values among ten individuals. Each missing draw is shifted upward by 6 units. By how much does the completed-data mean increase?
- A binary reference probability is 0.50 and the log-odds delta is \(\log(3)\). What is the adjusted probability?
- Why is it incorrect to add a binary delta of 0.4 directly to an imputed outcome equal to 1?
- In a binary imputation, should the delta be applied to the reference probability or to the completed-data event proportion?
Solutions. (1) The total increase across the ten outcomes is \(3(6)=18\) units, so the mean increases by \(18/10=1.8\) units. (2) The reference odds are \(0.50/(1-0.50)=1\). Multiplying by 3 gives odds of 3, so the adjusted probability is \(3/(1+3)=0.75\). (3) Adding 0.4 would produce 1.4, which is not a binary outcome. A log-odds delta changes the predictive probability before the Bernoulli draw. (4) Apply it to the reference predictive probability for each missing outcome, then draw the binary outcome; the completed-data event proportion is a summary, not the quantity to transform.