Pattern-Specific Delta Adjustments

Why distinguish missingness patterns?

A single delta adjustment applies the same departure from a missing-at-random (MAR) reference to every imputed value for a target variable. That can be too restrictive when records with different configurations of missing data have different reasons for being incomplete. For example, among people with a missing outcome, one pattern may have a particular covariate observed while another pattern has that covariate missing. A sensitivity analysis can assign these prespecified patterns different delta values, then combine the adjusted imputations according to how common each pattern is in the target sample.

The central calculation is not an unweighted average of the pattern-specific means. It is an average weighted by each pattern’s share of the target sample. The pattern definitions, the scale of delta, and the values assigned to patterns must be specified clearly: the observed data generally do not identify the pattern-specific departures from MAR.

Definition: A missingness pattern is a distinct configuration indicating which variables are observed and which are missing for a record. For a specified target outcome, a pattern-specific delta adjustment assigns a sensitivity parameter \(\delta_r\) to each prespecified pattern \(r\) in which that outcome is missing. For an additive adjustment on the outcome scale, an MAR-based imputed value for an individual in pattern \(r\) is shifted by \(\delta_r\). The pattern with the outcome observed is not shifted.

This tutorial focuses on a continuous target outcome \(Y\) and additive shifts on its original measurement scale. Thus, \(\delta_r\) has the same units as \(Y\). If the outcome is measured in points, a delta of \(-2\) means imputing values two points lower than the corresponding MAR-based values for that pattern. Other outcome types or delta scales require adjustments appropriate to their scale.

Definition and weighted estimate

Let \(n\) be the number of individuals in the target sample. Let \(n_0\) denote the number with \(Y\) observed, and let \(n_r\) denote the number with \(Y\) missing in pattern \(r\), for \(r=1,\ldots,R\). These categories must be mutually exclusive and cover the full sample, so \(n=n_0+\sum_{r=1}^{R}n_r\). Let \(\bar{Y}_{0}\) be the observed-outcome mean. For one imputed data set, let \(\bar{Y}_{r,\mathrm{MAR}}\) be the mean MAR-based imputation for the missing outcomes in pattern \(r\).

Under a constant additive shift within each pattern, the adjusted imputed mean in that pattern is \(\bar{Y}_{r,\mathrm{MAR}}+\delta_r\). The resulting completed-data estimate of the overall sample mean is therefore:

\hat{\mu}(\boldsymbol{\delta}) = \frac{ n_0\bar{Y}_{0} + \sum_{r=1}^{R} n_r\bigl(\bar{Y}_{r,\mathrm{MAR}}+\delta_r\bigr) }{n}.

The estimate includes the observed outcomes and all missing outcomes, with each pattern contributing in proportion to its count. Equivalently, the pattern weight is \(w_r=n_r/n\), and the adjustment to the MAR-reference estimate is the weighted sum of the pattern deltas:

\hat{\mu}(\boldsymbol{\delta}) = \hat{\mu}(\boldsymbol{0}) + \sum_{r=1}^{R}w_r\delta_r, \qquad w_r=\frac{n_r}{n}.

This identity follows by substituting the adjusted pattern mean into the first formula. The observed-outcome term does not change; each missing-outcome pattern adds \(n_r\delta_r\) to the numerator. Dividing the total change by \(n\) gives \(\sum_r(n_r/n)\delta_r\). In particular, a large delta in a rare pattern may have a modest effect on the overall mean, while a smaller delta in a common pattern may matter more.

Assumptions and interpretation: The pattern definitions are fixed before comparing delta scenarios, every target-sample record is counted exactly once, and the MAR reference imputations are obtained from a model suitable for the observed data and intended estimand. The additive formula assumes the specified delta is a constant shift on the outcome scale within its pattern. The deltas represent sensitivity assumptions, not values estimated from the observed outcomes. The resulting estimate is conditional on those choices; it does not establish that a particular pattern-specific departure is true.

Worked calculations

Worked Example: Two Patterns and a Pattern-Weighted Mean

Hypothetical data. A sample contains 100 individuals. The outcome is observed for 50, with an observed mean of 12 points. It is missing in pattern A for 20 individuals and in pattern B for 30. For one imputed data set, the MAR-based imputed means are 10 points in A and 14 points in B. Consider additive deltas \(\delta_A=-2\) points and \(\delta_B=-5\) points.

First calculate the adjusted means within the missing-outcome patterns: pattern A has \(10+(-2)=8\) points, and pattern B has \(14+(-5)=9\) points. The overall estimate is:

\hat{\mu}(\boldsymbol{\delta}) = \frac{50(12)+20(8)+30(9)}{100} = \frac{600+160+270}{100} = 10.30\text{ points}.

For comparison, the MAR-reference estimate is \(\{50(12)+20(10)+30(14)\}/100=1{,}220/100=12.20\) points. The pattern-weighted change is \( (20/100)(-2)+(30/100)(-5)=-0.40-1.50=-1.90\) points, giving \(12.20-1.90=10.30\) points. The two calculations agree.

Worked Example: Opposing Pattern Shifts

Hypothetical data. In a sample of 120 people, the outcome is observed for 80, with mean 15 units. The remaining 40 are split equally between patterns A and B. Their MAR-based imputed means are 13 units in A and 20 units in B. Set \(\delta_A=-3\) units and \(\delta_B=+4\) units.

The adjusted pattern means are \(13-3=10\) units for A and \(20+4=24\) units for B. Hence:

\hat{\mu}(\boldsymbol{\delta}) = \frac{80(15)+20(10)+20(24)}{120} = \frac{1{,}200+200+480}{120} = 15.67\text{ units}.

Under MAR, the corresponding estimate is \(\{1{,}200+20(13)+20(20)\}/120=1{,}860/120=15.50\) units. The weighted delta contribution is \((20/120)(-3)+(20/120)(4)=-0.50+0.6667=0.1667\) units, so the adjusted estimate is \(15.50+0.1667=15.67\) units. The positive overall change does not mean every pattern moved upward: pattern A moved downward, but pattern B’s upward contribution was larger.

Worked Example: Combining Pattern-Specific Estimates Across Imputations

Hypothetical data. There are 80 individuals: 50 have an observed outcome with sum 400 points; 10 are missing in pattern A; and 20 are missing in pattern B. Use \(\delta_A=-1\) point and \(\delta_B=+2\) points. Three imputed data sets produce the following MAR-based pattern means: A has 10, 12, and 11 points; B has 6, 7, and 8 points.

Apply each pattern’s delta within each imputation. For imputation 1, the adjusted means are 9 points in A and 8 in B, so the completed-data estimate is \((400+10(9)+20(8))/80=650/80=8.125\) points. For imputation 2, the adjusted means are 11 and 9, giving \((400+110+180)/80=690/80=8.625\) points. For imputation 3, they are 10 and 10, giving \((400+100+200)/80=700/80=8.750\) points.

The multiple-imputation point estimate at this fixed delta scenario is the average of the three completed-data estimates:

\bar{Q}(\boldsymbol{\delta}) = \frac{8.125+8.625+8.750}{3} = 8.500\text{ points}.

The pattern shifts are applied separately in every imputed data set; they are not applied once to an average of the MAR means in a way that discards imputation-to-imputation variation. For interval estimation, pool the completed-data estimates and their variances using the multiple-imputation combining rules at this specified delta scenario. Changing the delta vector defines a different sensitivity scenario, for which the analysis and uncertainty calculation should be repeated.

Specifying patterns and checking the analysis

Patterns are defined by the missingness configuration, not by the values of the outcome that an analyst hopes to obtain. In a data set with several incomplete variables, the number of possible configurations can grow quickly. A useful analysis prespecifies a manageable set of substantively relevant patterns, states which records belong to each, and explains any grouping of configurations. The pattern counts and weights should be reported so the overall estimate can be understood and reproduced.

The MAR reference and pattern-specific shift must also be coherent. The reference imputation model should use observed information appropriate to the target analysis. A delta then changes the imputed distribution for the designated pattern according to the chosen sensitivity assumption; it does not repair a poorly specified MAR reference model. If chained equations are used, the shifted imputations should be incorporated in the intended way while maintaining valid values and a usable imputation procedure.

Approach	What differs across missing outcomes?	When it is useful
MAR reference	No delta departure is imposed.	Provides the reference scenario under the stated imputation model.
One shared delta	The same shift is applied to every pattern with a missing outcome.	Useful when a common departure is plausible or as a simpler sensitivity comparison.
Pattern-specific deltas	A prespecified delta is applied to each missingness pattern.	Useful when missingness configurations plausibly imply different departures from MAR.

A missingness pattern is not the same thing as a subgroup defined by a measured characteristic. For example, “outcome missing and covariate missing” describes a missingness configuration; “older patients” describes a group based on observed or defined characteristics. The distinction matters because the pattern weights are determined by the frequencies of the missingness configurations, and the analysis is intended to represent the target sample across those patterns.

Several diagnostics can reveal an implementation problem or an unhelpful sensitivity scenario:

Check the partition and weights. Confirm that the pattern counts sum to \(n\), and that the weights for missing-outcome patterns plus the observed-outcome share sum to 1. Do not weight only by the number of missing outcomes if the target is the full-sample mean.
Inspect pattern-specific distributions. Compare MAR and adjusted imputed means, ranges, and distributions for each pattern. A mean shift can create implausible values or substantial distortion even when the weighted overall estimate looks reasonable.
Look for sparse patterns. Very small pattern counts can produce unstable MAR-based imputations and poorly supported pattern-specific assumptions. Combining patterns may be necessary, but it changes the sensitivity analysis and should be justified.
Verify the sign and units. A delta of \(-3\) on a points scale lowers imputations by 3 points; it is not a three-percent change. A sign error can reverse the intended scenario.
Compare direct and weighted calculations. For a constant additive shift, calculating the completed-data mean from adjusted values should match the MAR estimate plus \(\sum_r w_r\delta_r\), up to rounding.

Limitations and a common mistake

Pattern-specific deltas make sensitivity assumptions more flexible, but flexibility has a cost. With many patterns there may be little information to estimate stable MAR imputations in each one, and the number of delta choices can become difficult to interpret. Delta values are not identified by observed data alone; their plausibility requires subject-matter reasoning and transparent scenario reporting. A pattern-specific adjustment also does not automatically represent every form of departure from MAR: a constant additive shift changes a pattern’s location but leaves the shape and spread of its imputed distribution unchanged.

A common mistake is to average the pattern-specific imputed means equally. In the first worked example, this would average 8 and 9 points to obtain 8.5 points for the missing outcomes, then combine that with the observed mean using an incorrect implicit weighting. The patterns contain 20 and 30 individuals, not equal numbers. The correction is to weight each pattern by its count in the target sample, as in \(n_r/n\), and to include the observed-outcome contribution when estimating the full-sample mean.

A second mistake is to interpret the adjusted estimate as evidence that the chosen deltas describe the unobserved outcomes. It is instead a result conditional on those deltas. Report the assumptions and show how conclusions change across the scenarios considered; do not combine distinct delta scenarios as though they were repeated imputations from one specified model.

Key takeaway: For a pattern-specific additive adjustment, the overall mean changes from its MAR-reference value by the sum of each missingness pattern’s delta multiplied by that pattern’s proportion of the full target sample. Define patterns clearly, apply each delta consistently, and interpret the result as conditional on the chosen sensitivity assumptions.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

A sample of 50 has 30 observed outcomes with mean 6 units. Pattern A has 10 missing outcomes with MAR imputed mean 7 and delta \(-1\); pattern B has 10 with MAR imputed mean 9 and delta \(+2\). Calculate the adjusted full-sample mean.
Why is it incorrect to average the two pattern-specific deltas equally when their pattern counts differ?
What should be checked if the weights for the missing-outcome patterns sum to 0.7, but the estimate is intended to describe the full target sample?
Three imputed data sets yield adjusted estimates of 4.2, 4.5, and 4.8 units for a fixed delta vector. What is their multiple-imputation point estimate?
Does a pattern-specific delta estimated from observed data establish the unobserved outcomes’ true departure from MAR? Explain.

Solutions. (1) The adjusted means are \(7-1=6\) and \(9+2=11\) units. Thus the full-sample mean is \(\{30(6)+10(6)+10(11)\}/50=(180+60+110)/50=7.00\) units. (2) The contribution of a delta to the full-sample mean is weighted by its pattern’s proportion \(n_r/n\); equal weighting would ignore the different pattern frequencies. (3) The observed-outcome share and all missing-pattern shares must together sum to 1. If the missing-pattern weights sum to 0.7, the remaining 0.3 should be accounted for by the observed-outcome pattern if the categories partition the sample; otherwise, check for omitted records, overlapping categories, or a different denominator. (4) The point estimate is \((4.2+4.5+4.8)/3=4.50\) units. (5) No. The delta is a sensitivity assumption, not identified by the observed outcomes; the estimate is conditional on the chosen delta and should be compared with other defensible scenarios.