Concept Architecture
Concept
Theoretically, Delta Adjustment is a sensitivity analysis method used in missing data analysis to assess the robustness of statistical conclusions to departures from the Missing At Random (MAR) assumption. It is founded on pattern-mixture modelling, whereby the distribution of missing values is systematically shifted by a specified quantity (�) to represent plausible Missing Not At Random (MNAR) scenarios. The concept exists because the true values of missing observations cannot be verified and therefore require sensitivity analyses based on clinically or empirically justified assumptions.
Mathematically, delta adjustment modifies imputed values by adding or subtracting a prespecified constant or treatment-specific offset following multiple imputation under MAR. The adjusted imputations are then analysed using standard statistical methods, and results are combined using Rubin's rules. By varying � over a plausible range, investigators evaluate the sensitivity of treatment effect estimates to increasingly optimistic or pessimistic assumptions regarding missing outcomes.
In practice, delta adjustment is implemented after multiple imputation by modifying imputed values before model fitting. Separate delta values may be applied to different treatment groups or missing-data patterns depending on the assumed mechanism. In health economics, delta adjustment is widely used in sensitivity analyses of clinical trial outcomes, quality-of-life measures and cost data to evaluate the robustness of cost-effectiveness conclusions when missing data may be informative.
Purpose
Used to perform sensitivity analyses for missing data by systematically adjusting imputed values to evaluate the impact of departures from the Missing At Random assumption on statistical and economic conclusions.
Mathematical Formulae
Primary Formula
Adjusted imputed value:
Y* = Y? + �
where:
- Y? = imputed value under MAR
- � = sensitivity parameter
- Y* = adjusted imputed value
Supporting Formulae
Treatment-specific adjustment:
Y*T = Y?T + �T
Y*C = Y?C + �C
Incremental treatment effect:
? = ?*T ? ?*C
Rubin's pooled estimate:
Q? = (1/m) ? ????? Q?
Total variance:
T = W + (1 + 1/m)B
where:
- W = within-imputation variance
- B = between-imputation variance
Related Mathematical Methods
- Multiple Imputation
- Pattern-Mixture Models
- Missing Not At Random (MNAR) Analysis
- Rubin's Rules
- Sensitivity Analysis
- Reference-Based Imputation
Example
A quality-of-life score is imputed as 0.78 under the MAR assumption.
A sensitivity analysis assumes that patients with missing observations would have outcomes 0.05 units worse than predicted.
Adjusted value:
Y* = 0.78 + (?0.05)
Y* = 0.73
Repeating the economic evaluation for � values ranging from ?0.10 to +0.10 allows assessment of whether the incremental cost-effectiveness ratio remains robust to plausible MNAR assumptions.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUM | =B2+$E$1 | Applies the specified delta adjustment to each imputed value. |
| AVERAGE | =AVERAGE(C2:C501) | Calculates the adjusted mean outcome after delta adjustment. |
| DATA TABLE | =TABLE(,E1) | Performs one-way sensitivity analysis across alternative delta values. |
| IF | =IF(ABS(E1)>0.10,"Extreme assumption","Plausible assumption") | Flags implausible sensitivity assumptions. |
VBA (Optional)
Automate repeated delta-adjustment sensitivity analyses across a user-defined range of � values and summarise their effects on treatment estimates and cost-effectiveness results.
Sources
- Carpenter JR, Kenward MG. Multiple Imputation and its Application.
- National Research Council. The Prevention and Treatment of Missing Data in Clinical Trials.
- Little RJA. Pattern-Mixture Models for Multivariate Incomplete Data.
- Rubin DB. Multiple Imputation for Nonresponse in Surveys.
- NICE. Health Technology Evaluation Manual.
- ISPOR Good Practice Reports on Missing Data.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is delta adjustment?
A sensitivity analysis technique shifting the assumed outcome for patients with missing trial data by a specified amount to test conclusions' robustness.
Source: Carpenter & Kenward 2013
How does delta adjustment probe the effect of missing trial data?
Delta adjustment probes how much missing data could change a trial's conclusion by deliberately shifting the assumed outcomes of patients with missing values by a set amount, the delta, usually in a direction unfavourable to the treatment. By repeating the analysis under progressively harsher assumptions, it shows how far the missing outcomes would have to differ from the observed ones before the result overturned. This tests the robustness of the finding to what cannot be seen. Stress-testing conclusions against missing data is its purpose. Little and Rubin (2002) discuss such approaches.
Source: Little & Rubin 2002
Why is delta adjustment used?
Delta adjustment is used because the primary analysis of missing data usually relies on the missing at random assumption, which cannot be verified, so it is important to examine whether conclusions hold if that assumption is wrong, for example if patients with missing data had worse outcomes. Delta adjustment provides a structured way to test this. So delta adjustment is used to assess the robustness of trial conclusions to unverifiable assumptions about the missing data, since if the results remain favourable even when missing outcomes are assumed worse by a plausible amount, confidence is strengthened, whereas if they change readily, the conclusions are shown to depend heavily on the missing data assumption.
Source: Carpenter & Kenward 2013
How does delta adjustment work?
Delta adjustment works by imputing the missing outcomes, often under the missing at random assumption, and then shifting the imputed values by a specified amount, the delta, usually to make them less favourable to the treatment, before analysing the adjusted data. The size of delta can be varied to explore a range of departures. So delta adjustment works by systematically worsening the imputed missing outcomes by a chosen amount and repeating the analysis, which shows how the results change as the missing data are assumed to depart increasingly from the primary assumption, allowing the point at which conclusions would change to be identified.
Source: Carpenter & Kenward 2013
What does delta adjustment reveal?
Delta adjustment reveals how sensitive a trial's conclusions are to assumptions about the missing data: if the treatment effect remains significant and worthwhile even after plausibly unfavourable shifts to the missing outcomes, the conclusions are robust, whereas if a modest delta overturns them, the conclusions are fragile and depend heavily on the missing data being as the primary analysis assumed. So delta adjustment reveals the robustness or fragility of results to departures from the missing at random assumption, indicating how much the missing outcomes would have to differ to change the conclusions, which informs how much confidence to place in the primary analysis.
Source: Carpenter & Kenward 2013
How does delta adjustment relate to missing data assumptions?
Delta adjustment relates to missing data assumptions by directly probing them: the primary analysis typically assumes data are missing at random, and delta adjustment tests what happens under a missing not at random scenario in which those with missing data have systematically different, usually worse, outcomes. So delta adjustment is a way of examining the consequences of the missing at random assumption being false, exploring missing not at random possibilities in a controlled manner, which is valuable because the assumption cannot be checked from the data, and demonstrating that conclusions hold under reasonable departures strengthens their credibility.
Source: Carpenter & Kenward 2013
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 15 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/delta-adjustment
- Term code
- HE-ES-SA-045
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