Concept Architecture
Concept
Theoretically, Multiple Imputation is a statistical method for handling missing data by replacing each missing value with multiple plausible values drawn from an appropriate predictive distribution. Instead of producing a single completed dataset, the method generates several completed datasets that reflect uncertainty about the missing values. Statistical analyses are performed separately on each dataset, and the results are combined to obtain valid estimates and standard errors. Multiple Imputation is founded on Bayesian principles and Rubin's theory of missing data and is valid under the Missing at Random assumption when the imputation model is correctly specified.
Mathematically, Multiple Imputation consists of three sequential stages: imputation, analysis and pooling. Missing values are repeatedly sampled from their posterior predictive distributions to create multiple complete datasets. Each completed dataset is analysed using the statistical model of interest, and the resulting parameter estimates are combined using Rubin's Rules, which account for both within-imputation and between-imputation variability. This approach produces unbiased parameter estimates and appropriately reflects uncertainty arising from missing data.
In practice, Multiple Imputation is widely used in clinical trials, observational studies and health economic evaluations involving incomplete cost, utility and clinical outcome data. In health economics, it is routinely applied to missing healthcare costs, quality-adjusted life years, resource utilisation and patient-reported outcomes. The method is recommended by methodological guidance because it preserves statistical efficiency while accounting for uncertainty associated with missing values.
Purpose
Used to handle incomplete datasets by generating multiple plausible values for missing observations, reducing bias and producing statistically valid inference that incorporates uncertainty due to missing data.
Mathematical Formulae
Primary Formula
Rubin's pooled estimate:
?? = (1/m) ? ????? ???
where:
- ?? = pooled parameter estimate
- ??? = estimate from imputed dataset i
- m = number of imputations
Supporting Formulae
Within-imputation variance:
W = (1/m) ? ????? U?
Between-imputation variance:
B = (1/(m ? 1)) ? ????? (??? ? ??)�
Total variance:
T = W + (1 + 1/m)B
Relative increase in variance:
r = [(1 + 1/m)B] / W
Related Mathematical Methods
- Rubin's Rules
- Missing at Random
- Markov Chain Monte Carlo
- Fully Conditional Specification
- Multivariate Imputation by Chained Equations
- Maximum Likelihood Estimation
- Sensitivity Analysis
Example
A health economist analyses healthcare costs for 1,200 patients, with 12% of cost observations missing.
Five imputed datasets are created.
Estimated incremental costs are:
- �1,180
- �1,220
- �1,200
- �1,210
- �1,190
The pooled estimate is:
?? = (1,180 + 1,220 + 1,200 + 1,210 + 1,190) � 5
?? = �1,200
The within-imputation and between-imputation variances are subsequently combined using Rubin's Rules to obtain the final standard error and confidence interval for the incremental cost estimate.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| AVERAGE | =AVERAGE(B2:F2) | Calculate pooled parameter estimates across imputed datasets. |
| VAR.S | =VAR.S(B2:F2) | Estimate between-imputation variance. |
| SUM | =SUM(B2:F2) | Combine estimates before pooling. |
| COUNT | =COUNT(B2:F2) | Determine the number of imputations. |
| Solver | Estimate imputation model parameters by maximising predictive accuracy. | Support simplified multiple imputation workflows. |
VBA (Optional)
A VBA routine can automate pooling of estimates across multiple imputed datasets using Rubin's Rules and generate combined parameter estimates, standard errors and confidence intervals.
Sources
- Rubin DB. Multiple Imputation for Nonresponse in Surveys. Wiley.
- Little RJA, Rubin DB. Statistical Analysis with Missing Data. Wiley.
- Carpenter JR, Kenward MG. Multiple Imputation and Its Application. Wiley.
- van Buuren S. Flexible Imputation of Missing Data. Chapman & Hall/CRC.
- Faria R, Gomes M, Epstein D, White IR. A Guide to Handling Missing Data in Cost-Effectiveness Analysis Conducted within Randomised Controlled Trials. Pharmacoeconomics. 2014;32(12):1157?1170.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is multiple imputation?
A missing data technique generating several plausible complete datasets with different estimated replacement values, then combining separate analyses of each.
Source: Rubin 1987
How does multiple imputation reflect uncertainty about missing values?
Multiple imputation fills the gaps in a dataset several times over, each time drawing different plausible values for the missing entries, to produce a handful of complete datasets that differ where data were absent. Each is analysed separately and the results are combined, so the variation between them adds the uncertainty about what was missing into the final answer. This is what makes it better than single imputation, which inserts one guess and pretends it is certain. Building in the doubt about the missing values is its strength. Little and Rubin (2002) describe this technique.
Source: Little & Rubin 2002
How does multiple imputation work?
Multiple imputation works in three stages: imputation, creating several completed datasets by drawing plausible values for the missing data from a model based on the observed data; analysis, applying the intended analysis to each completed dataset; and pooling, combining the results using rules that account for the variation between the imputed datasets. So multiple imputation works by imputing, analysing, and pooling, with the variation across the multiple imputations capturing the uncertainty about the missing values, which is why the combined standard errors are larger than those from a single imputation and properly reflect the missing data, giving valid inference under the missing at random assumption.
Source: Rubin 1987
Why is multiple imputation better than single imputation?
Multiple imputation is better than single imputation because single imputation fills each missing value with one estimate, treating it as known and understating the uncertainty, which gives standard errors that are too small and over-confident inference, whereas multiple imputation creates several plausible values and combines the analyses, propagating the uncertainty. So multiple imputation is preferred for reflecting the uncertainty inherent in imputing missing data, giving valid standard errors and confidence intervals, unlike single imputation, which can bias estimates and falsely inflate precision, which is why multiple imputation is the standard principled approach when data are missing at random and the uncertainty must be properly conveyed.
Source: Rubin 1987
What assumptions does multiple imputation rely on?
Multiple imputation relies chiefly on the missing at random assumption, that the probability of missingness depends only on observed data, under which the observed variables can be used to impute the missing values validly; and on the imputation model being adequately specified, including the relevant variables. So multiple imputation depends on missing at random and a well-specified imputation model, and its validity rests on these, which is why the assumption is considered and the imputation model built carefully to include variables predictive of missingness and the outcomes, and why, because missing at random cannot be verified, sensitivity analyses examine robustness to departures from it.
Source: Rubin 1987
How many imputations are needed in multiple imputation?
The number of imputations needed in multiple imputation depends on the amount of missing data, with more missingness requiring more imputations for stable results; historically a small number was suggested, but more imputations, such as tens, are now often recommended to improve the precision and stability of the estimates. So the number of imputations is chosen to give stable, efficient results, increasing with the fraction of missing information, since too few imputations add variability to the estimates, which is why more imputations are used when much data are missing, balancing computational effort against the precision of the pooled results.
Source: Rubin 1987
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 19 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/multiple-imputation
- Term code
- HE-ES-SA-133
Stable URI · Machine-readable · Resolvable · CC BY 4.0