Multiple Imputation

Multiple Imputation: Concept Architecture

Orientation and learning roadmap

This page follows incomplete patient data through the construction of plausible completed datasets, separate analyses, and pooled inference. It then works through the variance calculation, examines missing-data assumptions, and connects the method to costs and quality-adjusted life-years (QALYs).

Why several completed datasets are needed

Replacing every missing value with one best guess treats uncertain predictions as though they were observed with certainty. Multiple imputation generates several plausible versions of the missing values conditional on an imputation model and the recorded data, so the final inference can reflect uncertainty both within and between completed datasets.

The target remains the originally specified treatment effect, population mean, or other analysis estimand, rather than the imputed values themselves. The imputation process should preserve observed entries while filling only genuinely missing entries; each completed dataset is analysed using the intended substantive model. Differences between the completed-data estimates carry information about how missing values affect the result.

Design the imputation and analysis together

Record what is missing, by group and time, and identify variables that predict missingness or the missing values before selecting an imputation model. Include the treatment assignment, variables in the substantive analysis, relevant auxiliary predictors, and features such as nonlinear terms or interactions needed to preserve the analysis relationship.

For example, a trial with incomplete follow-up utilities and costs may include baseline utility, disease severity, prior utilisation, treatment arm, and observed follow-up measurements when appropriate to the target estimand. Imputation methods should respect plausible ranges, skewed costs, structural zeroes, and the dependence of repeated observations; a single linear normal model is not automatically suitable for every variable. Do not impute values for outcomes that are undefined after death as if they were merely unobserved; first define the estimand and what each outcome means.

Generate, analyse, and pool

An imputation model generates $M$ completed datasets using uncertainty in its parameters and predicted missing values. The analysis model is fitted independently to each completed dataset, producing an estimate $Q_m$ and its estimated variance $U_m$ for the same scalar target on the same scale.

The pooled point estimate $\overline Q$, mean within-imputation variance $\overline U$, between-imputation variance $B$, and total variance $T$ under Rubin's standard scalar rules are

$$\begin{aligned} \overline Q&=\frac{1}{M}\sum_{m=1}^{M}Q_m,\\ \overline U&=\frac{1}{M}\sum_{m=1}^{M}U_m,\\ B&=\frac{1}{M-1}\sum_{m=1}^{M}(Q_m-\overline Q)^2,\\ T&=\overline U+\left(1+\frac{1}{M}\right)B. \end{aligned}$$

The pooled standard error is $\sqrt T$, with an appropriate degrees-of-freedom method used for interval estimates and tests. The factor $1+1/M$ accounts for the finite number of imputations; more imputations reduce simulation noise in the pooled result but cannot correct an implausible imputation model. For an odds ratio or another nonlinear effect, pooling is commonly performed on an appropriate transformed scale, such as log odds, before transforming back.

Worked example: treatment-effect uncertainty

Suppose three illustrative completed datasets yield treatment-effect estimates of $0.10$, $0.14$, and $0.12$ QALYs with estimated variance $0.0025$ QALYs squared in each dataset. Three imputations are used solely for transparent arithmetic and would usually be too few for a contemporary substantive analysis with appreciable missing information.

The mean estimate is $\overline Q=0.12$ QALYs and the mean within-imputation variance is $\overline U=0.0025$. The between-imputation variance is $B=[(0.10-0.12)^2+(0.14-0.12)^2+0]/2=0.0004$, giving $T=0.0025+(1+1/3)(0.0004)\approx0.0030333$ and $\sqrt T\approx0.0551$ QALYs. Reporting only $\sqrt{\overline U}=0.05$ would omit the extra uncertainty represented by the variation across imputed datasets.

Missingness assumptions and sensitivity

Standard multiple imputation is often justified under a missing-at-random (MAR) assumption: conditional on the included observed information, the probability that a value is missing does not further depend on its unobserved value. The imputation and analysis models also need to be compatible enough for the target estimate and uncertainty calculations to be credible.

MAR cannot generally be established from the observed values alone. If missingness plausibly depends on an unobserved cost or health state after conditioning on available data, a missing-not-at-random (MNAR) sensitivity analysis can shift or otherwise alter the distribution assigned to missing outcomes. Compare the downstream treatment effect and decision results across defensible assumptions, and continue efforts to collect outcomes after treatment changes whenever those outcomes remain meaningful.

Checks for health economic use

Inspect observed versus imputed distributions by treatment group, severity, and follow-up time, along with the share of missing information and stability across more imputations. Check that the imputation respects deterministic relationships, such as the way QALYs are constructed from survival and utility over time, rather than imputing incompatible derived and component variables independently.

Jointly missing costs and utilities can be associated, so the modelling strategy should preserve decision-relevant relationships where possible. Report the proportion and pattern of missing data, imputation variables and methods, number of imputations, pooling scale, diagnostics, and MNAR sensitivity assumptions. A narrow pooled confidence interval does not prove that unmeasured reasons for dropout or a misspecified model are harmless.

Source basis and further learning

The foundational text establishes repeated imputation and pooled uncertainty; the methodological paper addresses chained equations in practical analyses. Clinical-trial guidance emphasises preventing missing outcomes and testing the robustness of conclusions to assumptions that the data cannot verify.

• Rubin (1987), Multiple Imputation for Nonresponse in Surveys. This develops the repeated-imputation and pooling framework.
• White, Royston, and Wood (2011), “Multiple Imputation Using Chained Equations: Issues and Guidance for Practice”, Statistics in Medicine, 30. This discusses specification and diagnostics for chained equations.
• National Research Council (2010), The Prevention and Treatment of Missing Data in Clinical Trials. This supplies trial-specific prevention and sensitivity principles.