Multiple Imputation: A Practical Guide

Introduction

Missing data are almost unavoidable in clinical research. Patients may miss visits, withdraw from treatment, discontinue because of adverse events, fail to complete questionnaires, or have laboratory measurements that are unavailable at particular time points.

The statistical problem is not simply that some cells in the analysis dataset are blank. Missingness can change the population represented by the observed data and can introduce bias if the mechanism generating the missing values is related to the outcome, treatment, prognosis, or other variables.

Multiple imputation provides a general framework for handling missing data by replacing each missing value with several plausible values, creating multiple completed datasets, analyzing each dataset separately, and then combining the resulting estimates.

Why Not Simply Delete Patients With Missing Data?

The simplest approach to missing data is often complete-case analysis: retain only subjects whose required variables are completely observed.

Complete-case analysis can be reasonable under specific assumptions, but it can also discard substantial information.

Suppose a randomized clinical trial enrolls 500 patients and the primary endpoint is missing for 20% of them. A complete-case analysis may analyze only 400 patients. If the probability of having an observed outcome depends on baseline prognosis or treatment discontinuation, the 400 observed patients may no longer provide an unbiased representation of the randomized population for the estimand of interest.

There is also an efficiency cost. Baseline variables that remain fully observed may contain useful information about the missing outcomes, but a complete-case analysis may fail to use that information.

The Three Missing-Data Mechanisms

A central part of any missing-data analysis is understanding why observations are missing. The conventional framework distinguishes three mechanisms: MCAR, MAR, and MNAR.

Missing Completely at Random — MCAR

Data are missing completely at random when the probability that an observation is missing does not depend on either observed or unobserved data. Conceptually:

where \(R\) denotes the missingness indicator, \(Y_{\mathrm{obs}}\) represents observed data, and \(Y_{\mathrm{mis}}\) represents missing data.

MCAR is a strong assumption. For example, a laboratory sample might be lost because a freezer malfunctioned at random with respect to patient characteristics and laboratory values.

Missing at Random — MAR

Under MAR, the probability of missingness may depend on information that has been observed, but after conditioning on the observed information it does not depend on the missing value itself. The conceptual condition is:

For example, suppose patients with poorer baseline disease severity are more likely to discontinue. If baseline disease severity is observed, an imputation model can use that information when predicting missing outcomes.

Missing Not at Random — MNAR

MNAR refers to situations in which missingness remains related to the unobserved value even after conditioning on observed information. Conceptually:

For example, patients who discontinue treatment because their unobserved future outcome would have been particularly poor may generate a missingness mechanism that cannot be explained completely by observed covariates.

Standard multiple imputation is often developed under an MAR assumption. When MNAR is clinically plausible, a primary MAR analysis may still be used, but it should generally be accompanied by an appropriate sensitivity analysis.

What Multiple Imputation Actually Does

Suppose a dataset contains \(n\) subjects and some observations are missing. Multiple imputation creates \(M\) versions of the completed dataset.

The resulting inference incorporates both the ordinary sampling uncertainty within each completed dataset and the additional uncertainty caused by the fact that the missing observations were not actually observed.

Why One Imputation Is Not Enough

Suppose a patient's missing outcome could plausibly be 20, 25, or 30. A single imputation chooses one value and then treats it as though it had actually been observed. That can make the completed dataset look more certain than it really is.

Multiple imputation instead might produce several completed datasets:

The estimates from these analyses will differ slightly because the missing values differ. That variation is part of the information used to calculate the final standard error.

The Basic Mathematical Framework

Let \(M\) denote the number of imputed datasets. For imputed dataset \(m\), let:

be the estimated parameter of interest, and let:

be its estimated variance. The pooled point estimate is simply the average of the \(M\) estimates:

The within-imputation variance is:

The between-imputation variance is:

The total variance is:

The pooled standard error is therefore:

Rubin's Rules in Plain Language

Rubin's rules can be understood as three simple operations:

1. Average the estimates.
2. Average the within-dataset variances.
3. Add an inflation term based on how much the estimates vary across the imputed datasets.

If the estimates from the different imputations are nearly identical, the between-imputation variance \(B\) will be small. If the estimates differ substantially, \(B\) will be larger and the final standard error will increase.

This is exactly what we want: more uncertainty about the missing data should produce more uncertainty in the final inference.

A Worked Numerical Example

Suppose a clinical trial uses multiple imputation with \(M=5\) completed datasets. A regression coefficient for treatment is estimated as follows:

The pooled estimate is:

The average within-imputation variance is approximately:

The between-imputation variance is:

The total variance is therefore:

and the pooled standard error is approximately:

The important point is not the particular numerical result. It is the structure: the final uncertainty contains information from both the within-imputation analysis and the variation across imputations.

Multiple Imputation by Chained Equations — MICE

One of the most widely used approaches to multiple imputation is multiple imputation by chained equations, commonly abbreviated as MICE.

The basic idea is to specify a conditional model for each variable that contains missing values. For example, suppose a dataset contains:

• Age
• Treatment
• Baseline disease severity
• Week 4 outcome
• Week 8 outcome

If Week 4 outcome is partially missing, it can be modeled using the other variables. If Week 8 outcome is also missing, it can be modeled using other available variables, potentially including the imputed or observed Week 4 outcome.

The process cycles through the variables repeatedly.

Choosing the Imputation Model

The imputation model should respect the type of variable being imputed.

Predictive Mean Matching

Predictive mean matching, or PMM, is particularly useful for continuous clinical variables that are not well represented by a normal distribution.

Instead of generating an arbitrary value directly from a regression equation, PMM identifies observed subjects whose predicted values are similar to the predicted value of the subject with missing data. One of those observed values is then used as the imputed value.

This has an important practical advantage: the imputed value comes from the observed distribution.

For example, if a laboratory measurement is always positive and strongly right-skewed, PMM is less likely to generate implausible negative values than a simple normal linear-regression imputation.

How Many Imputations Should Be Used?

There is no universal number of imputations that is correct for every study. Historically, relatively small values such as 5 or 10 were common. Modern analyses often use substantially more imputations, particularly when the fraction of missing information is substantial.

The number \(M\) should be large enough that Monte Carlo error from the imputation process is negligible relative to the precision required for the analysis.

A practical strategy is to use more imputations when:

• The proportion of missing information is high.
• The sample size is large.
• Very precise confidence intervals are required.
• The analysis is being used for an important confirmatory endpoint.
• The imputation procedure is computationally inexpensive enough to support a larger \(M\).

What Variables Should Enter the Imputation Model?

A common mistake is to include only the variables that appear in the final analysis model. The imputation model should generally contain variables that help predict:

• The incomplete variable itself
• The probability that the variable is missing
• The primary outcome
• Treatment assignment
• Important baseline prognostic factors
• Previous and subsequent measurements in longitudinal studies

In clinical trials, the treatment indicator is particularly important because the relationship between treatment and the incomplete outcome should be preserved.

Auxiliary variables can also improve the plausibility and efficiency of the imputation model when they are strongly related to missingness or to the incomplete outcome.

Should the Outcome Be Included in the Imputation Model?

Generally, if the outcome is observed for some subjects and missing for others, the observed outcome information should be used to help predict other missing variables when appropriate.

For example, suppose the primary analysis is a treatment comparison adjusted for baseline disease severity. If baseline severity has missing values, the observed primary outcome can contain useful information for imputing those baseline values.

The imputation model should be compatible with the planned analysis and should avoid throwing away observed information simply because the variable is considered an "outcome."

Imputation and Randomized Clinical Trials

Randomization provides a powerful design feature, but it does not eliminate the missing-data problem. Suppose a trial randomizes patients to Treatment A or Treatment B and the primary outcome is missing for some patients. The imputation strategy should generally preserve the randomized treatment groups rather than ignoring treatment assignment.

One practical approach is to include treatment as a predictor in the imputation model. In some settings, separate imputation models by treatment group may be appropriate, particularly when the relationship between covariates and outcomes differs substantially between treatment arms.

The precise strategy should be driven by the estimand and the assumed missing-data mechanism rather than by an arbitrary software default.

Longitudinal Clinical-Trial Data

Multiple imputation becomes particularly useful when repeated measurements are collected. Consider:

The observed history can provide valuable information for predicting missing measurements. For example, Week 4 may be strongly predictive of Week 8, while Week 8 may help predict Week 12.

A longitudinal imputation model should therefore account for the temporal structure of the data rather than treating all observations as unrelated cross-sectional variables.

Passive and Derived Variables

Clinical datasets frequently contain variables that are mathematically derived from other variables. Examples include:

• Change from baseline
• Body-mass index
• Percent change
• Composite scores
• Log-transformed measurements
• Treatment-emergent indicators

A common mistake is to impute both the underlying variables and the derived variables independently. That can create internally inconsistent datasets.

For example, if:

then it is usually preferable to impute the underlying quantities and derive \(\Delta Y\) afterward, rather than independently imputing all three variables.

Respecting Bounds and Clinical Plausibility

An imputation model can produce statistically plausible but clinically impossible values if it is poorly specified. Examples include:

• Negative laboratory values that cannot be negative
• Probabilities below 0 or above 1
• Impossible dates
• Negative event counts
• Measurements outside physically possible ranges

This is one reason predictive mean matching and appropriately specified categorical models are useful.

After imputation, investigators should inspect distributions and ranges of the completed variables.

Diagnostics After Imputation

Multiple imputation should not be treated as a black box. Diagnostics should be performed before the final inferential model is accepted.

Compare Observed and Imputed Distributions

For a partially observed continuous variable, compare the distribution of observed values with the distribution of imputed values. The distributions do not need to be identical, but major discrepancies can signal a poorly specified model.

Check Extreme Values

Look for imputed values that are outside clinically plausible ranges or that occur much more frequently than expected.

Check Relationships

Examine whether important relationships between variables are preserved. For example, if higher baseline disease severity strongly predicts a worse outcome in the observed data, the completed data should not imply a completely different relationship without a substantive reason.

Check Convergence

For iterative imputation algorithms such as MICE, inspect traces of summary statistics across iterations. The chains should generally mix and stabilize rather than showing persistent trends.

Missingness Patterns Should Be Examined First

Before selecting an imputation method, characterize the missingness. Useful summaries include:

• Percentage missing for each variable
• Percentage missing by treatment group
• Percentage missing by visit
• Patterns of monotone versus intermittent missingness
• Reasons for missingness
• Withdrawal and discontinuation patterns
• Relationships between missingness and observed covariates

A missingness table might look like:

A pattern such as increasing missingness over time and greater missingness in one treatment group deserves investigation. It does not automatically prove that the data are MNAR, but it provides important evidence about the missing-data process.

Multiple Imputation Does Not Prove MAR

This point is extremely important. A software program can generate multiple imputations under an MAR model, but the software cannot establish that MAR is true.

MAR is fundamentally an assumption about the missing-data mechanism. The credibility of that assumption comes from clinical knowledge, study conduct, observed variables, reasons for dropout, previous outcomes, and subject-matter understanding.

MNAR Sensitivity Analysis

If the primary analysis assumes MAR, the analysis should be stress-tested when departures from MAR are scientifically plausible. One common strategy is a delta adjustment. Suppose the MAR-imputed value for a missing outcome is \(Y_{\mathrm{MAR}}\). An MNAR sensitivity analysis may shift the imputed value by a prespecified amount \(\delta\):

For example, investigators might examine increasingly unfavorable assumptions for treatment discontinuers.

The purpose is not to find the "correct" delta. The purpose is to determine whether the substantive conclusion remains stable under scientifically credible departures from the primary assumption.

Pattern-Mixture and Selection-Model Perspectives

More formal MNAR sensitivity analyses can be constructed using pattern-mixture models or selection models. A pattern-mixture approach conditions the distribution of outcomes on patterns of missingness. A selection model instead factors the joint distribution into an outcome model and a missingness model.

These frameworks can be useful when the clinical question requires explicit modeling of departures from MAR. For many applied analyses, however, a transparent sensitivity parameter such as a delta adjustment can provide a more interpretable way to communicate the robustness of the conclusion.

A Clinical-Trial Worked Example

Consider a randomized Phase III study comparing Treatment A with placebo. The primary endpoint is change from baseline in a continuous disease-severity score at Week 24. Suppose:

• 600 patients are randomized.
• 300 receive Treatment A.
• 300 receive placebo.
• Baseline severity is fully observed.
• Week 24 outcome is missing for 90 patients.
• Some missing outcomes occur after treatment discontinuation.

The primary estimand is defined before the missing-data analysis is selected. The investigators decide that the primary analysis should estimate the treatment effect under a specified hypothetical or treatment-policy framework, depending on the scientific objective.

A multiple-imputation strategy is then developed that includes:

• Treatment assignment
• Baseline severity
• Earlier post-baseline measurements
• Relevant demographic variables
• Prognostic clinical variables
• Variables associated with dropout
• Other strongly predictive auxiliary variables

The missing Week 24 outcomes are imputed repeatedly. Each completed dataset is analyzed using the prespecified treatment-effect model. The treatment estimates are then pooled using Rubin's rules.

Example Analysis Model

Suppose the primary analysis is an ANCOVA-style model:

where:

• \(Y_{24}\) is the Week 24 outcome.
• \(Y_0\) is baseline severity.
• \(\beta_1\) represents the adjusted treatment effect.

The same analysis model is fitted separately to each imputed dataset. The final treatment effect is then the pooled estimate \(\bar{\beta}_1\).

Example Pooled Results

This hypothetical example illustrates how sensitivity analysis can be more informative than reporting a single missing-data method. The conclusion may remain statistically significant across a reasonable range of assumptions, but the estimated effect becomes progressively smaller as more unfavorable assumptions are imposed.

R Example Using MICE

In R, the mice package is commonly used to implement multiple imputation by chained equations. A simple example for a continuous outcome using predictive mean matching is:

library(mice)

imp <- mice(
  data,
  m = 50,
  method = "pmm",
  seed = 2026
)

fit <- with(
  imp,
  lm(week24 ~ treatment + baseline)
)

pooled <- pool(fit)

summary(pooled, conf.int = TRUE)

Here:

• m = 50 creates 50 imputed datasets.
• method = "pmm" requests predictive mean matching for variables to which that method is assigned.
• with() fits the analysis model separately to each completed dataset.
• pool() combines the estimates using the multiple-imputation pooling framework.

In a real analysis, the imputation methods should be specified variable by variable rather than blindly assigning the same method to every incomplete variable.

R Example With Variable-Specific Methods

library(mice)

methods <- make.method(data)

methods["baseline"] <- "pmm"
methods["week4"]    <- "pmm"
methods["week8"]    <- "pmm"
methods["responder"] <- "logreg"

imp <- mice(
  data,
  m = 50,
  method = methods,
  seed = 2026
)

fit <- with(
  imp,
  glm(responder ~ treatment + baseline,
      family = binomial)
)

pooled <- pool(fit)

summary(pooled, conf.int = TRUE)

The exact method choices depend on the variable definitions and the analysis plan. The example illustrates the principle that different variables can require different imputation models.

Checking MICE Convergence in R

plot(imp)

densityplot(imp)

stripplot(imp, pch = 20, cex = 0.7)

Trace plots can be used to assess whether iterative chains have stabilized. Distribution plots can be used to compare observed and imputed values. These graphical checks are an important part of a reproducible multiple- imputation workflow.

SAS Example Using PROC MI

SAS provides multiple-imputation procedures through PROC MI and corresponding analysis and pooling procedures. For example:

proc mi data=analysis
        out=mi_data
        nimpute=50
        seed=2026;
   class treatment;
   fcs
      regpmm(week24)
      logistic(responder);
   var treatment baseline week4 week8 week24 responder;
run;

The completed datasets can then be analyzed with PROC GLM, PROC GENMOD, PROC MIXED, or another appropriate analysis procedure. The resulting parameter estimates and covariance information can then be combined using PROC MIANALYZE.

proc mianalyze data=estimates;
   modeleffects treatment_effect;
   stderr treatment_se;
run;

Do Not Pool Raw Data

A common conceptual mistake is to concatenate the \(M\) imputed datasets and analyze the resulting dataset as though it contained \(M\) times as many patients. That is incorrect.

If there are 500 patients and 50 imputations, there are still 500 patients, not 25,000 independent patients. The 50 completed datasets represent alternative plausible versions of the same 500-person study.

Multiple Imputation Versus Last Observation Carried Forward

Last observation carried forward, or LOCF, was historically used frequently in longitudinal clinical trials. For example, if a patient's Week 12 measurement is missing, the Week 8 measurement might be copied forward.

The problem is that LOCF makes a very strong assumption about the patient's future trajectory. A patient who discontinued treatment at Week 8 might have improved, worsened, or remained stable at Week 12. Simply copying the Week 8 measurement does not represent that uncertainty.

Multiple Imputation Is Not a Universal Cure

Multiple imputation can be extremely useful, but it does not automatically solve every missing-data problem. It can produce misleading inference if:

• The imputation model is badly misspecified.
• Important predictors of missingness are omitted.
• The temporal structure is ignored.
• Variable types are modeled incorrectly.
• Impossible values are generated.
• The MAR assumption is implausible and no sensitivity analysis is performed.
• The analysis model and imputation model are incompatible.

The phrase "we used multiple imputation" therefore does not by itself establish that a missing-data analysis is appropriate. The quality of the imputation model matters.

Congeniality and Compatibility

The imputation model and substantive analysis should be compatible. For example, if the final analysis contains important interactions or nonlinear relationships, an imputation model that ignores those relationships may fail to preserve them.

Suppose the final model contains:

If \(X\) is incomplete, the imputation model should generally account for the relationship between \(T\), \(X\), and the outcome rather than treating the interaction structure as irrelevant.

Missing Data and the Estimand

In modern clinical-trial practice, the missing-data strategy should be linked to the estimand. The same observed dropout pattern can require different analyses depending on the scientific question.

For example, investigators may ask:

• What would the treatment effect be if patients remained on treatment?
• What would the treatment effect be under a treatment-policy strategy that allows discontinuation?
• What would the effect be under a hypothetical scenario in which treatment discontinuation did not occur?

These are different questions. The imputation strategy should support the chosen estimand rather than being selected first and justified afterward.

Intercurrent Events

Treatment discontinuation, rescue medication, treatment switching, death, and other intercurrent events can make missing-data analysis particularly complex. For example, a Week 24 outcome may be unavailable because a patient:

• Discontinued study treatment
• Started rescue therapy
• Was lost to follow-up
• Died
• Withdrew consent

These situations should not necessarily be treated as equivalent. The clinical meaning of the missing value depends on why it is missing and on the estimand being targeted.

Common Mistakes

1. Using mean imputation as the primary method. Mean imputation generally distorts variance and relationships between variables.
2. Using one deterministic imputation. A single replacement fails to represent uncertainty about the missing values.
3. Ignoring treatment assignment. In randomized trials, the imputation model should generally preserve the role of treatment assignment.
4. Using the same imputation model for every variable. Continuous, binary, categorical, and count variables often require different models.
5. Ignoring auxiliary variables. Strong predictors of missingness or the incomplete outcome can improve the imputation model.
6. Imputing derived variables independently. This can produce internally inconsistent datasets.
7. Failing to inspect imputed values. Impossible or implausible imputed values can reveal model problems.
8. Assuming MAR has been proven. MAR is an assumption, not a property established by the imputation software.
9. Reporting only the MAR analysis. When departures from MAR are plausible, sensitivity analyses are important.
10. Treating imputed datasets as independent. The \(M\) datasets are alternative versions of the same study, not independent samples.
11. Choosing the number of imputations arbitrarily. The number should be sufficient for the amount of missing information and the precision required.
12. Ignoring the estimand. The missing-data strategy should answer the scientific question defined by the estimand.

A Practical Multiple-Imputation Workflow

What Should Be Prespecified?

For a confirmatory clinical trial, the statistical documentation should describe the missing-data strategy before the final outcome analysis. At minimum, specify:

• Primary estimand
• Primary analysis model
• Variables subject to imputation
• Expected missing-data patterns
• Variables included in the imputation model
• Imputation method for each variable type
• Number of imputations
• Number of iterations, where applicable
• Treatment-group handling
• Rules for derived variables
• Diagnostics
• Pooling procedure
• MNAR sensitivity analyses
• Rules for intercurrent events
• Handling of deaths or other structural missingness

How to Report a Multiple-Imputation Analysis

A concise but reproducible report should explain more than simply saying "missing data were handled using multiple imputation."

A useful description might state that:

The exact wording should of course match the actual statistical analysis. The purpose is to make the analysis reproducible and to communicate the assumptions clearly.

Multiple Imputation Versus Mixed Models

Multiple imputation and likelihood-based mixed models are not necessarily competing methods. For longitudinal continuous outcomes, a mixed model using all available outcomes may provide valid inference under an appropriate MAR assumption without explicitly imputing individual missing observations.

Multiple imputation may still be useful when:

• Multiple variables have missing values.
• Covariates are incomplete.
• The analysis requires a completed dataset.
• Derived endpoints need to be constructed.
• Sensitivity analyses require explicit imputed values.
• The substantive analysis is not naturally handled by a likelihood-based model.

The choice should be driven by the estimand, data structure, assumptions, and analysis model rather than by the simple rule that one method is always superior.

Multiple Imputation and Survival Analysis

Survival-analysis datasets create additional challenges because missingness can affect event indicators, event times, covariates, or longitudinal predictors. The event time itself should not generally be imputed using a simple continuous-outcome method without considering censoring and the survival-data structure.

For example, if a baseline covariate is missing in a Cox regression, multiple imputation of that covariate may be relatively straightforward. If the event process itself is incompletely observed, however, the missingness problem may require a substantially more specialized approach.

Multiple Imputation for Binary Clinical Endpoints

Suppose the primary endpoint is response versus nonresponse. A binary outcome can be imputed using an appropriate logistic model when the outcome is missing.

The imputation model may include treatment, baseline characteristics, prior measurements, and other predictors of response. The resulting completed datasets retain a binary response rather than generating continuous probabilities as patient-level outcomes.

For example:

methods <- make.method(data)

methods["response"] <- "logreg"

imp <- mice(
  data,
  m = 50,
  method = methods,
  seed = 2026
)

The final analysis might then use a logistic regression, risk difference, risk-ratio model, or another prespecified analysis appropriate to the estimand.

Relative Efficiency and Fraction of Missing Information

The impact of missing data can be summarized using the fraction of missing information. A commonly used conceptual quantity is:

where \(B\) is the between-imputation variance and \(T\) is the total variance. A larger value indicates that a greater fraction of the uncertainty in the estimate is attributable to missing information.

This provides another reason to use more imputations when the missing information is substantial: more imputations reduce Monte Carlo error in the representation of that uncertainty.

Interpreting the Final Confidence Interval

The confidence interval after multiple imputation incorporates uncertainty from both the statistical analysis and the missing-data imputation. For a pooled estimate \(\bar\theta\), the basic form is:

where \(T\) is the pooled variance and \(\nu\) represents an appropriate degrees-of-freedom calculation.

The exact small-sample degrees-of-freedom method can vary by implementation, so the software's pooling method and settings should be documented when they matter to the analysis.

What Makes a Good Imputation Model?

A strong imputation model should satisfy several principles.

The Most Important Concept

The most important conceptual point is that multiple imputation is an inferential framework, not merely a method for filling blank cells.

The objective is not to discover the one "best guess" for every missing value. The objective is to represent the uncertainty about those values in a way that produces valid inference under explicit assumptions.

A complete multiple-imputation analysis therefore has four connected components:

• A missing-data model that describes why values are absent.
• An imputation model that generates plausible values.
• A substantive analysis model that addresses the scientific question.
• A sensitivity-analysis strategy that examines departures from the primary missing-data assumptions.

If any of these components is poorly designed, simply increasing the number of imputations will not rescue the analysis.

Quick Reference

References

Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys. John Wiley & Sons.
Little, R.J.A. & Rubin, D.B. (2019). Statistical Analysis with Missing Data. 3rd ed. Wiley.
van Buuren, S. (2018). Flexible Imputation of Missing Data. 2nd ed. Chapman & Hall/CRC.
White, I.R., Royston, P. & Wood, A.M. (2011). Multiple imputation using chained equations: Issues and guidance for practice. Statistics in Medicine, 30, 377–399.
Carpenter, J.R. & Kenward, M.G. (2013). Multiple Imputation and its Application. Wiley.
National Research Council (2010). The Prevention and Treatment of Missing Data in Clinical Trials. National Academies Press.
European Medicines Agency. Guideline on Missing Data in Confirmatory Clinical Trials.