Missing Data in Research: MCAR, MAR, MNAR and How to Choose an Analysis Strategy

First: What Are You Trying to Estimate or Predict?

Missing-data decisions should follow the research question rather than precede it.

Before choosing complete case analysis, multiple imputation, or a likelihood-based method, identify:

• the outcome and predictors required for the analysis;
• whether the goal is estimation, hypothesis testing, or prediction;
• the population or sample to which the result is intended to apply;
• whether observations are independent, clustered, or repeatedly measured;
• which variables are incomplete; and
• whether missingness occurs once, intermittently, or through longitudinal dropout.

This matters because an observation can be complete for one analysis but incomplete for another. Complete-case analysis also does not merely remove blanks: it defines a subset of observations on which the model is fitted.

For prediction research, missing predictor values are especially important because excluding people who lack any one required predictor can rapidly reduce the development sample. Steyerberg describes complete-case analysis as inefficient because it discards information from subjects who have some, but not all, predictor measurements (Steyerberg, 2019).

MCAR, MAR and MNAR: What Do They Actually Mean?

MCAR, MAR and MNAR describe assumptions about the relationship between missingness and the data. They should not be treated simply as three labels that can be read directly from a missingness table.

MCAR: Missing Completely at Random

Under missing completely at random (MCAR), missingness does not depend on the observed or missing data relevant to the missingness process. Conceptually, the complete observations behave like a random subset with respect to the variables involved.

MCAR is therefore a strong condition. It is important because many simple deletion or available-data procedures require MCAR for generally valid inference. Little and Rubin note that when complete cases are not effectively a random sample of the full observations, complete-case estimates can lose both precision and validity for relevant estimands (Little & Rubin, 2020).

MAR: Missing at Random

Under missing at random (MAR), missingness may depend on information that has been observed, but conditional on the observed information it does not additionally depend on the values that remain missing.

In a longitudinal setting, for example, dropout may depend on responses recorded before dropout while satisfying MAR if, conditional on the relevant observed history, dropout does not additionally depend on subsequent unobserved responses. Fitzmaurice et al. distinguish this situation from MCAR and discuss its implications for longitudinal analysis (Fitzmaurice et al., 2011).

MNAR: Missing Not at Random

Missing not at random (MNAR) applies when the missingness process retains dependence on missing values after accounting for the relevant observed information. In longitudinal research, for example, dropout can be MNAR when it depends on the unobserved response even after conditioning on observed response history and other relevant measured variables.

MNAR creates a fundamental difficulty: the dependence involves information that, by definition, is not observed. Analyses therefore require additional modeling assumptions, and sensitivity analyses become particularly important (Little & Rubin, 2020; Fitzmaurice et al., 2011).

A practical distinction

MAR Is an Assumption, Not Something You Prove From the Missingness Table

One of the most important misconceptions about MCAR MAR MNAR is the idea that researchers can inspect their observed dataset and determine conclusively that the data are MAR.

They generally cannot.

Observed information can reveal useful facts. Researchers can determine which variables have missing values, examine patterns of missingness, and compare observed characteristics of complete and incomplete observations. Such comparisons may provide evidence against a simple MCAR explanation. Little and Rubin explicitly note, however, that these comparisons offer no direct evidence establishing the weaker MAR assumption (Little & Rubin, 2020).

The distinction between MAR and MNAR concerns how missingness relates to values that are not observed. Consequently, finding that missingness is associated with observed variables does not prove that there is no additional dependence on unobserved values.

Why Simply Deleting Incomplete Rows Can Change Your Answer

Complete-case analysis—also called listwise deletion—restricts the analysis to observations for which every required variable is observed.

Its attractions are obvious: it is simple, standard complete-data procedures can be applied, and every reported analysis based on those variables uses the same subset. But Little and Rubin identify two major consequences of discarding incomplete observations: loss of precision and potential bias, with the severity depending on the missingness mechanism, missingness pattern, differences between complete and incomplete observations, and the estimand being targeted (Little & Rubin, 2020).

Deleting rows can change an answer through several pathways.

1. It discards observed information. Someone missing one covariate may nevertheless have a recorded outcome and nearly every other predictor. Complete-case analysis throws all of that information away for that analysis.

2. It can change the effective analysis population. If becoming a complete case is related to variables relevant to the outcome or model, the retained subset can differ systematically from the original sample.

3. It usually reduces precision. Fewer observations contribute to estimation.

4. The effect can accumulate across variables. In a multivariable model, modest missingness in several different predictors can leave substantially fewer complete observations than any one variable-specific missing percentage suggests. Steyerberg highlights this problem in prediction modeling and emphasizes the inefficiency of discarding subjects who retain useful predictor information (Steyerberg, 2019).

Complete-Case Versus Available-Case Analysis

Available-case approaches can preserve more observed information than complete-case deletion, but they do not automatically solve the missing-data problem. Different quantities may be estimated from different subsets, and available-case covariance calculations can even produce problematic covariance structures. Little and Rubin do not generally recommend available-case analysis as a universal alternative to principled likelihood or imputation approaches (Little & Rubin, 2020).

The lesson is not that available-case analysis is always wrong. It is that using every available value is not, by itself, a statistical justification.

Single Imputation Versus Multiple Imputation

Imputation fills in missing values using information in the observed data. But not all imputation strategies treat uncertainty correctly.

Single imputation

Single-imputation procedures create one completed dataset.

Examples discussed in the approved sources include mean imputation, conditional regression imputation, and stochastic regression imputation. These approaches differ in sophistication, but once a single value replaces a missing observation, ordinary analysis of the completed dataset can easily treat that imputed value as though it had actually been observed.

Little and Rubin stress that imputation must not create the illusion that the data are genuinely complete. Missing values can be predicted using observed variables, but the uncertainty associated with those predictions remains part of the statistical problem (Little & Rubin, 2020).

Steyerberg likewise notes that simple imputation can underestimate variability, while regression-based methods can use correlations among predictors more effectively (Steyerberg, 2019).

Multiple imputation

With multiple imputation (MI), missing values are replaced multiple times using draws from an imputation model, producing multiple completed datasets. The substantive analysis is performed in each dataset, and the results are combined.

The crucial point is that multiple imputation does not simply produce several guesses and average them. Its inferential advantage is that variability between imputations contributes to uncertainty alongside the ordinary sampling variability estimated within imputations (Little & Rubin, 2020).

Steyerberg makes the same distinction in prediction modeling: variation across completed datasets represents uncertainty about the missing values, and combined variance includes both within- and between-imputation components. That between-imputation component is a fundamental difference between multiple and single imputation (Steyerberg, 2019).

Multiple imputation does not remove assumptions

MI should not be described as an assumption-free repair for missing data.

Its validity depends on the imputation model, the variables used to predict missing information, the relationships represented by that model, and the missingness assumptions under which the procedure is justified.

Missing Predictors and Missing Outcomes Are Not the Same Problem

The location of missingness matters.

Thus “missing data” should not be treated as a single homogeneous problem. Ask what is missing and what role that variable plays in the analysis.

When Likelihood-Based Methods Are Appropriate

Multiple imputation is not the only principled approach.

Likelihood-based methods can often use the observed components of incomplete records directly without first constructing a single filled-in dataset. Under appropriate ignorability conditions and correctly specified response models, likelihood-based inference can accommodate MAR missingness.

This is particularly important in longitudinal analysis. Fitzmaurice et al. explain that likelihood-based models can use partial response trajectories under MAR, whereas ordinary complete-case or available-data procedures generally require stronger conditions for valid inference. The advantage comes with a modeling requirement: the joint response model, including relevant mean and within-subject dependence structures, must be appropriately specified (Fitzmaurice et al., 2011).

Accordingly, a likelihood-based mixed model may be a natural strategy for incomplete repeated outcomes, while multiple imputation may be convenient when missingness affects several predictors and other analysis variables.

Longitudinal Dropout Needs Special Attention

Dropout creates a structured form of missingness because later measurements become unavailable after a participant leaves follow-up.

Suppose dropout is related to earlier observed outcomes. Such missingness is not naturally MCAR. It may nevertheless be compatible with MAR if, conditional on the observed history and relevant measured variables, dropout has no additional dependence on the later outcomes that become unobserved.

This distinction has direct consequences for analysis.

Fitzmaurice et al. note that complete-case analysis discards all measurements from people who later drop out, even measurements collected legitimately before dropout. Available-data methods preserve more of that information but are not automatically valid merely because they use all observed visits. Under MAR that is not MCAR, appropriately specified likelihood-based longitudinal models can provide valid inference under conditions that do not justify ordinary complete-case or standard available-data procedures (Fitzmaurice et al., 2011).

A Practical Workflow for Handling Missing Data in Research

1. Identify the missingness pattern

Begin descriptively.

Determine:

• how much information is missing for each analysis variable;
• how many observations have at least one missing analysis value;
• whether variables tend to be missing together;
• whether repeated measurements show intermittent gaps or monotone dropout; and
• how many observations a complete-case analysis would actually retain.

Do not reduce this description to one overall percentage.

2. Identify which variables are missing

Separate missing:

• outcomes;
• primary exposures or predictors;
• adjustment variables;
• auxiliary variables;
• repeated outcomes; and
• variables required only for secondary analyses.

The statistical consequences depend on the role of the missing variable.

3. Consider a plausible missingness mechanism

Ask what scientific or data-collection processes could have generated the missingness.

Then distinguish between what is observed and what must be assumed.

Observed associations between missingness and measured variables may make a simple MCAR explanation less plausible. They do not establish MAR over MNAR because the decisive distinction involves unobserved information (Little & Rubin, 2020).

4. Define the analysis target

Clarify the estimand or prediction goal before choosing the missing-data procedure.

For explanatory research, ask what population quantity the coefficient, mean difference, odds ratio, or trajectory contrast is intended to estimate.

For prediction, ask whether the task is model development, validation, or prediction for future individuals and whether missing predictors will also occur at the intended point of use.

5. Choose a method that matches the target and assumptions

6. Conduct diagnostics and sensitivity analysis

After fitting the primary analysis:

• examine whether the imputation or likelihood model adequately represents important relationships in the observed data;
• compare the effective sample and estimates with transparent alternatives where useful;
• inspect whether conclusions depend strongly on the treatment of missingness; and
• when MAR is scientifically uncertain and plausible MNAR departures could change the result, perform sensitivity analyses.

The purpose of sensitivity analysis is not to discover the “true” missing values. It is to determine whether the substantive conclusion survives reasonable alternative assumptions.

When Should You Consider Sensitivity Analysis?

Sensitivity analysis deserves particular attention when:

• substantial outcome data are missing;
• dropout is plausibly related to unobserved outcomes;
• the reason for missingness suggests that MAR may be questionable;
• results are close to a decision or inferential boundary;
• different reasonable missing-data methods give materially different estimates;
• a primary scientific conclusion depends strongly on an unverifiable missingness assumption; or
• the stakes of the inference make robustness to missing-data assumptions important.

Little and Rubin devote explicit attention to sensitivity analyses for MNAR models, including repeated-measures settings and multiple-imputation applications, reflecting the fact that missingness assumptions can be central to inference rather than a minor technical detail (Little & Rubin, 2020).

What Should Researchers Report?

A reproducible missing-data section should make clear what happened between data collection and final inference.

Report, where relevant:

• missingness by analysis variable;
• important missingness patterns;
• the number of observations with any missing analysis data;
• the number entering complete-case analyses;
• whether missingness occurred in predictors, outcomes, or both;
• longitudinal dropout patterns;
• the missing-data method used;
• the assumptions under which that method is interpreted;
• variables and structure used in an imputation model;
• how multiple-imputation estimates were combined;
• the likelihood model used for incomplete longitudinal outcomes;
• sensitivity analyses to alternative assumptions; and
• whether substantive conclusions changed across approaches.

Common Mistakes

Bottom Line: How to Handle Missing Data

There is no single best method for all missing data in research.

The defensible strategy is to begin with the research target and data structure, not with a software default.

Complete case analysis may sometimes be adequate, but it should be chosen rather than inherited automatically. Multiple imputation can preserve information and account for uncertainty from imputed values when its modeling assumptions are appropriate. Likelihood-based methods can be especially natural for incomplete longitudinal outcomes under suitable MAR assumptions. When conclusions depend materially on assumptions that distinguish MAR from MNAR, sensitivity analysis should make that dependence visible.

The central principle is simple: missing values are part of the statistical model and the scientific interpretation—not merely cells to delete before the “real” analysis begins.

FAQs

References

Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2011). Applied longitudinal analysis (2nd ed.). Wiley.

Harrell, F. E., Jr. (2015). Regression modeling strategies: With applications to linear models, logistic and ordinal regression, and survival analysis (2nd ed.). Springer.

Little, R. J. A., & Rubin, D. B. (2020). Statistical analysis with missing data (3rd ed.). Wiley.

Steyerberg, E. W. (2019). Clinical prediction models: A practical approach to development, validation, and updating (2nd ed.). Springer.