Missing at Random

Missing at Random: Concept Architecture

Orientation and learning roadmap

The page begins with the conditional missingness assumption and a patient follow-up example. It then compares missing-data mechanisms, explains what analyses can rely on the assumption, and finishes with a numerical illustration and sensitivity checks for departures from it.

What the assumption actually says

The word random can be misleading: missingness may be strongly associated with characteristics recorded in the dataset. MAR permits such associations if the observed information accounts for them. What it rules out is an additional dependence on the unseen value after that observed information is taken into account.

Let $Y$ be a value that may be missing, $X$ be relevant observed information, and $R=1$ indicate that $Y$ was observed. A simple conditional expression of MAR for this setting is:

$$P(R=1\mid Y,X)=P(R=1\mid X).$$

The condition is specific to the variables and missingness patterns in an analysis. For general multivariable data, MAR means that the probability of the observed missingness pattern can be expressed using the observed portions of the data, without dependence on the missing portions. Including an observed predictor in the dataset is not enough if an analysis omits it from the model that is meant to make MAR plausible.

A patient follow-up example

Suppose a study measures quality-of-life scores at baseline and six months. People with worse recorded baseline scores are less likely to return the six-month questionnaire. The follow-up data can be MAR conditional on baseline score and other relevant recorded characteristics if, among people with the same observed characteristics, return does not additionally depend on their unseen six-month score.

If patients experiencing a particularly poor six-month outcome are less likely to return even after conditioning on the recorded baseline and other available predictors, the missing outcomes may be missing not at random (MNAR). The observed records alone generally cannot establish which story is true. Knowledge of the follow-up process and a sensitivity analysis therefore matter as much as a statistical model.

How MAR differs from related mechanisms

The three labels describe assumptions about the missingness mechanism, rather than a visual pattern of blank cells. They are relative to the information observed and the particular variables being analyzed. A low percentage missing does not itself imply MAR, and a high percentage does not prove MNAR.

These are properties of a proposed data-generating process, not labels that can be read directly from a missingness table. Comparing observed characteristics of respondents and nonrespondents can reveal predictors that a MAR analysis should include, but cannot generally test away a dependence on an outcome that is never observed for nonrespondents.

What MAR permits in an analysis

Under MAR and appropriate additional assumptions, observed data can support likelihood-based estimation or multiple imputation without explicitly modeling the missingness process. MAR by itself does not guarantee unbiased results. The outcome or imputation model must be suitably specified, relevant predictors must be available and included, and there must be sufficient information across the observed data to estimate the relationships used to fill or integrate over missing values.

Multiple imputation repeatedly draws plausible missing values from an estimated predictive distribution, analyzes each completed dataset, and combines estimates while accounting for between-imputation variation. A mixed model for repeated measurements may also use incomplete outcome histories under its own model assumptions. Neither method recovers information that was never measured without relying on assumptions; a single mean fill-in or last observation carried forward does not automatically implement MAR and can understate uncertainty.

Complete-case analysis deserves a separate assessment. Even when MAR holds, discarding incomplete records can lose precision and can bias the particular estimate if selection into complete cases depends on observed variables related to the outcome and the analysis does not adequately account for that selection. Its validity depends on the estimand, analysis model, and missingness structure, rather than solely on the MAR label.

A small numerical illustration

Consider 100 patients in two recorded baseline-health groups. Of 50 patients with better baseline health, 40 return a follow-up questionnaire; of 50 with worse baseline health, 20 return. The observed response probabilities are $40/50=0.80$ and $20/50=0.40$. Response is therefore related to an observed characteristic, so MCAR is not a credible description of this example.

MAR could still hold if return depends only on baseline health and other modeled observed information, not on the unseen follow-up score within each group. If the returning patients have mean follow-up scores of 80 and 60 in the better and worse baseline groups, respectively, an illustrative standardized mean using the original equal group sizes is $(80+60)/2=70$, compared with the respondent-only mean $(40\times80+20\times60)/60\approx73.33$. The difference shows how unequal response across observed groups can alter an estimate. The standardized value is defensible as an estimate of the full cohort mean only if the relevant within-group missingness and measurement assumptions hold; this example does not prove MAR.

How to make a MAR analysis credible

The missing-data plan should begin with the target estimand and the reasons observations are missing. In a health economic evaluation, missing costs, resource use, utilities, and clinical outcomes can affect incremental estimates differently. An analyst should show the missingness by treatment arm and time point, explain the variables used to support the MAR assumption, and carry imputation uncertainty into the economic results.

1. Define the target analysis. State the population, outcome or model input, follow-up period, and estimand before choosing a missing-data method.
2. Describe the observed missingness. Report counts and proportions by variable, arm, time, and clinically relevant observed characteristics, including known reasons for loss to follow-up.
3. Identify predictive observed information. Include plausible predictors of the missing value and its missingness in an appropriate imputation or outcome model, while respecting timing and avoiding inappropriate post-outcome information.
4. Fit and inspect the model. Check distributions, relationships, convergence where relevant, plausibility of imputed values, and compatibility with the substantive analysis.
5. Test departures from MAR. Use justified MNAR scenarios, such as a stated shift in unobserved outcomes, and show whether the clinical or economic decision changes.

For a simple delta adjustment sensitivity analysis, one may add a prespecified shift $\delta$ to imputed missing outcomes to represent a departure from MAR. Repeating the analysis across plausible values of $\delta$ reveals the point at which a conclusion changes. The direction and range of those shifts should be justified using clinical knowledge or external evidence rather than chosen after seeing the desired conclusion.

How to interpret the conclusion

MAR is an assumption that can become more plausible when informative observed predictors and reasons for missingness are available. It cannot usually be verified from observed data alone because the needed comparison involves values that are missing. A clear report identifies the conditioning information, model, missingness extent, and sensitivity scenarios, then states how dependent the decision is on those choices.

Evidence used for this explanation

The original missing-data framework and applied guidance below support the conditional interpretation and its analytical implications. These sources also explain why sensitivity analysis remains necessary when the unseen values cannot be checked directly.

• Rubin DB. Inference and missing data. Biometrika (1976);63:581–592.
• Sterne JAC et al. Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls. BMJ (2009);338:b2393.
• CONSORT 2025 explanation and elaboration, especially the guidance on missing outcome data and sensitivity analyses.