1 Definition and framework
Missing not at random is a missing-data mechanism in which the chance that a value is absent depends on the value itself, even after other recorded variables have been taken into account. In practice, this means the missingness carries information about the unobserved data, making routine analyses that ignore the mechanism potentially misleading. MNAR is therefore treated as a substantive modeling problem rather than a purely technical nuisance.
1.1 Basic concept
The central idea is that the process generating missing values is linked to the unseen measurement. For example, respondents with very low incomes may be less likely to disclose their earnings, or patients with worsening symptoms may be more likely to leave a study. Because the unrecorded values influence the probability of being missing, the analyst cannot usually rely on the observed data alone to reconstruct the full distribution.
1.2 Comparison with other missing-data mechanisms
Missing-data mechanisms are often grouped into three broad categories. The distinctions matter because they determine whether the missingness can be ignored, modeled directly, or handled through additional assumptions. MNAR is the most demanding case because the missing values themselves help drive the absence pattern.
1.2.1 Missing completely at random
Missing completely at random describes situations in which the probability of missingness does not depend on either observed or unobserved data. Under this idealized mechanism, the missing subset behaves like a random sample of the full data. Analyses are simplest in this setting, since standard methods remain valid without special correction for the missingness process.
1.2.2 Missing at random
Missing at random means that, once the observed data are taken into account, the chance of missingness does not depend on the unobserved value itself. This mechanism is less restrictive than missing completely at random but still permits many standard modeling approaches. It is often assumed in multiple imputation and likelihood-based analyses, though the assumption is not directly verifiable from the observed data alone.
1.2.3 Missing not at random
Missing not at random refers to cases in which the missingness depends on the unseen value even after conditioning on observed information. This dependence makes the problem intrinsically harder, because the missing data mechanism and the data distribution are entangled. As a result, identification typically requires stronger structural assumptions, external information, or sensitivity analysis.
1.3 Statistical notation and terminology
Let the full data be separated into observed values and missing values, with a missingness indicator showing whether each entry is recorded. In common notation, the observed data are denoted by the available responses and the pattern of missingness, while the unobserved values are treated as latent quantities. Terms such as nonignorable missingness, informative missingness, and MNAR are often used in closely related ways, although their precise scope can vary by author.
2 Identification and assumptions
Identifying an MNAR model means distinguishing the underlying data distribution from the process that hides part of it. This is difficult because the same observed pattern can often be explained by multiple combinations of outcome behavior and missingness behavior. Consequently, analysts frequently rely on parametric assumptions, auxiliary variables, or prior information to obtain estimable models.
2.1 Nonignorable missingness
Nonignorable missingness is a common synonym for MNAR and emphasizes that the missingness process cannot be ignored in the analysis. If the mechanism is nonignorable, methods designed for MAR data may produce biased estimates. The term highlights the practical consequence that the absence itself must be modeled or examined through robustness checks.
2.2 Selection models
Selection models specify a joint distribution for the outcome and the missingness indicator by factoring the likelihood into a model for the data and a model for the probability of being missing. The missingness component is allowed to depend on the unobserved outcome. These models are flexible, but their conclusions can be sensitive to assumptions about the functional form and parameter values.
2.3 Pattern-mixture models
Pattern-mixture models organize the data according to observed missingness patterns and model the distribution of outcomes within each pattern. Rather than modeling the missingness directly, they describe how the data differ across patterns. This approach is useful for sensitivity analysis because it makes explicit how assumptions about unobserved subgroups affect inference.
2.4 Shared-parameter models
Shared-parameter models link the outcome process and the missingness process through latent variables or random effects. They are especially useful in longitudinal settings where dropout may be connected to an unobserved subject-specific trait. By introducing a common latent structure, these models can capture dependence between the outcome and the dropout mechanism in a compact way.
3 Causes and examples
MNAR arises when the reasons for missingness are tied to the hidden value itself. This occurs in many real settings, especially where answers are sensitive, health status is changing, or participation becomes burdensome. The pattern is often subtle, and the presence of missing data may reflect genuine differences rather than random loss.
3.1 Survey nonresponse
In surveys, people with extreme opinions, lower incomes, or less stable living situations may be less likely to answer certain questions. If the missingness is related to the unreported response after accounting for observed characteristics, the data are MNAR. Such cases are common in questionnaires involving private, embarrassing, or complex information.
3.2 Clinical dropout
In clinical research, participants may discontinue treatment or stop follow-up because their condition worsens, improves, or becomes difficult to tolerate. When dropout depends on the unobserved health outcome, later measurements are no longer representative of the original cohort. This can distort estimated treatment effects if the dropout process is treated as unrelated to the outcome.
3.3 Income and sensitive-item missingness
Questions about income, debt, substance use, or sexual behavior often have nonresponse tied to the value being asked about. Higher, lower, or otherwise sensitive values may be systematically underreported or omitted. In such situations, the missingness is informative and cannot be assumed to be random without supporting evidence.
3.4 Longitudinal data attrition
Attrition in longitudinal studies occurs when participants leave a study over time. If departure is related to their unobserved trajectory, such as declining health or decreasing motivation, the missingness mechanism becomes MNAR. Attrition of this kind can create a selective sample of those who remain, altering the apparent trend over time.
4 Modeling approaches
Modeling MNAR data usually requires an explicit description of both the outcome process and the missingness mechanism. Different approaches represent this dependence in different ways, and each has its own strengths. The choice often depends on study design, available auxiliary variables, and the substantive question being asked.
4.1 Likelihood-based methods
Likelihood-based methods build the statistical model for the observed data under a joint formulation of the outcome and missingness process. They are attractive because they can be efficient when correctly specified. However, the missingness component must be modeled carefully, since the observed likelihood may not uniquely determine the parameters.
4.1.1 Full maximum likelihood
Full maximum likelihood estimates all unknown parameters by maximizing the likelihood of the observed data under the chosen MNAR model. It can be implemented in specialized software and may yield efficient estimates under correct specification. Its performance depends heavily on the validity of the assumed model, especially the structure of the missingness equation.
4.1.2 Expectation-maximization methods
Expectation-maximization methods iteratively estimate parameters by alternating between imputing latent quantities and updating the model. They are useful when direct maximization is difficult because of incomplete data or latent structure. In MNAR settings, the algorithm may require careful initialization and can converge to local optima if the model is complex.
4.2 Bayesian methods
Bayesian methods combine the data with prior distributions for the model parameters and, when needed, for unidentified components. They are well suited to MNAR problems because prior information can help stabilize inference when the data alone are insufficient. The resulting posterior distribution can also support transparent sensitivity analysis.
4.2.1 Prior specification
Prior specification involves choosing distributions for parameters governing the outcome, the missingness process, and any latent quantities. In MNAR settings, priors may have a stronger influence than in better-identified models, so their choice should be deliberate and well documented. Informative priors can encode expert knowledge, previous studies, or plausible bounds.
4.2.2 Sensitivity priors
Sensitivity priors are alternative prior choices used to examine how conclusions change under different assumptions about the missingness process. They are especially valuable when certain parameters are weakly identified or not identified at all. Comparing posterior results across several priors can reveal which conclusions are robust and which are assumption-dependent.
4.3 Multiple imputation under MNAR
Multiple imputation under MNAR fills in missing values using a model that allows the imputed distribution to differ from the observed-data distribution. Unlike standard imputation under MAR, these methods incorporate extra parameters or shifts to reflect nonignorable missingness. The resulting datasets are analyzed separately and then combined to reflect uncertainty about the missing values.
4.4 Heckman-type selection correction
Heckman-type selection correction methods model the process that determines whether a value is observed and adjust the outcome equation accordingly. They are widely known in econometrics and related fields, especially for sample selection problems. These methods can correct bias under specific assumptions, though results depend on identifying restrictions and model form.
5 Estimation and inference
Inference under MNAR is more complicated than under MAR because the mechanism generating missingness must be estimated or assumed. Even large datasets may leave important parameters weakly identified. As a result, reported results often need to emphasize model dependence and uncertainty rather than a single definitive estimate.
5.1 Parameter identifiability
Parameter identifiability refers to whether the model parameters can be recovered uniquely from the observed data. In MNAR settings, some parameters may be only partially identified or not identified without auxiliary information. When identifiability is weak, estimates can be unstable and highly sensitive to small changes in assumptions.
5.2 Bias and inconsistency
If MNAR data are analyzed as though they were MAR or MCAR, estimates may be biased. The bias can persist even in large samples, producing inconsistent results that do not converge to the true value. This is one reason MNAR is considered a serious threat to valid inference in observational and follow-up studies.
5.3 Standard errors and uncertainty quantification
Standard errors under MNAR must reflect both sampling variability and uncertainty about the missingness model. When parameters are weakly identified, conventional formulas may understate the true uncertainty. Bootstrap methods, Bayesian credible intervals, or model-based variance calculations are often used, but each depends on the assumed structure.
5.4 Sensitivity analysis
Sensitivity analysis examines how conclusions change when the assumptions about missingness are varied. Because MNAR mechanisms are rarely verifiable from the observed data alone, sensitivity analysis is often an essential part of reporting. It helps distinguish findings that are stable from those that depend on a narrow set of assumptions.
5.4.1 Delta-adjustment approaches
Delta-adjustment approaches shift the imputed or modeled values by a specified amount to represent systematic differences between observed and missing cases. The adjustment parameter encodes how much the unobserved data are assumed to deviate from the MAR-based estimate. By varying this parameter, analysts can assess the range of plausible results.
5.4.2 Tipping-point analysis
Tipping-point analysis identifies the point at which a conclusion changes under progressively stronger departures from MAR. It is often used to show how extreme the missingness behavior would need to be to alter a substantive finding. This provides a clear summary of robustness without claiming to determine the true mechanism.
6 Diagnostics and model assessment
Because MNAR assumptions are not directly observable, model assessment relies on indirect checks, auxiliary data, and comparison of fitted patterns. Diagnostics are less definitive than in fully observed settings, but they remain important for judging whether a model is reasonable. Careful assessment can reveal implausible assumptions or overly rigid specifications.
6.1 Assessing missing-data mechanisms
Assessing the mechanism involves studying whether missingness is associated with observed covariates, prior outcomes, or external indicators. Patterns in response rates can sometimes suggest likely sources of nonresponse. However, because the unobserved values are unavailable, such assessments can only provide indirect evidence rather than proof.
6.2 Model checking
Model checking examines whether the fitted model reproduces key features of the observed data and the missingness pattern. Residual analysis, posterior predictive checks, and comparison of observed versus predicted distributions are common tools. If the model fails to match the available data well, its MNAR conclusions are unlikely to be reliable.
6.3 Comparing competing assumptions
Comparing competing assumptions involves fitting several plausible MNAR specifications and contrasting their implications. This may include alternative selection functions, different pattern-mixture restrictions, or several priors for unidentified parameters. The goal is not to prove one model correct, but to understand the range of results supported by the data and assumptions.
7 Applications
MNAR methods are used in many fields where missingness is informative and cannot be safely ignored. The details differ by discipline, but the underlying challenge is the same: the absence of data is related to the quantity being studied. This makes careful modeling especially important in applied work.
7.1 Social science surveys
In social science surveys, nonresponse can depend on income, attitudes, family situation, or other sensitive characteristics. MNAR methods help adjust for the fact that certain groups may be underrepresented among respondents. These models are often used with auxiliary demographic variables to reduce, though not eliminate, uncertainty about the missing responses.
7.2 Epidemiology and biostatistics
In epidemiology and biostatistics, MNAR arises in treatment studies, symptom tracking, and disease progression research. Dropout and intermittent missingness may be linked to health outcomes, medication side effects, or severity of illness. Accounting for these patterns can change estimates of risk, progression, or treatment benefit.
7.3 Economics and labor studies
Economic surveys and labor-market studies frequently encounter nonresponse in earnings, employment status, and hours worked. If the likelihood of missingness depends on the unreported economic variable, standard estimators may misstate averages or inequalities. Selection models and related corrections are therefore common in empirical economics.
7.4 Psychology and education research
In psychology and education, missing test scores, survey items, or follow-up measures may depend on performance, motivation, or well-being. Students with lower achievement or participants experiencing distress may be more likely to skip items or leave studies. MNAR analysis helps address the selective nature of these missing observations.
8 Limitations and practical considerations
MNAR methods are useful, but they do not remove uncertainty about what was not observed. Results can depend strongly on assumptions that cannot be fully tested, so practical analysis requires caution. Good study design and transparent reporting are often as important as the statistical model itself.
8.1 When MNAR assumptions are plausible
MNAR assumptions are most plausible when there is a clear substantive reason for missingness to depend on the unobserved value. Examples include sensitive questions, health-related dropout, and situations where extreme outcomes discourage response. In such cases, treating the data as MAR may be less credible than explicitly modeling the nonresponse process.
8.2 Data collection strategies to reduce MNAR
Design choices can reduce the severity of MNAR problems. These include follow-up contact, improved questionnaire wording, incentives for response, collection of auxiliary variables, and recording reasons for dropout or refusal. Although these steps do not guarantee random missingness, they can make the mechanism easier to model and assess.
8.3 Reporting and reproducibility
Reports of MNAR analyses should describe the missingness pattern, the modeling assumptions, and the sensitivity checks performed. Reproducible work often includes code, parameter choices, and a clear account of how alternative assumptions affect conclusions. Because the results may depend on nonverifiable structures, transparency is essential.
9 Related concepts
MNAR is one part of a broader family of ideas about incomplete data. It overlaps with statistical notions of selection, censoring, and imputation, but each term has a distinct meaning. Understanding the differences helps prevent misuse of methods across similar-looking problems.
9.1 Missing data
Missing data refers broadly to any dataset in which one or more values are not observed. The term covers many mechanisms and problem types, including MCAR, MAR, and MNAR. It is the umbrella concept under which specific missingness models are studied.
9.2 Imputation
Imputation is the process of filling in missing values with estimated or simulated replacements. Under MNAR, imputation must account for the fact that the missing values are systematically different from the observed ones. Simple imputation methods that ignore this feature can distort variability and bias estimates.
9.3 Selection bias
Selection bias occurs when the observed sample is not representative of the target population because inclusion depends on characteristics related to the outcome. MNAR can produce a form of selection bias when missingness acts like an unobserved selection process. The concepts are related, though selection bias is broader and includes many mechanisms beyond missing data.
9.4 Censoring and truncation
Censoring and truncation are related but distinct forms of incomplete observation. In censoring, a value is only partially known; in truncation, some observations are entirely excluded from the dataset by design or rule. Although these settings can resemble MNAR problems, they are usually modeled with separate statistical frameworks.