What Missingness Mechanisms Can Be Identified

What Can the Observed Availability Pattern Tell Us?

For each participant, an availability indicator records whether a modality was measured. That indicator is observed even when the modality itself is not. This makes it possible to study whether availability varies with information that was recorded for everyone, such as recruitment site, age, or a baseline clinical measurement. It does not, by itself, reveal whether availability depends on the unobserved values in the missing modality.

The distinction matters because MAR and MNAR describe how missingness relates to the full data, including values that are unavailable. The introductory material on MCAR, MAR, and MNAR for missing modalities defines those mechanisms. Here the central question is which parts of that distinction can actually be learned from observed records.

The condition is about dependence on the values that would be available under complete measurement, not just about whether the observed availability indicator is associated with some recorded covariate. MAR can allow strong association between \(R\) and \(X\). Conversely, an apparent absence of such association does not rule out dependence between \(R\) and the missing values.

Why MAR and MNAR Are Generally Not Identifiable

A property is identifiable from observed data if distinct values of that property cannot produce the same distribution of observed records. To see the difficulty here, consider a binary modality \(Y\), an observed covariate \(X\), and an availability indicator \(R\). For a fixed value of \(X\), the observed data identify the probabilities of \(R=1,Y=1\), of \(R=1,Y=0\), and of \(R=0\). They do not reveal how the \(R=0\) probability is divided between people whose unobserved \(Y\) is 1 and people whose unobserved \(Y\) is 0.

The observed probabilities for the two available groups combine the distribution of \(Y\) with the probability of availability. The missing group contributes only its total probability, obtained by adding over the unknown values of \(Y\). Thus different full-data distributions and different availability mechanisms can produce the same observed probabilities.

The first equation gives two observed products, not separate knowledge of the full-data outcome probabilities and the outcome-specific availability probabilities. The second gives their combined probability for the missing group. Without additional information or restrictions, these quantities do not determine how many missing participants had each value of \(Y\). The same reasoning applies to a continuous modality, with the sum replaced by an integral.

Checks Based on Observed Covariates

Although MAR versus MNAR is generally not identified, some implications of simple missingness assumptions can be checked. If a modality is subject to MCAR, its availability indicator is independent of every measured variable, including covariates observed for all participants. A clear association between availability and a fully observed covariate is evidence against MCAR. It does not distinguish MAR from MNAR, because either mechanism can generate that association.

For a binary covariate, compare availability proportions across its groups. For several categorical covariates, use a contingency table and an association test, provided its sampling assumptions are reasonable. For multiple or continuous covariates, a regression model for the availability indicator can summarize conditional associations. These checks are informative only for variables that are actually observed for participants with missing modalities. A covariate measured only as part of the unavailable block cannot be used to test its association with that block’s availability.

For more than one covariate, a model such as logistic regression can examine availability conditional on the included covariates. If \(R\) is binary, one possible model is \(\log\{P(R=1\mid X)/P(R=0\mid X)\}=\beta_0+\beta_1X_1+\cdots+\beta_qX_q\). The model requires suitable functional forms and a reasonable treatment of dependence among participants. A statistically significant coefficient indicates an association under that model; a non-significant coefficient does not establish independence. Nonlinearity, interactions, sparse groups, limited sample size, and covariates omitted from the model can all obscure or distort patterns. Examining plots, cross-tabulations, and plausible interactions can complement a single model-based test.

Observed Data Can Bound Some Quantities, Not Reveal the Mechanism

When an outcome is bounded, the observed records can sometimes give bounds on its overall mean without identifying the missing-value distribution. The width of such bounds makes the practical consequence of nonidentifiability visible. Any point estimate that fills in the missing group using a particular assumption should be understood as depending on that assumption.

Practical Diagnostics, Alternatives, and Limitations

Availability checks are most useful when treated as diagnostics rather than mechanism classifiers. First establish which participants were eligible for a modality. A modality not collected by design for a defined group is structural absence, not necessarily intermittent missingness; combining the two can create misleading availability associations. Among eligible participants, summarize availability by important covariates and by relevant combinations of covariates. For multiple modalities, examine each indicator and, where useful, the joint availability patterns.

A strong observed association can guide the description of the missingness process and make an MCAR assumption implausible. It may motivate modeling availability using those recorded predictors. A weak or undetected association is less informative: small samples, sparse cells, measurement error, poor functional form, or unexamined interactions can conceal dependence. Testing many covariates also raises multiplicity concerns, so isolated small \(p\)-values should be interpreted with the analysis plan and the overall pattern in mind.

There are ways to gain information beyond ordinary observed records, but each requires additional design information or assumptions. A validation subsample with the modality measured despite ordinary nonavailability can reveal something about the missing values in that subsample. External data may provide relevant information if their population and measurement process are comparable. An instrument or parametric restriction can also support identification, but its assumptions need substantive justification. When such information is unavailable, sensitivity analysis makes the dependence on unverifiable assumptions explicit by examining conclusions across plausible missing-value distributions or availability relationships.

None of these options makes a general MAR-versus-MNAR test appear from the availability indicator alone. Observed covariate checks assess observable implications and help describe who has data; validation or external information can add evidence about missing values; sensitivity analysis shows how conclusions may vary when the unobserved part is handled differently. These are different tasks and should not be conflated.

Common Pitfalls

• Calling any observed association evidence for MAR. Both MAR and MNAR can produce an association between availability and observed covariates. The association alone does not distinguish the mechanisms.
• Treating a non-significant test as proof of MCAR. A test can fail to detect an association because of limited information or model misspecification, and even genuine independence from measured covariates does not rule out MNAR dependence on missing values.
• Testing a covariate that is unavailable for missing cases. If a proposed predictor is part of the absent modality, it is not observed for the very participants whose availability it would need to help explain.
• Ignoring design-driven absence. Before interpreting availability patterns, distinguish participants who were never eligible for a modality from those with an eligible but unmeasured modality.
• Equating prediction with explanation. A covariate may predict availability without causing it, and a prediction model cannot determine the unobserved values without additional assumptions.