Why Missingness Mechanisms Matter
A multi-omics data set can contain subjects with different combinations of measured modalities. Whether a modality is available may be unrelated to the subject’s measurements, may depend on information already observed, or may depend on the value that is missing. These possibilities are not interchangeable: they describe different probability models for how the observed data arose.
The mechanism labels MCAR, MAR, and MNAR describe relationships between modality availability and the data values. They are assumptions about the data-generating process, not labels that can be assigned just by looking at an availability table. In particular, an observed association between availability and a recorded variable can be informative, but observed rates alone do not establish which mechanism generated the data.
Let \(Y_{ij}\) denote the complete data for subject \(i\) in modality \(j\), and let \(R_{ij}=1\) if that modality is observed for the subject and \(R_{ij}=0\) otherwise. Write \(Y_i=(Y_{i1},\ldots,Y_{iK})\) for the subject’s complete set of modality values and \(R_i=(R_{i1},\ldots,R_{iK})\) for the availability pattern. Let \(X_i\) denote covariates observed for every subject. A modality value is treated as missing here only when it belongs to the target measurement but was not observed; a value that is structurally inapplicable is not automatically a missing value.
The definition is about the probability of a whole pattern, not merely a separate probability for each modality. For example, in a three-modality study the pattern \(r=(1,0,1)\) means modalities 1 and 3 are observed and modality 2 is missing. Its MAR probability may depend on the observed values \(Y_{i1}\) and \(Y_{i3}\), and on \(X_i\), but not on the missing value \(Y_{i2}\). The dependence can differ across patterns.
This formulation also makes the role of \(X_i\) explicit. If availability depends on a fully observed covariate, the mechanism is not MCAR under the definition above. It can still be MAR because the covariate is available for every subject. If a covariate itself is missing, it cannot be treated as fully observed in this definition without extending the model to include its missingness.
How the Definitions Relate
MCAR is the strongest of the three assumptions: it rules out dependence of availability on both observed and unobserved data. MAR allows dependence on observed information, but rules out dependence on the values missing under the particular pattern. MNAR allows that remaining dependence on unobserved values.
The implication from MCAR to MAR follows directly from the definitions. If a pattern probability is constant in \(y\) and \(x\), it cannot depend on any of the missing values. It therefore satisfies the MAR requirement, which permits dependence on observed values but does not require it.
Here \(y_{\mathrm{obs}}(r)\) denotes the modality values observed under pattern \(r\). The reverse implication does not hold: a mechanism that depends on observed covariates can be MAR without being MCAR. And a mechanism that depends on a missing value is MNAR even if it also depends on observed covariates.
Worked Example: Availability Assigned Independently of Omics Values
Hypothetical data-generating setup. A study measures transcriptomics and proteomics on each eligible subject. Before the measurements are inspected, a data manager assigns each subject one of four patterns, with probabilities \(0.10\) for neither modality, \(0.20\) for transcriptomics only, \(0.30\) for proteomics only, and \(0.40\) for both. The assignments are independent across subjects and independent of \(Y_i\) and \(X_i\).
The probabilities sum to one: \(0.10+0.20+0.30+0.40=1.00\). For each pattern \(r\), the stipulated mechanism is \(P(R_i=r\mid Y_i=y,X_i=x)=\pi_r\), with \(\pi_r\) equal to that pattern’s assigned probability. For example, the chance of observing both modalities is \(0.40\) for every possible omics profile and covariate value.
Because every pattern probability is constant in both the complete omics data and the fully observed covariates, the mechanism is MCAR. It is consequently MAR as well. A finite sample need not show the four pattern proportions in exactly these percentages: random variation in the assignments can produce different observed proportions without contradicting the stipulated MCAR mechanism.
Worked Example: Availability Depends on an Observed Covariate
Hypothetical setup. A study measures RNA and protein for subjects in two baseline severity groups, both recorded for everyone. Suppose the probability that the protein modality is observed is stipulated to be \(0.90\) in the low-severity group and \(0.60\) in the high-severity group, regardless of the true protein value or other omics values. For simplicity, assume RNA is observed for all subjects.
For the pattern in which protein is missing, the conditional probabilities are \(P(R_{\mathrm{protein}}=0\mid X=\text{low})=1-0.90=0.10\) and \(P(R_{\mathrm{protein}}=0\mid X=\text{high})=1-0.60=0.40\). These probabilities depend on a covariate observed for every subject, but not on the missing protein value. The mechanism is therefore MAR.
It is not MCAR under the stipulated mechanism: the probability of protein availability differs by the observed severity group. Suppose, for illustration, that a sample contains 10 low-severity and 10 high-severity subjects, with 9 and 6 protein measurements observed, respectively. Those sample fractions are \(9/10=0.90\) and \(6/10=0.60\), consistent with the stipulated probabilities. However, observed fractions that differ would not, by themselves, establish non-MCAR; random samples can show different fractions even if underlying availability probabilities are equal. Here the classification follows from the explicitly stipulated group-specific probabilities, not merely from the observed counts.
Worked Example: Availability Depends on the Missing Value
Hypothetical setup. A metabolomics study records a fully observed clinical covariate \(X\) and a transcriptomics profile, but some metabolomics measurements are unavailable. Suppose the probability that metabolomics is observed depends on the subject’s true, potentially unobserved metabolite concentration \(Y_{\mathrm{met}}\): it is \(0.20\) for a low concentration and \(0.80\) for a high concentration. Assume these probabilities do not otherwise depend on \(X\) or the transcriptomics profile.
For a low concentration, the probability of a missing metabolomics measurement is \(1-0.20=0.80\). For a high concentration, it is \(1-0.80=0.20\). Thus, even after fixing the observed covariate and transcriptomics values, the probability of the pattern with missing metabolomics differs according to the unobserved metabolite concentration.
The mechanism is MNAR: availability depends on a value that is unobserved for subjects whose metabolomics measurement is missing. The fact that the study also records other modalities does not make this dependence MAR. Those observed modalities would matter to the classification only if conditioning on them removed the mechanism’s dependence on the missing concentration.
Applying the Labels Carefully
For a study with several modalities, classify a mechanism pattern by pattern. First specify the target population and which modality values count as defined measurements. Then identify which values and covariates are observed for each availability pattern. Finally ask what the probability of that pattern depends on. A concise audit can be organized as follows.
State the modality values \(Y_i\) that would be defined for each eligible subject and the covariates \(X_i\) recorded for everyone.
For each availability pattern \(r\), list which modality values are observed and which are missing.
Ask whether the pattern probability changes with any complete-data value, only with observed values and \(X_i\), or with a value missing under that pattern.
Constant pattern probabilities give MCAR; dependence only on observed values and fully observed covariates gives MAR; dependence on a missing value gives MNAR.
A real mechanism can be more complicated than a single example. For instance, protein availability might depend on a recorded severity group and on the unobserved protein abundance. The observed-group dependence alone would be compatible with MAR, but the additional dependence on the missing abundance makes the mechanism MNAR. The labels describe the full conditional mechanism, not just one of its predictors.
Evidence, Diagnostics, and Limits
Availability patterns can be summarized by modality and related to observed covariates. Marked differences in availability across recorded batches, sites, or clinical groups can be evidence against MCAR, subject to sampling variability and the study design. Such summaries can also reveal data-processing errors or protocol differences worth investigating. They do not determine whether availability depends on the unobserved modality values.
The central limitation is that, for subjects with a missing modality, the value that could reveal a direct dependence of availability on that value is unavailable. Consequently, MCAR, MAR, and MNAR are assumptions about the mechanism, supported by design knowledge, measurement procedures, observed-data patterns, and scientific reasoning. They are not generally proven by an availability table alone. The distinction between MAR and MNAR is especially challenging because both can produce similar observed records.
A practical comparison is useful, provided it is read as a comparison of assumptions rather than a recipe for selecting a label from sample counts.
| Mechanism | What pattern probability may depend on | What it excludes |
|---|---|---|
| MCAR | Neither complete-data values nor fully observed covariates | Any systematic dependence on measured or unmeasured values |
| MAR | Values observed under that pattern and fully observed covariates | Dependence on values missing under that pattern |
| MNAR | At least one value missing under a pattern, possibly alongside observed information | Nothing in the definition requires dependence to be only on observed data |
The most common reasoning error is to infer a mechanism directly from the observed availability fractions. For example, if protein is observed in 90% of one group and 60% of another, the tempting conclusion is that the mechanism cannot be MCAR. That conclusion is warranted only if the underlying probabilities are known or otherwise justified to differ; observed sample fractions can differ by chance. Conversely, similar fractions do not prove MCAR, since missingness could depend on an unobserved modality value without producing a clear difference across the recorded groups.
Check Your Understanding
For each question, identify the mechanism or explain what the information does and does not establish. Solutions follow each question.
- A lipidomics measurement is observed with probability \(0.75\) for every subject, independently of all modality values and fully observed covariates. What mechanism is stipulated?
Solution: MCAR, because the availability probability is constant in both the complete data and the fully observed covariates. It is also MAR, since MCAR implies MAR. - Suppose protein availability depends on a baseline diagnosis recorded for every subject, but not on protein abundance after conditioning on that diagnosis. Which mechanism is described?
Solution: MAR. The pattern probability can depend on a fully observed covariate, but not on the missing protein value. Because availability probabilities vary with diagnosis, this is not MCAR under the stated mechanism. - In a hypothetical sample, an assay is observed for 42 of 50 subjects in group A and 31 of 50 in group B. Does this alone prove that missingness is not MCAR?
Solution: No. The sample fractions are \(42/50=0.84\) and \(31/50=0.62\), but sample proportions can differ by chance even when underlying probabilities are equal. The counts alone do not establish the mechanism. - For subjects missing metabolomics, suppose the chance that metabolomics is missing remains higher when the true, unobserved metabolite concentration is high, even after conditioning on all observed values. How is the mechanism classified?
Solution: MNAR, because availability still depends on a value unobserved under that pattern after conditioning on observed information. - A modality is not defined for some subjects because a protocol made the assay inapplicable to them. Should that absence automatically be treated as MCAR, MAR, or MNAR?
Solution: No. The mechanism definitions here concern missing values that belong to the target measurements. A structurally inapplicable modality should first be distinguished from an unobserved measurement in the target population; it is not automatically a missingness event.