Why the Source of Absence Matters
A multi-omics data table can lack a modality for two fundamentally different reasons. A cohort may never have been scheduled to undergo a particular assay, or an assay may have been scheduled but omitted for some individual subjects. The first is a feature of the study design; the second is a subject-level gap in data collection. Treating both as generic missing values can lead to invalid counts, inappropriate imputation targets, and claims of cross-modality evidence unsupported by the observed subjects.
The distinction is especially important when studies contain design groups, such as cohorts enrolled under different protocols or phases. A modality may be available throughout one group and absent by design throughout another. Before analyzing a modality block, therefore, establish what was planned for each group and what was actually observed for each subject.
A zero in \(D_{gm}\) means the group was not designed to supply that modality. It does not mean that each person in the group has an unmeasured value awaiting recovery. A zero in \(R_{im}\), by contrast, means a planned subject-level measurement was not observed. The distinction is made using design records and collection history, not merely by inspecting whether a column happens to contain values.
Counting Planned, Observed, and Omitted Measurements
Let \(n_g\) be the number of subjects in group \(g\). The number of modality-\(m\) measurements planned for that group is \(D_{gm}n_g\). The number observed is the sum of the overall indicators \(O_{im}\) for subjects in the group. This definition remains valid whether the modality was planned or not, because \(O_{im}\) is defined for every subject and modality.
If \(D_{gm}=0\), every term in this sum is zero, so \(n^{\mathrm{obs}}_{gm}=0\). If \(D_{gm}=1\), then \(O_{im}=R_{im}\) for subjects in the group, and the observed count is the number of planned subject-level measurements that were obtained. In that planned case, define the omitted count as \(n^{\mathrm{omit}}_{gm}=\sum_{i:g(i)=g}(1-R_{im})\). It follows that planned measurements partition exactly into observed and omitted measurements:
This identity is a useful audit check. It should not be applied by summing \(R_{im}\) when \(D_{gm}=0\), because \(R_{im}\) is not defined for that design-group and modality combination. The overall indicator \(O_{im}\) avoids that problem.
Worked Example: Counting Structural and Intermittent Absence
Suppose a hypothetical study has five subjects in group A and four in group B. RNA and protein assays were planned for group A; metabolite profiling was not. RNA and metabolite profiling were planned for group B; protein was not. In group A, RNA was observed for all five subjects and protein for four. In group B, RNA was observed for three subjects and metabolite profiles for all four.
| Group | Subjects \(n_g\) | Modality | Design indicator \(D_{gm}\) | Observed \(n^{\mathrm{obs}}_{gm}\) | Interpretation |
|---|---|---|---|---|---|
| A | 5 | RNA | 1 | 5 | All planned measurements observed |
| A | 5 | Protein | 1 | 4 | One planned measurement omitted |
| A | 5 | Metabolites | 0 | 0 | Not collected by design |
| B | 4 | RNA | 1 | 3 | One planned measurement omitted |
| B | 4 | Protein | 0 | 0 | Not collected by design |
| B | 4 | Metabolites | 1 | 4 | All planned measurements observed |
For group A protein, five measurements were planned and four observed, so \(n^{\mathrm{omit}}_{A,\mathrm{protein}}=5-4=1\). For group B RNA, four were planned and three observed, so \(n^{\mathrm{omit}}_{B,\mathrm{RNA}}=4-3=1\). In contrast, no metabolite measurement was planned for group A: \(D_{A,\mathrm{metabolites}}n_A=0\times5=0\). The five group-A subjects are not five subjects with individually omitted metabolite assays; the modality is absent from that group's design.
Structural Absence Is a Design Statement
“Structural absence” describes a modality not included in the design for a defined cohort or group. It is not established just because a modality has no observed values in a group. For example, if a protocol planned metabolite profiling for a group but every sample failed quality control, then \(D_{gm}=1\), while \(R_{im}=0\) for each subject. That is a group-wide failure of planned measurements, not absence by design. The all-zero subject-level pattern may have different implications from a protocol that never collected the modality.
Conversely, the availability of at least one value does not establish that every other subject was planned for that modality. Design records should identify which groups were assigned which assays. A useful representation keeps the design table \(D\) separate from the observed-data indicators \(O\), or from \(R\) restricted to planned measurements. This prevents a single missing-value code from hiding whether the measurement was outside the design or omitted after being planned.
Worked Example: An Entirely Failed Planned Assay
In a hypothetical cohort of six subjects, the protocol specified RNA and protein assays for everyone. The protein assay failed for all six samples, so no protein values were retained. Determine the design and observation indicators and the counts.
The protocol establishes \(D_{g,\mathrm{protein}}=1\), and the group size is \(n_g=6\). Since none of the planned protein data were observed, \(R_{i,\mathrm{protein}}=0\) for all six subjects. Thus:
The planned count is \(1\times6=6\), and the omitted count is \(\sum_i(1-0)=6\), so \(6=0+6\). Although the observed count is zero, the design indicator is 1. Calling this structural absence would erase the fact that six planned measurements failed. The distinction matters for quality-control reporting and for deciding whether the group can contribute protein information.
How Absence Constrains Integration
Integration requires more than having several modality files. To make a direct subject-level comparison between modalities \(m\) and \(k\), there must be subjects for whom both modalities were planned and observed. The number of such subjects in group \(g\) is:
Each product is 1 only when both measurements are observed for that subject. Structural absence in a group forces the corresponding \(O_{im}\) to zero for every subject in that group, so that group contributes no directly paired observations for any comparison requiring modality \(m\). Intermittent omissions can reduce the paired count without necessarily making it zero. If no subjects have both measurements, their empirical cross-modality relationship cannot be estimated directly from paired observations in that data set.
When some groups have both modalities and others do not, the data can support direct cross-modality comparisons only in the groups with observed overlap. Extending those relationships to a group with a structurally absent modality requires additional assumptions or external information; it is not a consequence of the absent block itself. This is a reason to inspect the overlap graph from the earlier material on overlap graphs, while keeping the design explanation behind each missing edge visible.
Worked Example: Finding the Subjects Available for a Paired Comparison
Consider a hypothetical group of five subjects for whom both RNA and protein assays were planned. RNA is observed for subjects 1, 2, 3, and 4; protein is observed for subjects 1, 3, 4, and 5. The overall indicators for these modalities are:
The number with both modalities observed is the sum of the subject-wise products:
Therefore, subjects 1, 3, and 4 can contribute directly to a paired RNA–protein comparison; subjects 2 and 5 cannot. Both assays were planned for all five, so these exclusions are intermittent subject-level omissions. If instead protein had not been included in this group's protocol, the relevant design indicator would be zero and there would be no planned protein measurements to pair with RNA. The paired count would then be zero for a design reason, not because five protein measurements were individually omitted.
A Short Justification for the Indicator Accounting
The accounting identities follow from the definitions, not from a probabilistic model. For a subject in a group with \(D_{gm}=0\), the definition gives \(O_{im}=0\). For a subject in a group with \(D_{gm}=1\), it gives \(O_{im}=R_{im}\), where \(R_{im}\) is either 0 or 1. Thus, for every subject and modality, \(O_{im}\) is a well-defined indicator of an observed measurement.
The final equality uses \(R_{im}+(1-R_{im})=1\) for every subject with a planned assay. No subject-level \(R_{im}\) value is required for a structurally absent group-modality combination.
Diagnostics, Alternatives, and Limitations
A practical audit separates design information from observed values. For each group and modality, record whether collection was planned, how many measurements were planned, how many were observed, and—when planned—why any were omitted. Compare that table with the subject-level indicator matrix. If a group's entire block is empty, check the protocol and assay records before classifying the pattern.
- Warning sign: using observed counts to infer design. A zero observed count does not distinguish a modality never planned from a planned assay that failed for every subject. Check study documentation.
- Warning sign: assigning subject-level zeros where no indicator exists. For a structurally absent block, \(R_{im}\) is not defined. Use \(D_{gm}\) and \(O_{im}\) as defined above instead of treating undefined subject indicators as zero and summing them.
- Warning sign: confusing an observed zero with an absent measurement. A measured abundance that equals zero is a data value; an uncollected or unavailable assay is not. Preserve that distinction in the data representation.
- Warning sign: claiming cross-modality evidence without overlap. If \(n^{\mathrm{both}}_{g,mk}=0\), no within-group paired observations support a direct comparison of those modalities.
The main alternative is a single missingness indicator for every unobserved entry, regardless of cause. It can be convenient for some software, but it discards design information and can obscure which values were never targeted for collection. A better analysis retains the design indicator and subject-level observation status separately, even if a software interface later requires a unified missing-value representation.
This classification alone does not determine whether subject-level omissions are related to observed or unobserved characteristics; that question belongs to the analysis of missingness mechanisms. Nor does it supply values for a structurally absent block. Imputation or model-based integration cannot create direct observations where the design provided none. Any extrapolation across groups must be justified by a scientific model, relevant external data, or additional collection, and its assumptions should be made explicit.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A modality was never included in group C's protocol. What are \(D_{C,m}\) and \(O_{im}\) for subjects in C? Is \(R_{im}\) defined?
- A modality was planned for eight subjects, but only six measurements were observed. How many planned measurements were omitted, and which indicator records that omission?
- A planned assay failed for every subject in a group of four. Explain why this is not structural absence and give the planned, observed, and omitted counts.
- For two planned modalities, the subject-level observation pairs are \((1,1),(1,0),(0,1),(1,1),(0,0)\). How many subjects have both measurements observed?
Solutions. (1) \(D_{C,m}=0\) and \(O_{im}=0\) for every subject in C; \(R_{im}\) is undefined because the modality was not planned. (2) Six of eight were observed, so two were omitted; \(R_{im}=0\) for those two subjects, while \(R_{im}=1\) for the six observed subjects. (3) The assay was planned, so \(D_{gm}=1\). The planned count is 4, the observed count is 0, and the omitted count is 4, with \(R_{im}=0\) for all four. (4) Multiply the entries within each pair and sum: \(1+0+0+1+0=2\). Thus two subjects have both modalities observed.