Incomplete Modality Measurements in Joint Models

Why Missing Modalities Change a Joint Analysis

A cell may have RNA counts but no protein measurements, or may be missing an accessibility profile. This can happen because assays were applied to different subsets, technical failures removed some measurements, or quality-control rules excluded a modality. The resulting data are not simply a smaller version of a complete multimodal matrix: the cells available for analysis, and the evidence available for comparing any two cells, can differ.

Two practical strategies are to restrict analysis to cells measured in every modality, or to compare cells using only the modalities observed for both members of each pair. The first gives a consistent set of modalities but discards incomplete cells. The second can retain more cells, but comparisons may rely on different evidence from pair to pair. Neither strategy automatically corrects bias caused by why measurements are missing.

What Complete-Case Analysis Assumes

Complete-case analysis uses only cells with every selected modality observed. It is attractive because every retained cell has the same set of inputs, so standard joint procedures can be applied without defining how to compare cells with different modality patterns. Its cost is reduced sample size, sometimes substantial, and possible selection bias.

The missingness mechanism describes how the observation indicators \(R_{im}\) relate to the data. Missing completely at random (MCAR) means the probability of a cell's observation pattern does not depend on measured or unmeasured data. Missing at random (MAR) means that, conditional on observed information, the pattern does not depend on the values that are missing. Missing not at random (MNAR) means dependence remains on unobserved values, even after conditioning on observed information.

Under MCAR, complete cases are a random subset with respect to the data values. Thus, for example, a complete-case mean can estimate the corresponding population mean without selection bias, although it has less precision. Under MAR, an unadjusted complete-case analysis is not generally unbiased: it may be possible to account for selection using a suitable model or weighting, but that requires additional assumptions and information. Under MNAR, observed data alone generally cannot establish or remove the bias; sensitivity analysis or external information may be needed.

These statements concern the target quantity and analysis, not just the missingness label. For example, a complete-case mean of a protein feature describes the complete cells directly. To interpret it as the mean for all cells, the inclusion mechanism must support that extrapolation. MCAR is a sufficient, not necessary, condition for unbiasedness in some settings; other particular designs or models can also justify it.

Modality-Aware Pairwise Dissimilarities

A modality-aware comparison can use cells that lack some measurements by calculating a comparison from modalities observed for both cells. Let \(d_m(i,j)\) be a nonnegative component dissimilarity between cells \(i\) and \(j\) in modality \(m\), computed in a representation appropriate to that modality. Let \(w_m\geq 0\) be a prespecified modality weight, with at least one positive weight. Define the shared set \(S_{ij}=\{m:R_{im}=R_{jm}=1\}\). If the total weight in the shared set is positive, one common construction is

The denominator renormalizes the weights over the modalities available to that pair. If \(S_{ij}\) has no positive-weight modality, \(D_{ij}\) is undefined; it should not be silently set to zero. This construction is a pairwise dissimilarity. It is symmetric and nonnegative when each component dissimilarity is symmetric and nonnegative, but those facts do not establish that it is a metric.

For the result to be interpretable, component values must be on scales that make their weighted combination meaningful. The scaling and representation of each modality should be chosen consistently with the intended comparison. The fixed weights should also express a defensible relative contribution, rather than compensate invisibly for missing measurements. Renormalizing over \(S_{ij}\) makes each pair's available weights sum to one, but it does not make comparisons based on different modality subsets equally informative or directly interchangeable.

Why the Pairwise Result May Not Be a Metric

A metric must satisfy nonnegativity, identity of indiscernibles, symmetry, and the triangle inequality. In particular, for any three cells, their metric distances must satisfy \(D_{AC}\leq D_{AB}+D_{BC}\). Pairwise renormalization can violate this condition because each pair may be compared using a different modality. The following construction shows that this is not merely a technical possibility.

The formula's limited guarantees follow directly. If all \(w_m\) are nonnegative and the denominator is positive, each \(D_{ij}\) is a weighted average of the available \(d_m(i,j)\), so it lies between the smallest and largest available component values. If the component dissimilarities are symmetric, then \(S_{ij}=S_{ji}\) and \(D_{ij}=D_{ji}\). Neither fact relates \(D_{ij}\) to \(D_{jk}\), however, so neither proves a triangle inequality. Further, distinct cells can have zero dissimilarity if their shared observed features coincide, so identity of indiscernibles is not assured either.

Choosing and Checking an Approach

The appropriate strategy depends on the target and the missingness pattern. Complete-case analysis is often preferable when the complete subset is large enough, its selection is defensible for the target, and the joint method requires a common feature set. A pairwise dissimilarity is useful when retaining partially measured cells matters and there is adequate overlap across modalities. It does not, by itself, estimate missing measurements or repair selection bias.

Before relying on either of the first two approaches, inspect the cell-by-modality observation pattern. Tabulate complete cells and each missingness pattern; check whether missingness is concentrated in a batch, sample, or observed cell group; and summarize which modalities each pair shares. A graph with cells as vertices and an edge when two cells share at least one positively weighted modality can reveal isolated cells or disconnected groups that cannot be compared directly.

For pairwise calculations, report or inspect the shared modality set and total available weight for each pair. Also examine whether component dissimilarities are comparable after the chosen scaling, whether some pairs depend on just one modality, and whether the dissimilarity matrix is suitable for the intended downstream method. If that method assumes metric geometry, do not infer that requirement is met just because a symmetric matrix has been produced. Check metric properties or choose a method that accepts nonmetric dissimilarities.

No diagnostic based only on observed data can prove that missingness is MCAR, MAR, or MNAR in every case. Associations between missingness indicators and observed variables can expose some concerns, but an absence of detected association does not establish MCAR; dependence on missing values cannot generally be tested directly from those values when they are unobserved. State the assumed mechanism, inspect what is observable, and assess whether the substantive process of measurement supports the assumption.

Common Mistakes and Practical Limits

• Calling every pairwise result a distance. A shared-modality weighted average can violate the triangle inequality. Call it a dissimilarity unless the needed metric properties have been verified.
• Assuming that retaining more cells removes bias. Pairwise comparison avoids excluding every incomplete cell, but it does not make missingness ignorable or recover information from an unmeasured modality.
• Treating different pairs as equally supported. A pair based on three modalities and a pair based on one may have different precision and meaning even if both produce a number.
• Ignoring pairs with no overlap. If no positively weighted modality is shared, the formula is undefined. Imputing zero would falsely treat the pair as identical.
• Assuming complete cases represent all cells. A complete-case result directly describes retained cells. Generalization to the full set requires an appropriate missingness assumption or a justified adjustment.

These methods also have practical limits. Complete-case analysis can become unstable when few cells remain, while pairwise calculations can be uneven or disconnected when modality overlap is sparse. Model-based methods may use more information, but their conclusions depend on their model assumptions. The choice should therefore be tied to the scientific target, the observation design, and the requirements of the downstream analysis—not only to which calculation is easiest to run.