Why Diagnose Modality Dominance?
A joint clustering model can produce a clear posterior partition even when most of the assignment information comes from just one omics block. That may be appropriate: one modality can genuinely provide a stronger cluster signal. But it can also indicate that the joint result is fragile, driven by a block with many features, or inconsistent with the intended integration. Diagnosing dominance asks a focused question: how much do the inferred assignments depend on each block?
No single diagnostic answers that question. Blockwise likelihood contributions show how the fitted model’s evidence for an assignment is distributed across modalities. Refitting after removing a block shows whether assignments remain similar when that modality is unavailable. These are complementary checks: the first attributes conditional assignment odds within a fit, while the second assesses the effect of changing the data and fitting the model again.
Model and Likelihood Contributions
Suppose \(n\) samples are measured on \(M\) omics blocks. Write \(y_i^{(m)}\) for the measurement vector for sample \(i\) in block \(m\), \(z_i\) for its cluster label, and \(\theta_{mk}\) for the parameters describing block \(m\) in cluster \(k\). Let \(\pi_k\) be the prior mixture weight for cluster \(k\). For the diagnostics below, assume that samples are conditionally independent given their labels and parameters, and that the blocks are conditionally independent given the labels and their block-specific parameters. Thus, for a sample assigned to cluster \(k\), the sampling density is \( \prod_{m=1}^{M} f_m(y_i^{(m)}\mid \theta_{mk}) \). This conditional independence assumption is a modeling choice, not a property guaranteed by having multiple data blocks.
For a transparent blockwise decomposition, take the block-specific parameters as fixed at the current fitted values. If instead they have been integrated out, a corresponding decomposition requires the marginal likelihood to factor across blocks; independent block-specific priors and no shared parameters are sufficient for that factorization. With shared parameters or dependent blocks, the contributions may not separate in this way.
Conditional on the other labels and the fitted parameters, the allocation probability is proportional to the mixture weight times the product of the block densities. Therefore,
Here “rest” denotes the other labels and model quantities held fixed for this conditional comparison. In a collapsed sampler, the corresponding block terms are predictive log-density contrasts conditional on the remaining data. The same logic applies, but the terms should not be mistaken for the fixed-parameter likelihood contributions above.
Worked Example: Decomposing Assignment Odds
Worked Example: Decomposing Assignment Odds
Consider a hypothetical sample and two candidate clusters, \(k\) and \(l\), in a three-block model. Suppose the fitted mixture weights are equal, so \(\log(\pi_k/\pi_l)=0\). The sample’s blockwise contrasts are \(+3.0\), \(-1.2\), and \(+0.4\) log units for blocks 1, 2, and 3, respectively.
The total log posterior odds for \(k\) over \(l\), conditional on the stated fitted quantities, are \(0+3.0-1.2+0.4=2.2\). The odds are \(\exp(2.2)=9.025\) to 1, and the conditional probability of \(k\) among these two alternatives is \(9.025/(1+9.025)=0.9002\).
Block 1 supplies a \(+3.0\) log-unit contribution, while blocks 2 and 3 together supply \(-0.8\). Block 1 strongly favors \(k\) and overcomes the combined counter-evidence from the other blocks. Removing block 1 from this conditional calculation leaves log odds \(-0.8\), corresponding to probability \(1/(1+\exp(0.8))=0.3100\) for \(k\) among \(k\) and \(l\). This is evidence that this sample’s assignment contrast depends heavily on block 1 under the fitted model; it is not by itself evidence that the block is erroneous or that the model is misspecified.
Comparing Fits With and Without a Block
A blockwise contrast is conditional on the fitted model. To test whether a modality is driving the overall result, also fit the model after excluding that block. Use the same samples, the same clustering specification, comparable prior choices, and the same convergence checks. Re-estimation matters: simply deleting one term from the full fit does not allow the remaining block parameters or cluster structure to adapt.
Cluster labels can be permuted without changing a mixture fit, and the preferred number of occupied clusters may differ between fits. Consequently, compare label-invariant summaries rather than raw label numbers. One useful summary is the posterior co-clustering probability \(C_{ij}\), the fraction of posterior draws in which samples \(i\) and \(j\) share a cluster. Let \(C_{ij}^{\mathrm{full}}\) denote this quantity from all blocks and \(C_{ij}^{(-m)}\) the quantity from a refit without block \(m\). An average absolute discrepancy across sample pairs is
This summary ranges from 0 to 1. A value near 0 indicates similar pairwise co-clustering probabilities; a larger value indicates more change. It does not say which fit is correct. Examine the individual pair differences as well: a moderate average can hide a major change for a small, scientifically important subgroup. Posterior uncertainty also matters. If both fits have diffuse co-clustering probabilities, a difference in a single hard partition can exaggerate how much the evidence changed.
Worked Example: Comparing Co-Clustering Probabilities
In a hypothetical study with three samples, the posterior co-clustering probabilities are as follows. The reduced fit omits one omics block and is refitted.
| Pair | All blocks | Block omitted | Absolute difference |
|---|---|---|---|
| 1 and 2 | 0.95 | 0.55 | 0.40 |
| 1 and 3 | 0.10 | 0.45 | 0.35 |
| 2 and 3 | 0.12 | 0.40 | 0.28 |
There are three sample pairs, so \(D_m=(0.40+0.35+0.28)/3=1.03/3=0.3433\). The reduced fit substantially changes pairwise clustering support in this example: the full fit strongly supports samples 1 and 2 sharing a cluster, whereas the reduced fit gives that pair a probability of only 0.55. The result suggests sensitivity to the omitted block. It does not establish which partition better represents biology, and it should be interpreted alongside posterior uncertainty and fit diagnostics.
How to Diagnose Dominance in Practice
A useful assessment combines assignment-level and fit-level evidence. For important samples or ambiguous cluster boundaries, inspect the \(\Delta_{im}(k,l)\) values for the leading competing clusters. Summarize how often each block favors each alternative, whether one block repeatedly overwhelms the others, and whether that pattern is concentrated in a subset of samples. Then perform leave-one-block-out refits and compare posterior co-clustering probabilities. If several blocks are considered, repeat the removal analysis for each one.
A larger absolute log-likelihood contribution does not automatically mean a block dominates in a problematic sense. Blocks can differ greatly in feature count, measurement scale, likelihood family, and residual variability. A block with many features can accumulate a large total log likelihood because it contains more modeled observations. Conversely, comparing contributions per feature can obscure the fact that a block with more measurements may genuinely provide more information. Report the chosen scale and use per-feature summaries only as supplementary descriptions, not as replacements for the model’s likelihood.
Worked Example: Feature Count and Apparent Dominance
Suppose a hypothetical two-block model has 1,000 features in block A and 10 features in block B. For a particular assignment contrast, the total contributions are \(+120\) log units from A and \(+8\) log units from B. The total is \(+128\), so the conditional log odds favor the first cluster by 128 units if the prior odds are equal and no other blocks are present. The corresponding odds are \(\exp(128)\) to 1, an extremely large number.
For a descriptive per-feature calculation, the averages are \(120/1000=0.120\) log units per feature for A and \(8/10=0.800\) for B. These averages show that A’s large total is not evidence of a larger average contrast per feature; A has many more features. However, neither the total nor the average alone settles whether A is unduly influential. Check whether the model’s conditional independence assumptions are plausible, whether the features were appropriately transformed and scaled for their likelihoods, and whether assignments remain stable when A is removed and the model is refitted.
Derivation and Interpretation
The assignment-odds decomposition follows directly from the mixture model. For a candidate label \(k\), conditional independence gives an unnormalized allocation probability \( \pi_k\prod_m f_m(y_i^{(m)}\mid\theta_{mk}) \). Divide this expression by the analogous expression for \(l\). Taking logarithms turns the product across blocks into a sum:
This identity justifies attributing the conditional log-odds contrast to additive block terms under the stated model. It does not make the terms causal effects, nor does it show that a block is independently informative in every possible model. If block dependence is ignored, one may count correlated evidence more than once. If a block uses a poorly fitting likelihood or has extreme outliers, its contribution can also distort assignments; diagnosing such cases requires model checking rather than interpreting a large contribution as a biological signal.
Diagnostics, Alternatives, and Limitations
A leave-one-block-out comparison is meaningful only if the fits are computationally trustworthy. Check chain convergence, effective sampling, and posterior uncertainty in each fit; a poorly mixed reduced fit can look different simply because it explored a different part of the posterior. Use the same clustering assumptions and comparable priors, while recognizing that hyperparameters may require principled adjustment when the data dimension changes. Record whether the number of occupied clusters changes, but do not treat label counts alone as a measure of stability.
The main alternatives answer related but different questions:
| Approach | What it assesses | Main caution |
|---|---|---|
| Blockwise assignment contrasts | Which blocks favor competing labels within a fitted model | Conditional attribution; sensitive to model assumptions and feature structure |
| Leave-one-block-out refit | How posterior clustering changes when a block is unavailable | Requires reliable refits; removal changes the model’s available information |
| Compare hard partitions | Whether selected summary partitions agree | Discards posterior uncertainty and can depend on label matching or a partition summary rule |
| Posterior predictive checks | Whether replicated data under the model resemble observed data in chosen features | Checks model fit, not the contribution of each block to assignments by itself |
These diagnostics have limits. A block may appear influential because it is genuinely more informative, because it contains more measured features, or because the model’s assumptions favor its likelihood scale. Removing a modality also changes the fitted posterior and may change cluster separation, parameter uncertainty, and the effective interpretation of a cluster. Block dominance is therefore a sensitivity description, not a binary test and not proof of a modeling error.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- With equal mixture weights, two blocks contribute \(+1.5\) and \(-0.5\) log units to the odds of cluster \(k\) over cluster \(l\). What are the total log odds and the conditional probability of \(k\) among these two clusters?
- Why should a full fit and a fit without one block be compared using posterior co-clustering probabilities rather than cluster label numbers?
- A block has a larger total assignment contrast than another block but also has 50 times as many features. What can you conclude from the total alone?
- Suppose removing a block changes hard cluster labels, but the posterior co-clustering probabilities remain close. How should this result be interpreted?
Solutions. (1) The total log odds are \(1.5-0.5=1.0\). The conditional probability is \(\exp(1)/(1+\exp(1))=0.7311\). (2) Mixture labels can be permuted without changing the clustering, so the same numeric label need not refer to the same group across fits; co-clustering probabilities are label-invariant. (3) The total alone does not establish problematic dominance: feature count affects the scale, and the contrast is conditional on the model. Inspect blockwise contrasts and refit without the block. (4) The hard summaries differ, but the similar co-clustering probabilities suggest that posterior pairwise clustering support is largely stable; the label changes may reflect label permutation or the partition-summary rule rather than substantial posterior change.