Merging Similar Modules With Eigengene Correlations

Why Merge Co-Expression Modules?

Dynamic tree cutting assigns genes to modules by cutting a gene-clustering tree. That cut can produce two modules whose expression summaries are strongly correlated across the same samples. Merging such modules is a post-processing step: it replaces their separate labels with one combined label when their module eigengenes are sufficiently similar. This can yield a more concise summary of the expression patterns without rebuilding the gene network.

The merge decision is based on the correlation between module eigengenes, not on whether every gene in one module is correlated with every gene in the other. A merge threshold is therefore an operational rule for grouping module-level patterns. It is not, by itself, a hypothesis test or evidence that two modules have the same biological function.

Definition: A module eigengene is a single expression summary for a module, commonly defined as the first principal component of its gene-expression profiles across samples. For merging, each module eigengene must have one value for each sample, with the same samples in the same order across modules. The eigengene correlation for two modules is the correlation between those two vectors of sample-level values.

The details of deriving and interpreting an eigengene are a separate topic. Here, treat the eigengenes as aligned vectors summarizing the modules. Before computing their correlations, use a consistent sign convention: a principal component and its negative represent the same component. One common convention orients each eigengene to correlate positively with the average expression profile of its module. Without a consistent orientation, a sign flip can change a positive correlation to a negative one and alter the merge decision.

From Correlation to a Merge Rule

For signed Pearson correlations, define the dissimilarity between module eigengenes \(M_i\) and \(M_j\) as one minus their correlation. A smaller dissimilarity means a stronger positive correlation. If the dissimilarity cut height is \(\tau\), modules are joined at that cut when their dissimilarity is at most \(\tau\).

d_{ij}=1-\operatorname{cor}(ME_i,ME_j)

d_{ij}\leq \tau \quad\Longleftrightarrow\quad \operatorname{cor}(ME_i,ME_j)\geq 1-\tau

For example, a dissimilarity cut of \(0.20\) corresponds to a required positive correlation of at least \(0.80\). Signed dissimilarity ranges from \(0\) to \(2\), because correlations range from \(-1\) to \(1\). A strongly negative correlation is not a close match under this rule. Using an absolute correlation instead would treat strongly negative and strongly positive patterns alike, which is a different merging convention and needs a substantive justification.

With more than two modules, the pairwise dissimilarities are usually clustered hierarchically and the resulting tree is cut at a chosen height. The linkage rule matters: single linkage uses the smallest cross-cluster distance, complete linkage the largest, and average linkage the average. Consequently, a cut height alone does not specify every detail of a multi-module merge. State the dissimilarity, linkage rule, and cut height together. Keep those choices fixed when comparing thresholds.

Key result: For a fixed hierarchical tree and linkage rule, increasing the cut height can join existing clusters but cannot split one. Thus the number of modules after merging is nonincreasing as the cut height increases. The genes in each merged module are the union of the genes in its constituent modules.

Worked Correlation and Threshold Calculation

Worked Example: Does a Pair of Modules Meet the Cut?

Suppose two hypothetical modules have eigengene values across five aligned samples \(ME_A=(-2,-1,0,1,2)\) and \(ME_B=(-2,0,0,1,1)\). Both vectors have mean zero. Calculate their Pearson correlation and decide whether they merge at dissimilarity cut \(\tau=0.10\).

For mean-zero vectors, the Pearson correlation is their cross-product sum divided by the square root of the product of their sums of squares:

r_{AB}= \frac{\sum_{k=1}^{5} ME_{A,k}ME_{B,k}} {\sqrt{\left(\sum_{k=1}^{5}ME_{A,k}^2\right) \left(\sum_{k=1}^{5}ME_{B,k}^2\right)}} = \frac{(-2)(-2)+(-1)(0)+(0)(0)+(1)(1)+(2)(1)} {\sqrt{(4+1+0+1+4)(4+0+0+1+1)}}

The numerator is \(4+0+0+1+2=7\); the sums of squares are \(10\) and \(6\). Therefore \(r_{AB}=7/\sqrt{60}=0.9037\), to four significant figures. The dissimilarity is \(d_{AB}=1-0.9037=0.0963\). Since \(0.0963\leq0.10\), the two modules meet this cut and are merged. At a stricter cut of \(0.05\), they would not merge because \(0.0963>0.05\).

This calculation assumes the eigengene values refer to the same five samples and that their signs have already been oriented consistently. The example is hypothetical; its purpose is to show the arithmetic and the decision rule, not to suggest that five samples would generally provide a reliable correlation estimate.

How Thresholds Change Module Count and Membership

The following example uses single linkage to make the effect of the clustering rule explicit. Single linkage merges two clusters when at least one pair of eigengenes across the clusters is within the cut height. Other linkage rules can yield different memberships at the same threshold.

Worked Example: A Cut Height Can Join a Chain of Modules

Suppose four hypothetical modules \(A,B,C,D\) have the following eigengene correlations: \(r_{AB}=0.92\), \(r_{BC}=0.84\), and \(r_{CD}=0.93\). All other pairwise correlations are below \(0.70\). Convert these three correlations to dissimilarities.

d_{AB}=1-0.92=0.08,\qquad d_{BC}=1-0.84=0.16,\qquad d_{CD}=1-0.93=0.07

At cut height \(\tau=0.08\), \(A\) joins \(B\), and \(C\) joins \(D\). The clusters are \(\{A,B\}\) and \(\{C,D\}\), so the module count falls from four to two. At cut height \(\tau=0.16\), the \(B\)-to-\(C\) distance also meets the cut. Under single linkage, that joins the two clusters into \(\{A,B,C,D\}\), leaving one module.

This is a chaining effect: the correlation between \(B\) and \(C\) connects two groups even though the other cross-group correlations are low. A single-linkage merge does not imply that all pairs within the final cluster meet the threshold. Complete or average linkage may avoid this particular chain, but their cut outcomes differ. Choose a linkage rule that is appropriate for the intended meaning of “similar,” and report it with the threshold.

Worked Example: Tracking Genes Across Two Thresholds

Suppose a hypothetical analysis starts with four modules: \(M_1\) contains genes \(g_1\)–\(g_4\), \(M_2\) contains \(g_5\)–\(g_7\), \(M_3\) contains \(g_8\)–\(g_{12}\), and \(M_4\) contains \(g_{13}\)–\(g_{14}\). Their relevant eigengene correlations are \(r_{12}=0.83\), \(r_{23}=0.76\), and \(r_{13}=0.65\); the correlations involving \(M_4\) are all below \(0.60\). Use single linkage.

At \(\tau=0.18\), the required correlation is \(1-0.18=0.82\). Since \(r_{12}=0.83\geq0.82\), \(M_1\) and \(M_2\) merge. The other stated correlations among \(M_1,M_2,M_3\) do not meet the cutoff, so the resulting modules contain \(g_1\)–\(g_7\), \(g_8\)–\(g_{12}\), and \(g_{13}\)–\(g_{14}\). The count is three.

At \(\tau=0.25\), the required correlation is \(0.75\). Now both \(r_{12}=0.83\) and \(r_{23}=0.76\) meet the cutoff. Under single linkage, they connect \(M_1,M_2,M_3\), so those modules become one module containing \(g_1\)–\(g_{12}\). \(M_4\) remains separate, and the count is two. The genes are assigned as whole original modules; this rule does not select only the genes that happen to correlate with the other module’s eigengene.

After a merge, the combined module’s eigengene should be recalculated from its combined set of genes for subsequent analyses. Its value is not generally the arithmetic average of the former eigengenes. The calculations above determine which original module labels are grouped; they do not claim that the new eigengene has a particular correlation without recomputing it.

Assumptions, Diagnostics, and Alternatives

A useful merge requires more than a numerical correlation. The expression data should be processed consistently across samples, module eigengenes should use exactly the same sample set and ordering, and each eigengene should have enough variation for its correlation to be defined. Missingness handled differently across pairs can make correlations incomparable. A module with an almost constant eigengene can also produce an unstable or undefined correlation. Check the vectors and the resulting correlation matrix before cutting the tree.

Inspect the eigengene dendrogram and the correlation matrix, not just the final labels. Look for unexpected negative correlations, near-constant eigengenes, or a small number of unusually influential samples driving a high correlation. Recheck the sign convention if correlations appear to change sign under an equivalent eigengene calculation. Sample-quality problems, batch structure, or cell-mixture differences can create similarity between module summaries even when the underlying biological interpretation is not shared.

Correlation estimates are affected by sampling uncertainty, especially when the number of samples is modest. A module pair just above a threshold should not be treated as categorically different from a pair just below it. The threshold is a practical clustering choice, not a \(p\)-value cutoff; crossing it does not establish that the correlation is statistically significant. Useful reporting includes the correlation method, eigengene orientation convention, dissimilarity, linkage rule, cut height, and resulting module count. Compare a small set of reasonable cut heights to see whether the broad grouping is sensitive to the choice.

Prepare comparable summaries.
Confirm that eigengenes describe the same samples, have consistent orientation, and are not constant or corrupted by unresolved data problems.

Choose the merge convention.
Specify signed or otherwise justified correlation, the corresponding dissimilarity, and the hierarchical linkage rule.

Cut and inspect.
Apply the stated height, record the new module count and gene unions, and inspect the dendrogram and pairwise correlations for surprising results.

Recompute summaries for downstream use.
Calculate eigengenes for merged gene sets anew and retain the original-to-merged module mapping for interpretation.

Eigengene-based merging is a compact alternative to returning to gene-level clustering and rebuilding modules. It operates on a small set of module summaries, so it can combine modules that show similar sample-level patterns even if their genes were separated by the original tree cut. In contrast, gene-level clustering revisits similarities among individual genes, while a topological-overlap-based procedure uses network neighborhood structure. These choices answer different questions: high eigengene correlation alone does not show that two modules have the same gene membership, network structure, or biological role.

Common Pitfall and Limitations

Common mistake: “The eigengene correlation is \(0.82\), so the modules are significantly associated and biologically equivalent.” This conclusion does not follow. A merge threshold is a clustering rule, not a significance test, and correlation of two summaries does not establish shared biological function. State instead that the modules were grouped under a specified correlation-based dissimilarity cut and linkage rule.

Merging also sacrifices resolution: once labels are combined, differences between the original modules may be hidden by the single merged label. Keep the original module assignments when those distinctions may matter. A threshold that is too strict can leave redundant summaries separate; one that is too permissive can combine distinct patterns, particularly under single-linkage chaining. No threshold can correct poor expression preprocessing, confounding, or an unsuitable initial network.

Check Your Understanding

Answer each question, then compare your reasoning with the solution that follows it.

A pair of eigengenes has signed correlation \(0.78\). What is their dissimilarity, and do they meet a cut height of \(0.25\)?
Solution: \(d=1-0.78=0.22\). Since \(0.22\leq0.25\), they meet the cut.
With signed dissimilarity, do eigengenes correlated at \(-0.90\) count as close at cut height \(0.20\)?
Solution: No. Their dissimilarity is \(1-(-0.90)=1.90\), which exceeds \(0.20\). Signed merging treats strong negative correlation as dissimilar.
For a fixed hierarchical tree and linkage rule, can increasing the cut height increase the number of clusters?
Solution: No. Increasing the height can retain the same clusters or join them, but cannot split a cluster in the fixed tree. The number is nonincreasing.
Two modules each contain six genes and are merged. How many genes are in the merged module if the modules are disjoint, and is its new eigengene necessarily the average of the two old eigengenes?
Solution: The union contains \(6+6=12\) genes because the modules are disjoint. Its eigengene must be recalculated from the combined genes and is not generally the arithmetic average of the old eigengenes.
Why is a correlation just above the chosen cutoff not evidence that the modules are statistically or biologically equivalent?
Solution: The cutoff defines a grouping rule, not a hypothesis test or equivalence procedure. Sampling uncertainty and biological context still matter; the merge only records that the specified algorithm grouped the modules.