We introduce the Clustermatch Correlation Coefficient (CCC), an efficient not-only-linear clustering-based statistic. Applying CCC to GTEx v8 revealed that it was robust to outliers and detected linear relationships as well as complex and biologically meaningful patterns that standard coefficients missed. In particular, CCC alone detected gene pairs with complex nonlinear patterns from the sex chromosomes, highlighting the way that not-only-linear coefficients can play in capturing sex-specific differences. The ability to capture these nonlinear patterns, however, extends beyond sex differences: it provides a powerful approach to detect potentially complex relationships where a subset of samples or conditions are explained by other factors (such as differences between health and disease). We found that top CCC-ranked gene pairs in whole blood from GTEx were replicated in independent tissue-specific networks trained from multiple data types and attributed to cell lineages from blood, even though CCC did not have access to any cell lineage-specific information. This suggests that CCC can disentangle intricate cell lineage-specific transcriptional patterns missed by linear-only coefficients. In addition to capturing nonlinear patterns, the CCC was more similar to Spearman than Pearson, highlighting their shared robustness to outliers. The CCC results were concordant with MIC, but much faster to compute and thus practical for large datasets. Another advantage over MIC and standard coefficients is that CCC can also process categorical variables together with numerical values. CCC is conceptually easy to interpret and has a single parameter that controls the maximum complexity of the detected relationships while also balancing compute time.
Datasets such as Anscombe or "Datasaurus" highlight the value of visualization instead of relying on simple data summaries. While visual analysis is helpful, for many datasets examining each possible relationship is infeasible, and this is where more sophisticated and robust correlation coefficients are necessary. Advanced yet interpretable coefficients like CCC can focus human interpretation on patterns that are more likely to reflect real biology. The complexity of these patterns might reflect heterogeneity in samples that mask clear relationships between variables. For example, genes UTY - KDM6A (from sex chromosomes), detected by CCC, have a strong linear relationship but only in a subset of samples (males), which was not captured by linear-only coefficients. This example, in particular, highlights the importance of considering sex as a biological variable (SABV) [@doi:10.1038/509282a] to avoid overlooking important differences between men and women, for instance, in disease manifestations [@doi:10.1210/endrev/bnaa034; @doi:10.1038/s41593-021-00806-8]. More generally, a not-only-linear correlation coefficients that support categorical variables like CCC could identify significant differences between variables (such as genes) that are explained by a third factor (beyond sex differences), that would be entirely missed by linear-only coefficients.
It is well-known that biomedical research is biased towards a small fraction of human genes [@pmid:17620606; @pmid:17472739]. Some genes highlighted in CCC-ranked pairs (Figure @{fig:upsetplot_coefs}b), such as SDS (12q24) or PRSS36 (16p11), were previously found to be the focus of fewer than expected publications [@pmid:30226837]. It is possible that the widespread use of linear coefficients may bias researchers away from genes with complex coexpression patterns. A beyond-linear gene co-expression analysis on large compendia might shed light on the function of understudied genes. For example, gene KLHL21 (1p36) and AC068580.6 (a long non-coding RNA gene in 11p15) have a high CCC value and are missed by the other coefficients. KLHL21 was suggested as a potential therapeutic target for hepatocellular carcinoma [@pmid:27769251] and other cancers [@pmid:29574153; @pmid:35084622]. Its nonlinear correlation with AC068580.6 might unveil other important players in cancer initiation or progression, potentially in subsets of samples with specific characteristics (as suggested in Figure @{fig:upsetplot_coefs}b).
Not-only-linear correlation coefficients might also be helpful in the field of genetic studies. In this context, genome-wide association studies (GWAS) have been successful in understanding the molecular basis of common diseases by estimating the association between genotype and phenotype [@doi:10.1016/j.ajhg.2017.06.005]. However, the estimated effect sizes of genes identified with GWAS are generally modest, and they explain only a fraction of the phenotype variance, hampering the clinical translation of these findings [@doi:10.1038/s41576-019-0127-1]. Recent theories, like the omnigenic model for complex traits [@pmid:28622505; @pmid:31051098], argue that these observations are explained by highly-interconnected gene regulatory networks, with some core genes having a more direct effect on the phenotype than others. Using this omnigenic perspective, we and others [@doi:10.1038/s41467-023-41057-4; @doi:10.1186/s13040-020-00216-9; @doi:10.1101/2021.10.21.21265342] have shown that integrating gene co-expression networks with genetic studies could potentially identify core genes that are missed by linear-only models alone like GWAS. Our results suggest that building these networks with the latest approaches [@doi:10.1093/bib/bbab495] and advanced and efficient correlation coefficients could better estimate gene co-expression profiles and thus more accurately identify these core genes. Approaches like CCC could play a significant role in the precision medicine field by providing the computational tools to focus on more promising genes representing potentially better candidate drug targets.
Our analyses have some limitations.
We worked on a sample with the top variable genes in a single tissue from GTEx to keep computation time feasible.
Although CCC is much faster than MIC, Pearson and Spearman are still the most computationally efficient since they only rely on simple data statistics.
Our results, however, reveal the advantages of using more advanced coefficients like CCC for detecting and studying more intricate molecular mechanisms that are replicated in independent datasets.
The application of CCC on larger compendia, such as recount3 [@pmid:34844637] with thousands of heterogeneous samples across different conditions, can reveal other potentially meaningful gene interactions.
We compute
While linear and rank-based correlation coefficients are exceptionally fast to calculate, not all relevant patterns in biological datasets are linear. For example, patterns associated with sex as a biological variable are not apparent to the linear-only coefficients that we evaluated but are revealed by not-only-linear methods. Beyond sex differences, being able to use a method that inherently identifies patterns driven by other factors is likely to be desirable. Not-only-linear coefficients can also disentangle intricate yet relevant patterns from expression data alone that were replicated in models integrating different data modalities. CCC, in particular, is highly parallelizable, and we anticipate efficient GPU-based implementations that could make it even faster. The CCC is an efficient, next-generation correlation coefficient that is highly effective in transcriptome analyses and potentially useful in a broad range of other domains.