Global association between tables

Test the tables as a whole before looking at feature pairs.

• Mantel test (Mantel 1967): correlation between the two sample distance matrices; significance by permutation
• RV coefficient (Robert and Escoufier 1976): a “correlation between tables”, between 0 and 1

\[ RV(X, Y) = \frac{\mathrm{tr}(XX^\top YY^\top)} {\sqrt{\mathrm{tr}\big((XX^\top)^2\big)\,\mathrm{tr}\big((YY^\top)^2\big)}} \]

• Co-inertia analysis (Dolédec and Chessel 1994): shared axes of co-variation between two tables measured on the same samples

Review: Josse and Holmes (2016)

Global association in R

# Mantel test: Euclidean distances on clr / log10 data
vegan::mantel(dist(x), dist(y), permutations = 999)

# RV coefficient with a permutation test
ade4::RV.rtest(as.data.frame(x), as.data.frame(y), nrepet = 999)

In this data:

• Mantel r = 0.23 (p = 0.001)
• RV = 0.48 (p = 0.001; expected under permutation 0.20)

➡️ Microbiota and metabolite profiles are associated, but which features drive this?

Many more features than samples

• Canonical correlation analysis (CCA) finds linear combinations of the features in each table that are maximally correlated
• Classical CCA needs more samples than features: it overfits omics data
• Sparse CCA adds an L1 penalty and keeps only a few features from each table (Witten, Tibshirani, and Hastie 2009)
• Use it as a screening step, then visualize the selected features (e.g. PCA biplot)

Example: microbiota and metabolites

Mouse gut, two diets, wild type vs. knockout (Kashyap et al. 2013):

• 12 samples; 637 metabolite features; 20,609 taxa (filtered first)
• Sparse CCA selected 5 taxa and 15 metabolites
• Correlation between the two selected combinations: 0.97
• PCA of the 20 selected features separated the two diets

Analysis from Holmes and Huber (2019, Ch. 9).

Sparse CCA in R

cc <- PMA::CCA(x, y,
               typex = "standard", typez = "standard",
               penaltyx = 0.15, penaltyz = 0.15)

# Selected features
colnames(x)[cc$u[, 1] != 0]
colnames(y)[cc$v[, 1] != 0]
cc$cors

# PCA of the selected features only
selected <- cbind(x[, cc$u[, 1] != 0], y[, cc$v[, 1] != 0])
pca <- prcomp(selected, scale. = TRUE)

In HintikkaXOData: 10 genera and butyrate selected; correlation 0.88

A “compromise” across tables

• Compute the RV coefficient between every pair of tables
• Weight each table by how well it agrees with the others
• The weighted combination is the compromise: ordinate it and project each table onto it to see where tables agree or differ
• Classical methods: STATIS (Lavit et al. 1994; Abdi et al. 2012), multiple factor analysis (Escofier and Pagès 1994), multiple co-inertia analysis (Chessel and Hanafi 1996)
• Latent factor models (MOFA2, Joint-RPCA) pursue a related goal in a probabilistic or robust low-rank framework

The phylogenetic tree

• Taxa are not independent features: related taxa often behave alike
• DPCoA (double principal coordinate analysis) uses phylogenetic distances between taxa in the ordination (Pavoine, Dufour, and Chessel 2004; Purdom 2011)
• Adaptive generalized PCA tunes how strongly the tree is used, from ignoring it (PCA) to relying on it fully (DPCoA) (Fukuyama 2019)
• Example: repeated antibiotic courses in three adults (Dethlefsen and Relman 2011); tree-aware ordination separated disturbed from undisturbed samples in some subjects (Fukuyama et al. 2017)

DPCoA in mia

data("GlobalPatterns", package = "mia")
tse <- agglomerateByRank(GlobalPatterns, rank = "Genus", update.tree = TRUE)
tse <- tse[rowSums(assay(tse, "counts")) > 0, ]

# Uses the phylogenetic tree stored in rowTree(tse)
tse <- addDPCoA(tse, name = "DPCoA")

scater::plotReducedDim(tse, "DPCoA", colour_by = "SampleType")

Summary

• Global tests (Mantel, RV, co-inertia): are two tables associated?
• Sparse CCA: which features drive the association?
• Compromise methods (STATIS, MFA, MCIA): more than two tables
• Tree-aware ordination (DPCoA, adaptive gPCA): use prior knowledge

➡️ Multi-omics slides · Joint-RPCA slides