Going Further: Advanced Topics

This page points toward methods beyond the core course. It is organised by category, with field-specific reading paths at the end.

Advanced Classical Methods

MANOVA (Multivariate Analysis of Variance) tests for differences across groups when there are multiple correlated response variables simultaneously. It is more powerful than running separate ANOVAs, but its strict assumptions (multivariate normality, homogeneity of covariance matrices) mean it is often better replaced by PERMANOVA for ecological and biological data.

Canonical Correlation Analysis (CCorA) identifies relationships between two sets of variables, for example relating a set of environmental variables to a set of species abundances, or a set of metabolites to a set of phenotypic traits. It finds linear combinations of each variable set that are maximally correlated with each other.

Partial Least Squares (PLS) is a regression method for situations where predictors greatly outnumber samples. It finds components that simultaneously summarise predictor variance and predict the response, handling collinearity better than standard regression. PLS-DA is the discriminant variant used for classification.

Modern Dimensionality Reduction

t-SNE (t-Distributed Stochastic Neighbor Embedding) is a non-linear method optimised for visualisation of high-dimensional data. It reveals local cluster structure that PCA may miss, making it popular for single-cell RNA-seq data. However, distances between clusters are not meaningful, results vary between runs, and the method is computationally expensive. Always run PCA first.

UMAP (Uniform Manifold Approximation and Projection) is faster than t-SNE and better preserves global structure alongside local structure. It is currently the preferred method for single-cell data visualisation. The same interpretational caveats apply: axes have no biological meaning and inter-cluster distances should not be over-interpreted.

Both methods are tools for visualisation only. Do not use t-SNE or UMAP instead of PCA; use them when PCA clearly fails to reveal structure you know or suspect is present.

Machine Learning Methods

Random Forests build an ensemble of decision trees on bootstrapped subsets of the data. They handle non-linear relationships and interactions without tuning, provide variable importance scores, and are robust to outliers. A natural next step after LDA or logistic regression when linear boundaries are insufficient.

Support Vector Machines (SVM) find an optimal decision boundary between classes, with kernel functions allowing non-linear boundaries. They work well for small to medium datasets with clear margins between classes.

Regularised Regression (LASSO, ridge, elastic net) is covered in the logistic regression and logistic regression: going further pages. The same regularisation principles extend to linear regression for continuous outcomes via glmnet.

Specialised Topics

Compositional Data Analysis addresses data that sum to a constant: microbiome relative abundances, proportions, percentages. Standard distances and correlations on compositional data produce spurious results. The Aitchison framework (ALR, CLR, ILR transformations and Aitchison distance) provides a principled solution. CLR transformation before PCA is already used in this course; this topic extends that foundation.

Network Analysis represents relationships between variables or taxa as graphs. Common applications include microbial co-occurrence networks and gene co-expression networks. Methods such as SparCC and SPIEC-EASI estimate correlations from compositional data before network construction. Covered briefly in Zero-Inflated Data.

Spatial Statistics accounts for spatial autocorrelation, which violates the independence assumption in standard tests. When samples are collected across geographic space, nearby samples are often more similar than distant ones for reasons unrelated to your variable of interest. Partial RDA with spatial eigenvectors (Moran's eigenvector maps) is one approach within the ordination framework.

Longitudinal and Time Series Analysis covers multivariate data with repeated measurements over time. Methods include mixed-effects models for correlated observations, multivariate autoregressive models, and trajectory analysis. Temporal structure violates independence assumptions in all methods covered in this course.

Field-Specific Pathways

Microbiome

Priority	Topic	Reason
1	Compositional data analysis	Relative abundance data require it
2	Network analysis	Co-occurrence and keystone taxa
3	Random forests	Classify samples, identify important taxa
4	LASSO logistic regression	Already in course; extend to more taxa

Genomics and Transcriptomics

Priority	Topic	Reason
1	Regularised regression	Essential when p >> n
2	PLS and PLS-DA	Dimensionality reduction for prediction
3	t-SNE and UMAP	Single-cell data visualisation
4	Random forests	Non-linear classification of subtypes

Ecology

Priority	Topic	Reason
1	Variance partitioning and partial RDA	Already introduced; worth deepening
2	Spatial statistics	Geographic samples are rarely independent
3	MANOVA	Multiple response variables across treatments
4	Compositional methods	If working with relative abundance data

Clinical and Medical

Priority	Topic	Reason
1	Cross-validation strategies	Rigorous model validation for clinical data
2	Regularised regression	Many biomarkers relative to sample size
3	Random forests	Robust non-linear classification
4	Survival analysis	Time-to-event outcomes common in clinical research

Package	Topic
`Rtsne`	t-SNE
`umap`	UMAP
`randomForest`	Random forests
`e1071`	SVM
`compositions`	Aitchison transformations
`mixOmics`	PLS and multi-omics integration
`ade4`	Comprehensive multivariate methods