Multivariate Statistics

PCA, cluster analysis, ordination, discriminant analysis, and multivariate hypothesis testing for environmental data

Overview

Multivariate statistics analyze datasets with multiple measured variables simultaneously, revealing patterns, groupings, and relationships that univariate methods miss. In ocean and environmental science, multivariate techniques are essential for analyzing water quality parameters, community composition, environmental gradients, and spatiotemporal patterns. Principal Component Analysis (PCA) reduces dimensionality; cluster analysis groups similar samples; PERMANOVA tests multivariate hypotheses; and canonical correspondence analysis (CCA) relates species composition to environmental drivers.

Key Methods

Principal Component Analysis (PCA)

PCA finds orthogonal axes (principal components) that maximize variance in the data. Eigenvalues measure explained variance; loadings show variable contributions. A scree plot identifies the number of meaningful components. PCA assumes linear relationships and is sensitive to variable scaling.

Σv = λv (eigenvalue problem)
PC1 = w₁₁X₁ + w₁₂X₂ + ... + w₁ₚXₚ

Cluster Analysis

Hierarchical clustering (Ward's, UPGMA) produces dendrograms; partitional methods (K-means, PAM) optimize within-cluster similarity. Silhouette scores and gap statistics determine optimal cluster number. Applied to group water masses, stations, or species assemblages.

Non-metric MDS (nMDS)

nMDS ordinates samples in low-dimensional space preserving rank-order dissimilarities. Stress < 0.1 indicates good representation. Uses Bray-Curtis dissimilarity for ecological data. Complements ANOSIM/PERMANOVA for group testing.

Canonical Correspondence Analysis

CCA constrains ordination axes to be linear combinations of environmental variables, directly relating species composition to gradients (temperature, salinity, depth). Variance partitioning quantifies unique and shared explanatory power.

PERMANOVA & ANOSIM

Permutational MANOVA tests multivariate group differences using distance matrices and permutation p-values. ANOSIM (Analysis of Similarities) provides a complementary R statistic. Both are robust to non-normality.

Discriminant Analysis (LDA/QDA)

Linear and quadratic discriminant analysis find coordinate systems that best separate known groups. LDA assumes equal covariance; QDA allows different covariances per group. Used for species classification from environmental features.

Interactive Visualizations

PCA Biplot — Water Quality Parameters

Scree Plot — Variance Explained by Components

Cluster Dendrogram & Silhouette Analysis

Key References

  1. Clarke, K.R. & Warwick, R.M. (2001). Change in Marine Communities: An Approach to Statistical Analysis and Interpretation. PRIMER-E.
  2. Legendre, P. & Legendre, L. (2012). Numerical Ecology. 3rd ed. Elsevier.
  3. Anderson, M.J. (2001). A new method for non-parametric multivariate analysis of variance. Austral Ecology, 26(1), 32–46.
  4. Jolliffe, I.T. (2002). Principal Component Analysis. 2nd ed. Springer.
  5. Ter Braak, C.J.F. (1986). Canonical correspondence analysis: A new eigenvector technique for multivariate direct gradient analysis. Ecology, 67(5), 1167–1179.