Multivariate statistics examines relationships among multiple variables simultaneously, extending beyond univariate and bivariate methods to reveal complex patterns in data. These techniques are essential across fields from psychology to ecology, enabling researchers to reduce dimensionality, detect latent structures, test group differences, and predict outcomes when multiple responses or predictors are involved. The key distinction from running separate univariate tests is that multivariate methods account for correlations among variables, preventing inflated error rates and uncovering relationships that single-variable analyses miss. Modern practice increasingly combines classical methods like PCA and MANOVA with nonlinear manifold techniques such as UMAP, making it essential to understand both the assumptions and the appropriate use context of each approach.
What This Cheat Sheet Covers
This topic spans 15 focused tables and 87 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Foundational Concepts
| Concept | Example | Description |
|---|---|---|
Σ = [[σ₁², σ₁₂], [σ₂₁, σ₂²]] | • Square matrix storing covariances between all pairs of variables • diagonal contains variances. | |
R = [[1, r₁₂], [r₂₁, 1]] | • Standardized covariance matrix with 1s on diagonal and correlation coefficients off-diagonal • scales variables to −1 to +1 range. | |
X ~ N(μ, Σ) | • Joint distribution of variables follows multivariate Gaussian • required assumption for parametric tests like MANOVA, MANCOVA, and LDA. | |
λ₁ = 3.2, λ₂ = 1.5 | • Scalar indicating variance explained by a principal component or factor • larger values represent more important dimensions. |