Multivariates

Multivariates, in statistics and data analysis, refers to methods that involve multiple variables simultaneously. The focus is on understanding relationships among several variables, detecting patterns, and modeling outcomes that depend on more than one measurement. Data are typically organized as an n-by-p matrix, with n observations and p measured variables. The term contrasts with univariate analyses, which treat each variable separately, and with broader multivariate data analysis that emphasizes joint distributions and correlations among variables.

Common techniques include multivariate analysis of variance (MANOVA), principal component analysis (PCA), factor analysis, canonical correlation analysis, discriminant analysis, cluster analysis, multidimensional scaling, partial least squares, multivariate regression, and structural equation modeling. Some methods focus on reducing dimensionality and revealing latent structure (PCA, factor analysis), others on prediction and association across several dependent variables (multivariate regression, MANOVA), and others on grouping or organizing observations (cluster analysis, discriminant analysis).

Applications span psychology, genomics, finance, marketing, social sciences, and engineering. They are used to study correlated outcomes, identify underlying constructs, classify subjects, or forecast multiple related responses. Practical work requires careful attention to sample size, measurement scales, and assumptions such as multivariate normality, linearity, and homoscedasticity; missing data and multicollinearity pose additional challenges.

Interpretation of multivariate results often relies on visualization of loadings, scores, and canonical variates, together with inferential tests and cross-validation. The field continues to evolve with advances in high-dimensional data, regularization methods, and machine learning approaches that extend traditional multivariate techniques.