Hotelling T^2 Statistic — hotellingsT2 • metabom8

Computes the Hotelling T^2 statistic defining a multivariate confidence region for a set of observations. The region corresponds to an ellipsoid in p-dimensional space and is commonly visualised as an ellipse in two-dimensional (OPLS) score plots.

Usage

hotellingsT2(X, alpha = 0.95)

Arguments

X: Numeric matrix with observations in rows and variables (dimensions) in columns.
alpha: Numeric scalar. Confidence level for the T^2 region. Default is 0.95.

Value

A named list with elements:

center: Numeric vector of column means.
cov: Sample covariance matrix.
c2: Squared radius of the Hotelling T^2 region.

Details

The Hotelling T^2 region is defined as $$(x - \mu)^T S^{-1} (x - \mu) \le c^2$$ where $c^2 = (p (n - 1) / (n - p)) F_{p, n-p}(\alpha)$.

Examples

set.seed(1)
X <- matrix(rnorm(200), ncol = 2)
t2 <- hotellingsT2(X)
ell <- ellipse2d(t2)
plot(X, asp = 1)
lines(ell$x, ell$y, col = "red", lwd = 2)