Wilksféle

Wilksféle, known in English as Wilks' lambda, is a multivariate test statistic used in MANOVA and discriminant analysis to assess whether multiple dependent variables have the same mean vectors across groups. It is named after Samuel S. Wilks.

Definition: Let H be the hypothesis sum of squares and cross-products for between-group effects, and E be the within-group SSCP. Wilks' lambda is Λ = det(E) / det(H + E). Equivalently, if the eigenvalues of E^{-1}H are λ1,...,λp, then Λ = ∏_{i=1}^p 1/(1+λ_i). A smaller value of Λ indicates stronger evidence against H0; a value near 1 suggests little or no group differences.

Inference: Under H0, the exact distribution of Λ is complex, so researchers rely on approximations or transform Λ to an F-like statistic for large samples. Common approaches include Bartlett, Box, or other modern approximations to obtain p-values or to construct confidence regions for the multivariate mean vectors.

Relation and alternatives: Wilks' lambda is one of several multivariate test statistics used in MANOVA. Other choices include Pillai's trace, Lawley-Hotelling trace, and Roy's largest root. Pillai's trace is often cited as more robust to certain violations of covariance assumptions.

History and use: The statistic is named for Samuel S. Wilks, who contributed to likelihood ratio tests in multivariate analysis. Wilks' lambda remains widely used in psychology, education, and social science research to test multivariate mean differences across groups.