Wilks's lambda distribution

Wilks's Lambda distribution is a statistical distribution used in multivariate hypothesis testing, particularly in the analysis of variance. It is named after Samuel S. Wilks, who introduced the measure in the context of multivariate analysis. The distribution is central to multivariate statistical methods, especially in the fields of multivariate analysis and statistical hypothesis testing, where it is used to determine the significance of a set of multivariate data.

Definition

Wilks's Lambda (Λ) is defined as the ratio of the determinant of the within-groups sum of squares and cross-product matrix to the determinant of the total sum of squares and cross-product matrix. Mathematically, it can be represented as:

\[\Lambda = \frac{|W|}{|T|}\]

where |W| is the determinant of the within-groups matrix and |T| is the determinant of the total matrix. The value of Λ ranges from 0 to 1, where values close to 0 indicate that the group means are very different in at least one of the variables, suggesting that the independent variable has a strong effect on the dependent variables.

Applications

Wilks's Lambda is widely used in the analysis of multivariate data, particularly in the context of MANOVA (Multivariate Analysis of Variance), to test the hypothesis that the means of multiple groups are equal in some population. It is also used in discriminant analysis to evaluate the discriminant functions that differentiate between groups.

Distribution

The exact distribution of Wilks's Lambda under the null hypothesis is complex and depends on several parameters, including the number of groups, the number of variables, and the sample size. However, various approximations can be used to determine the significance of the test statistic. The most common method is to use the F approximation, which converts the Wilks's Lambda statistic into an F statistic, making it easier to determine the p-value for the test.

See also

• Multivariate analysis
• Analysis of variance
• MANOVA
• Discriminant analysis
• Statistical hypothesis testing