Wilks Lambda as a Product of Independent Beta Variables

Why Wilks’ Lambda Has an Exact Product Law

Wilks’ lambda summarizes a MANOVA test by comparing the error sum-of-products matrix with the total sum of-products matrix. Its determinant ratio is compact, but its null distribution is not generally a single familiar univariate distribution. Under the central null, Bartlett decomposition gives a more revealing description: Wilks’ lambda has the distribution of a product of independent beta variables.

The dimensions matter. The number of response variables determines how many beta factors appear; the rank of the tested hypothesis determines one beta parameter; and the residual degrees of freedom determine the other. Keeping these quantities distinct prevents a common error: substituting the number of hypothesis restrictions or the sample size where the error degrees of freedom belong.

Here \(W_p(m,\Sigma)\) denotes a \(p\)-dimensional Wishart distribution with \(m\) degrees of freedom and scale matrix \(\Sigma\). The condition \(\nu\ge p\) ensures that \(E\) is positive definite with probability one, so both determinants in the ratio are well-defined and the denominator is positive. The matrix \(H\) may be singular when \(q<p\); that does not make \(E+H\) singular. We consider \(q\ge1\). If \(q=0\), there is no tested effect and \(\Lambda=1\).

The beta distribution is parameterized by shape parameters \(a,b>0\), with density proportional to \(x^{a-1}(1-x)^{b-1}\) on \(0<x<1\). The theorem’s shapes are positive: \(\nu\ge p\) implies \(\nu-j+1\ge1\) for every factor, and \(q\ge1\). The product representation is an equality in distribution; it does not say that the beta variables are observed separately from the MANOVA data.

Deriving the Product Representation

First, a common nonsingular change of response coordinates does not alter \(\Lambda\). If \(E\) and \(H\) are replaced by \(C'EC\) and \(C'HC\), for nonsingular \(C\), each determinant is multiplied by \(|C|^2\), so the ratio is unchanged. We can therefore reduce the derivation to \(\Sigma=I_p\).

The central-null matrices can be represented as Gram matrices of independent standard normal observations: \(E=X'X\) for a \(\nu\times p\) standard normal matrix \(X\), and \(H=Z'Z\) for an independent \(q\times p\) standard normal matrix \(Z\). Equivalently, stack the rows to form \(G=\begin{pmatrix}X\\Z\end{pmatrix}\), an \((\nu+q)\times p\) standard normal matrix. Then \(E+H=G'G\).

Bartlett decomposition, applied to this Gaussian Gram-matrix construction, gives the determinant-ratio moment identity \[ E[\Lambda^t] = \frac{\Gamma_p(\nu/2+t)\,\Gamma_p((\nu+q)/2)} {\Gamma_p(\nu/2)\,\Gamma_p((\nu+q)/2+t)},\qquad t\ge0, \] where \(\Gamma_p(a)=\pi^{p(p-1)/4}\prod_{j=1}^{p}\Gamma\bigl(a-(j-1)/2\bigr)\) is the multivariate gamma function. For \(q\ge p\), the identity also follows by transforming the two Wishart matrices to their sum and their matrix-beta ratio: the ratio has matrix-beta normalizing constant \(\Gamma_p(\nu/2)\Gamma_p(q/2)/\Gamma_p((\nu+q)/2)\), and multiplying its density by \(|U|^t\) replaces \(\nu/2\) by \(\nu/2+t\). When \(q<p\), \(H\) is singular and that full-dimensional density argument does not apply; the Gaussian Gram-matrix Bartlett decomposition gives the same moment identity in this case as well.

This derivation also explains the indexing. Expanding \(\Gamma_p(\nu/2+t)\) gives scalar gamma factors with arguments \((\nu-j+1)/2+t\), one for each response dimension \(j\). The second beta shape is \(q/2\) for every factor. It is the rank of the tested hypothesis, not the number of beta factors, that supplies this shape.

Worked Examples

What the Dimensions Tell You

The product representation is useful for more than naming an exact distribution. It makes the model’s dimensional structure visible. Increasing \(p\) adds beta factors; changing \(q\) changes the common second shape; changing \(\nu\) changes the first shape of each factor. For example, the mean follows immediately from independence: \[ E[\Lambda]=\prod_{j=1}^{p} \frac{\nu-j+1}{\nu+q-j+1}. \] Each denominator exceeds its numerator when \(q>0\), so this null mean is strictly between zero and one.

The beta factors are a distributional device. They need not be independent components of the observed error and hypothesis matrices, and the product representation does not imply that each factor corresponds to an individual response. Its purpose is to specify the exact null law, with the dimensions and degrees of freedom made explicit.

Before applying the result, check the assumptions and the bookkeeping: the null is central, the hypothesis and error matrices are independent Wishart matrices with a common scale, \(q\) is the hypothesis rank, and \(\nu\ge p\) ensures a nonsingular error matrix. If these conditions fail, the stated exact law need not apply. In particular, noncentral alternatives do not have this central-null distribution.