SAS uses four different test statistics based on the MANOVA table:
- Wilks Lambda
\(\Lambda^* = \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\)
Here, the determinant of the error sums of squares and cross-products matrix E is divided by the determinant of the total sum of squares and cross-products matrix T = H + E. If H is large relative to E, then |H + E| will be large relative to |E|. Thus, we will reject the null hypothesis if Wilks lambda is small (close to zero).
- Hotelling-Lawley Trace
\(T^2_0 = trace(\mathbf{HE}^{-1})\)
Here, we are multiplying H by the inverse of E; then we take the trace of the resulting matrix. If H is large relative to E, then the Hotelling-Lawley trace will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.
- Pillai Trace
\(V = trace(\mathbf{H(H+E)^{-1}})\)
Here, we are multiplying H by the inverse of the total sum of squares and cross products matrix T = H + E. If H is large relative to E, then the Pillai trace will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.
- Roy's Maximum Root: Largest eigenvalue of HE-1
Here, we multiply H by the inverse of E and then compute the largest eigenvalue of the resulting matrix. If H is large relative to E, then Roy's root will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.
Recall: The trace of a p x p matrix
\(\mathbf{A} = \left(\begin{array}{cccc}a_{11} & a_{12} & \dots & a_{1p}\\ a_{21} & a_{22} & \dots & a_{2p} \\ \vdots & \vdots & & \vdots \\ a_{p1} & a_{p2} & \dots & a_{pp}\end{array}\right)\)
is equal to
\(trace(\mathbf{A}) = \sum_{i=1}^{p}a_{ii}\)
Statistical tables are not available for the above test statistics. However, each of the above test statistics has an F approximation: The following details the F approximations for Wilks lambda. Details for all four F approximations can be found on the SAS website.
1. Wilks Lambda
\begin{align} \text{Starting with }&& \Lambda^* &= \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\\ \text{Let, }&& a &= N-g - \dfrac{p-g+2}{2},\\ &&\text{} b &= \left\{\begin{array}{ll} \sqrt{\frac{p^2(g-1)^2-4}{p^2+(g-1)^2-5}}; &\text{if } p^2 + (g-1)^2-5 > 0\\ 1; & \text{if } p^2 + (g-1)^2-5 \le 0 \end{array}\right. \\ \text{and}&& c &= \dfrac{p(g-1)-2}{2} \\ \text{Then}&& F &= \left(\dfrac{1-\Lambda^{1/b}}{\Lambda^{1/b}}\right)\left(\dfrac{ab-c}{p(g-1)}\right) \overset{\cdot}{\sim} F_{p(g-1), ab-c} \\ \text{Under}&& H_{o} \end{align}