# 8.3 - Test Statistics for MANOVA

8.3 - Test Statistics for MANOVASAS 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}