Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 681
Multivariate Details
Multivariate Tests
Test statistics in the multivariate results tables are functions of the eigenvalues λ of E – 1 H . The following
list describes the computation of each test statistic.
Note: After specification of a response design, the initial E and H matrices are premultiplied by M'
postmultiplied by M.
and
Table A.16 Computations for Multivariate Tests
Wilks’ Lambda
Pillai’s Trace
Hotelling-Lawley Trace
Roy’s Max Root
n
det( E)
Λ --------------------------- 1
= = -------------
det( H+
E)
∏ 1 + λ
i = 1 i
V = Trace[ HH ( + E) – 1 ] =
n
U = Trace E 1 H = λ i
i = 1
n
λ
------------- i
1 + λ
i = 1 i
Θ = λ 1 , the maximum eigenvalue of E – 1 H .
The whole model L is a column of zeros (for the intercept) concatenated with an identity matrix having the
number of rows and columns equal to the number of parameters in the model. L matrices for effects are
subsets of rows from the whole model L matrix.
Approximate F-Test
To compute F-values and degrees of freedom, let p be the rank of H+ E. Let q be the rank of L( X'X) – 1 L' ,
where the L matrix identifies elements of X'X associated with the effect being tested. Let v be the error
degrees of freedom and s be the minimum of p and q. Also let m = 0.5( p – q – 1)
and n = 0.5( v– p–
1)
.
Table A.17 on page 681, gives the computation of each approximate F from the corresponding test statistic.
Table A.17 Approximate F-statistics
Test Approximate F Numerator DF Denominator DF
Wilks’ Lambda
F
=
1 – Λ 1 ⁄ t
--------------------
Λ 1 ⁄ t
---------------
rt – 2u
pq
pq rt – 2u
Pillai’s Trace
F
=
----------
V ------------------------
2n+ s + 1
s–
V2m + s + 1
s( 2m+ s + 1) s( 2n+ s + 1)