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 Introduction to Regression Procedures

## Multivariate Tests

Multivariate hypotheses involve several dependent variables in the form

where L is a linear function on the regressor side, is a matrix of parameters, M is a linear function on the dependent side, and d is a matrix of constants. The special case (handled by PROC REG) in which the constants are the same for each dependent variable is written

where c is a column vector of constants and j is a row vector of 1s. The special case in which the constants are 0 is

These multivariate tests are covered in detail in Morrison (1976); Timm (1975); Mardia, Kent, and Bibby (1979); Bock (1975); and other works cited in Chapter 6, "Introduction to Multivariate Procedures."

To test this hypothesis, construct two matrices, H and E, that correspond to the numerator and denominator of a univariate F test:

Four test statistics, based on the eigenvalues of E-1 H or (E+H)-1 H, are formed. Let be the ordered eigenvalues of E-1 H (if the inverse exists), and let be the ordered eigenvalues of (E + H)-1 H. It happens that and , and it turns out that is the ith canonical correlation.

Let p be the rank of (H+E), which is less than or equal to the number of columns of M. Let q be the rank of L(X' X)- L'. Let v be the error degrees of freedom and s = min(p,q). Let m = (|p-q|-1)/2, and let n=(v-p-1)/2. Then the following statistics have the approximate F statistics as shown.

### Wilks' Lambda

If

then

is approximately F, where

The degrees of freedom are pq and rt-2u. The distribution is exact if .(Refer to Rao 1973, p. 556.)

### Pillai's Trace

If

then

F = [(2n+s+1)/(2m+s+1)] ·[(V)/(s-V)]

is approximately F with s(2m+s+1) and s(2n+s+1) degrees of freedom.

### Hotelling-Lawley Trace

If

then

F = [(2(sn+1)U)/(s2 (2m+s+1))]

is approximately F with s(2m+s+1) and 2(sn+1) degrees of freedom.

### Roy's Maximum Root

If

then

where r = max(p,q) is an upper bound on F that yields a lower bound on the significance level. Degrees of freedom are r for the numerator and v-r+q for the denominator.

Tables of critical values for these statistics are found in Pillai (1960).

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