## X

29x6

experiment is given in Table 1.7. The structural matrix is coded as follows:

where p is the number of levels of treatment A and h is the number of cell means. In order for the null hypothesis CA ( = 0 to be testable, the CA matrix must be of full row rank. This means that each row of CA must be linearly independent of every other row. The maximum number of such rows is p — 1, which is why it is necessary to express the null hypothesis as Equation 1.1 or 1.2.An estimator of the null hypothesis, CA ( — 0, is incorporated in the formula for computing a sum of squares. For example, the estimator appears as CA ( — 0 in the formula for the treatment A sum of squares ssa = (CA( — 0)'[CA(X'X)-:Ca]—:(CA(X — 0), (1.3)

where ( is a vector of sample cell means. Equation 1.3 simplifies to

1, if an observation is in a1b\

0, otherwise

1 , if an observation is in a1 b2

0, otherwise

1, if an observation is in a2b1 0, otherwise

1, if an observation is in a3b2 0, otherwise

For the weight-loss data, the sum of squares for treatment A is

because 0 is a vector of zeros. In the formula, CA is a coefficient matrix that defines the null hypothesis, ( = [(X/X)-1(X'y)] = [Y.11, Y.12 • • • Y.23]', and X is a structural matrix. The structural matrix for the weight-loss with p — 1 = 2 degrees of freedom. The null hypothesis for treatment B is

0 0