The elements in the upper triangle of the decomposition matrix are used to obtain the sum of
sguares for each effect in the model. Consider a 2 x 3 factorial design, where T is the upper
triangle of the decomposition matrix.

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The first row of T represents the CONSTANT effect, the second row represents the effect of A, the
third and fourth rows are the effects of B, and the last two rows are the effects of AB. lfh' < (h,h,
h; h, h; h,) is the least-sguares estimate of the contrasts of effects, then the seguential sums of
sguares for the effects are as shown in Table 1.13.

Table 1.13

Source Sum of Sguares

CONSTANT (tih,t, hatha, At ashseE treh)?

A (th tao tah, taste)?

B adjusted A (usb, tah, tast tebe) (uh, t,shs--t,;h,)?

AB adjusted A.B (t;zh,t tsshi)?--(t;she)?

If the DESIGN specification for this example is
DESIGN-A,B,A BY B/

then the bias matrix is a 4 X 4 upper triangular matrix, since the order of the bias matrix is the
number of effects in the model (in this case, CONSTANT, A, B, and A BY B). The (i,j)th element
of this matrix is obtained by summing the sguared elements of the T matrix, which are in the rows
of effect i and the columns of effect j. The bias matrix for this example is

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The bias matrix can be used as a measure of the degree of the confounding among effects. For
example, the coefficients corresponding to h; and h, (factor B) in the calculation of sum of sguares
Of A are ta and t,,; thus ta" - t,,' (sguaring is to avoid the Sign) can be used as a confounding index
between A and B.

1.14 Redundant Effects

If there are empty cells in the design, some effects in the model may not be estimable. MANOVA
determines the redundant effects by orthonormalization of the design matrix and prints the
information. Figure 1.14 indicates that the interaction effects in columns 10 and 12 in the design
matrix are not estimable because of empty cells.

Figure 1.14

o RE O O EVRE OEEO EVE OEEPEtttsstm

REDUNDANCIES IN DESIGN MATRIX

COLUMN EFFECT
10 A BY B
12 (SAME)

A o o oo oo ooo a a o a NA NNNNNNNI]

1.15 Solution Matrices

For any connected design, the hypotheses associated with the seguential sums of Sguares are
weighted functions of the population cell means with weights depending on the cell freguencies
(e.g. see Searie(1971), pp. 306-313). For designs with every cell filled, it can be shown that the
hypotheses corresponding to the regression model sums of sguares are the unweighted hypotheses
about the celi means. With empty cells the hypotheses will depend on the pattern of the
missingness. In such cases, one can reguest that the solution matrix, which contains the coefficients
of the linear combinations of the cell means being tested, be printed by specifying

MANOVA 9