12 SPSS UPDATE 7-9

1.16 Orthogonal Contrasts
for Unegual Numbers of Replicates

For balanced designs, two treatment contrasts are orthogonal if the cross products of the contrast
coefficients sum to zero. When treatments have unegual numbers of replicates, for contrasts to be
orthogonal the weighted sum of cross products, where the weights are the reciprocals of the
numbers of replicates, must be zero. For example, suppose the numbers of replicates for five
treatments are 4, 2, 1, S, and 1 respectively; then contrasts (4,2,-6, 0, 0) and (4, 2, 1, 5, -12) are
orthogonal, since 4 x 4/4 t 2 x 2/2 t (-6)(1)/1 < 0.

Figure 1.16a illustrates the use of the orthogonal contrasts in a one-way unbalanced design in
which the numbers of observations for treatments are 4, 4, 1, and 1, respectively. Note that
specification of the PARTITION command without degrees of freedom results in single-degree-of-
freedom partitions.

Figure 1.16a

JA RENER MEEEEEMEREEEMAE EEA WE

RUN NAME ORTHOGONAL CONTRASTS FOR UNBALANCED DESIGN.
VARIABLE LIST  TREATMNT,Y
N OF CASES 10
INPUT MEDIUM CARD
INPUT FORMAT FIXED(F1.0,F2.0)
MANOVA Y BY TREATMNT(1,4)/

PRINT<DESIGNI BIAS) /

CONTRAST( TREATMNT)-SPECIAL(l lll

bl « (0)

PARTITION( TREATMNT) /
DESIGN<TREATMNT (1), TREATMNT( 2), TREATMNT(3) /
READ INPUT DATA
1

YAVDNE-UONNE
hu
ZDODAROCOO0NBO

im!

ISH

a Va S A RE WE EDEM

In this example, TREATMNT(1) defines a comparison between treatments | and 2;
TREATMNT(2) is the contrast between treatment 3 and the combination of treatments | and 2;
and TREATMNT(3) can be used to test the hypothesis that the average of the first three treatment
effects is egual to the last treatment effect. All pairs of contrasts are orthogonal since (1)(4)/4
(-1)(4)/4 < 0, (1)(4)/4 £ (-1)(4)/4 < 0, and (4)(4)/4 - (4)(4)/4 4 (-8)(1)1 — 0. The F tests are
therefore independent. The bias matrix and the ANOVA table corresponding to Figure 1.16a are
given in Figure 1.16b.

Figure 1.16b
nn

BIAS COEFFICIENTS FOR SEGUENTIAL ORDERING

EFFECT

EFFECT 1 2 3 4

1 10.00000 0.0 00434 .00278

2 0.0 2.00000 s -0

3 0.0 0.0 .01389 o

4 0.0 0.0 4. OI111
TESTS OF SIGNIFICANCE FOR Y USING SEGUENTIAL SUMS OF SOUARES
SOURCE OF VARIATION SUM OF SOGUARES DF MEAN SOUARE F SIG. OF F
MITHIN CELLS 6.75000 6 1.12500
CONSTANT 608 .40000 1 608 .40000 540.80000 0.0
TREATMNT( 1) 1.12500 l 1.12500 1.00000 .356
TREATMNT(2) 6. 12500 1 6.12500 5.44444 058
TREATMNT( 3) 1.60000 1 1.60000 1.42222 278

a A a Em EMEA AE EVEEEEEWEMEEEEER ARE ER EEEE VA EEEEEEE MOORE