24 SPSS UPDATE 7-9

Figure 1.25b

aj DODA A A A A RE ME EVE VE EEA MAAE EME OSE AM OE EDS EEENEE EVEN

TESTS OF SIGNIFICANCE FOR RESP USING SEGUENTIAL SUMS OF SOUARES

SOURCE OF VARIATION SUM OF SOUARES DF MEAN SOUARE F SIG. OF F
WITHIN CELLS 41.58993 18 2.31055

CONSTANT 13455 .99445 1 1345599443 5823.71557 0.0
ERROR 1 10.72164 6 1.78694

METHOD 651 95062 l 651.95062 364. 84200 0.0
METHOD BY GROUP 1.1872) 2 59361 33219 -T30
ERROR 2 39.25829 6 6.54305

GROUP . 16.05166 2 8.02583 1.22662 .358

a ROMA a a EVE EME RR ESEM EEEEEEEEEEMW EME EREEEEEEEEEEEEA

1.26 Confounding Designs

In some factorial designs it may not be possible to apply all factor combinations in every block.
Two methods can be used to handle this problem. The first one is the BIB designs discussed in
Section 1.21. Another method for circumventing this difficulty is to reduce the size of a block by
sacrificing the estimation of certain higher-order interactions. Consider a 2 X 2 x 2 factorial
. experiment, with factors A, B, and C. Let abc denote the experimental unit with all three factors at
the high level (since each factor has two levels, one is low and one is high), ab denote the unit
where A and B are at the high level and c is at the low level. Thus if a letter appears, that factor is
at the high level; otherwise, it is at the low level. When aH factors appear at the low level it is
designated by (1). Suppose we arrange the 2 x 2 x 2 factorial in two blocks as in Table 1.26a.

Table 1.26a

Block

Bi
»|

NEU
e/E
8 S
KANE MI poe

|
dil

—
(mej
—

0 The effect o£ A is estimated by comparing the observations receiving high and low levels of A,
ie.,

abc ta tab £ac- b—-c- be- (1)

and so on.
Note that the ABC interaction is estimated from the comparison '

abc tatb tc- ab —- ac- be - (1)

which is the same as the difference between blocks 1 and 2. Hence we cannot distinguish between
the block effects and the ABC interaction. The ABC interaction is said to be confounded with the
block effect.

If this experiment were replicated four times, the layout might be as shown in Table 1.26b.

Table 1.26b
Replication 1 Replication 2 Replication 3 Replication 4
Block Block Block Block

2| 1 2 1 2
abc ab abc (1) bc c sb |
(a | ac | b be ac b| ac |

—-a —

b [be ] a ac ab abc | (1)
[oe (0) | c ab () a | be |

Since the confounded effect (ABC) is the same for all four replications, ABC is completely
confounded with blocks. The MANOVA specifications needed for this example are

MANOVA Y BY REPLIC(1,4), BLOCK(l,2), A, B, C(1,2)/
DESIGNEREPLIC, BLOCK W REPLIC, A, B, C, A BY B, A BY C, B BY C/