Note that the model does not include A BY B BY C, which is confounded with BLOCK W
REPLIC.

It is possible to test the ABC interaction if some interaction other than ABC is confounded in
some of the replications. One possible layout would be that given in Table 1.26c.

Table 1.26c
Replication 1 Replication 2 Replication 3 Replication 4
Block Biock Block Biock
l 2 1 | 2| 1 2
— 4 H 7]
abc ab | b | ab ac ab
4 —

a ac H a c () be
b be | ac (i) abc a
c 0) le [abc | b | ce | D |

In replication 1, ABC is confounded with blocks. in replication 2, the AB interaction is
confounded with blocks. For replications 3 and 4, AC and BC are confounded.

For this example, A, B, and C are free of the block effects and three-fourths information for
AB, AC, BC, and ABC can be obtained, since the unconfounded interactions can be estimated in
three out of four of the replications. Hence we say AB, AC, BC, and ABC are partially
confounded with blocks. The MANOVA specifications for this 2 x 2 x 2 factorial with partial
confounding are

MANOVA Y BY REPLIC(l.4), BLOCK(l,2).A, B, cf. 2)/

DESIGN<REPLIC,BLOCK W REPLIC, A, B, C, A BY B, A BY C,
B BY.C, A BY B BY C/

More complex confounding designs can be found in Davies (1954) and Cochran and Cox (1957).

Another Example The following example is taken from Cochran and Cox (1957, p. 205). The

data are a 3 X 3 x 2 factorial in blocks of six units with three blocks in each of four replications.
Interactions AB and ABC are partially confounded with blocks. The SPSS commands for this
analysis are given in Figure 1.26a.

Figure 1.26a
RUN NAME CONFOUNDING IN MIXED SERIES.
COMMENT CONFOUNDING IN MIXED SERIES. 3"3"2 FACTORIAL
COMMENT FROM COCHRAN AND COX(1957) P. 205
COMMENT SECOND ANALYSIS GIVES AB TWO-WAY TABLE ADJUSTED FOR BLOCK
COMMENT THIRD ANALYSIS GIVES AC TWO-WAY TABLE ADJUSTED FOR BLOCKS
COMMENT
COMMENT FACTOR A : 8—8-6 FERTILIZER APPLIED IN THE ROW
COMMENT 3 LEVELS —- 0 (NONE), l (200 LB.), 2 (400 LB.)
COMMENT FACTOR B : MEALS, 3 LEVELS —- O (NONE), l (TUNG MEAL),
COMMENT 2 ( COTTONSEED MEAL).
COMMENT FACTOR C : a 8-6 FERTILIZER APPLIED AS STIE DRESGINA,
COMMENT 2 LEVELS — O (NONE), 1 (200 LB.).

VARIABLE LIST NREPLICS,BLOCKS,A,B,C,DEP
INPUT MEDIUM CARD
INPUT FORMAT  FIXED(2X,5Fl.0,8X,F3.0)
N OF CASES T2
MANOVA DEP BY REPLICS(1,4),BLOCKS(1,3),A(0,2),B(0,2),C(0,1)/
DESIGN < REPLICS, BLOCKS WITHIN REPLICS,A,B,C,
A BY B,A BY C, B BY C, A BY B BY C/

DESIGN < REPLICS,BLOCKS W REPL.ICS, CONSPLUS A AND B/
DESIGN - REPLICS,BLOCKS W REPLICS,CONSPLUS A AND C/
READ INPUT DATA

1l0ll 82

11020 To

11100 BC

1ll2l 86

11201 T4

1l2l0o 86

12001 67

12010 55

42210 66

43001 S0

43010 58

43100 8I

43121 67

43211 68 -

43220 56

FINISH N