Statistical Methods for Psychology

means to produce row and column means. Before exploring that issue, however, we must

first examine the competing methods.

Method III(or Type III Sum of squares)is the method we used in the preceding

section. In this case, each effect is adjusted for all other effects. Thus we obtain SSAB

as , SSA as , and SSBas

. In terms of Figure 16.1, each effect is defined as the part
of the area that is unique to that effect. Thus, SSAis represented by area “1,” SSBby area
“2,” and SSABby area “3.”
Method II(or Type II SS)breaks up the pie differently. We continue to define SSAB
as area “3.” But now that we have taken care of the interaction, we still have areas
“1,” “2,” “4,” “5,” “6,” and “7,” which can be accounted for by the effects of A
and/or B. Method II essentially redefines the full model as and obtains
, and SSBas. Thus, Ais allot-
ted areas “1” and “4,” whereas Bis allotted areas “2” and “5.” Methods II and III are sum-
marized in Table 16.4.
Both of these methods make a certain amount of sense when looked at from the point
of view of the Venn diagram in Figure 16.1. However, the diagram is only a crude approxi-
mation and we have pushed it about as far as we can go.^2

SSA=SSregressiona,b 2 SSregressionb SSregressiona,b 2 SSregressiona

SSregressiona,b

SSregressiona,b,ab 2 SSregressiona,ab

SSregressiona,b,ab 2 SSregressiona,b SSregressiona,b,ab 2 SSregressionb,ab

594 Chapter 16 Analyses of Variance and Covariance as General Linear Models

Table 16.4 Alternative models for solution of nonorthogonal designs

Method III

Portion of

Source df SS Diagram

A 1

B 2

AB 3

Error

Total

Method II

and

Portion of

Source df SS Diagram

A

B

AB 3

Error

Total N 21 SSY

N 2 ab SSresiduala,b,ab

(a 2 1)(b 2 1) SSregressiona,b,ab 2 SSregressiona,b

b 21 SSregressiona,b 2 SSregressiona 215

a 21 SSregressiona,b 2 SSregressionb 114

Yijk=m1ai1bj 1 eijk

Yijk=m1ai1bj1abij 1 eijk

N 21 SSY

N 2 ab SSresiduala,b,ab

(a 2 1)(b 2 1) SSregressiona,b,ab 2 SSregressiona,b

b 21 SSregressiona,b,ab 2 SSregressiona,ab

a 21 SSregressiona,b,ab 2 SSregressionb,ab

Yijk=m1ai1bj1abij 1 eijk

(^2) From this discussion you could easily get the impression that Method II will always account for more of the
variation than Method III. This is not necessarily the case, since the degree of overlap represents the correlation
between effects, and suppressor relationships might appear as “black holes,” canceling out accountable variation.
Method III
Method II