Statistical Methods for Psychology

Testing the Interaction Effects

First, we delete the predictors associated with the interaction term and calculate. For

these data, representing a drop in R^2 of about .05. If we examine the pre-

dictable sum of squares (SSregression), we see that eliminating the interaction terms has pro-

duced a decrement in SSregressionof

This decrement is the sum of squares attributable to the ABinteraction (SSAB).

In the case of unequal ns, it is particularly important to understand what this term rep-

resents. You should recall that , for example, equals. Then

The final term in parentheses is the squared semipartial correlation between the criterion

and the interaction effects, partialling out (adjusting for) the effects of Aand B. In other

words, it is the squared correlation between the criterion and the part of the ABinteraction

that is orthogonal to Aand B. Thus, we can think of SSABas really being SSAB(adj), where the

adjustment is for the effects of Aand B. (In the equal-ncase, the issue does not arise because

A, B, and ABare independent, and therefore there is no overlapping variation to partial out.)^4

Testing the Main Effects

Because we are calculating Method III SS, we will calculate the main effects of Aand Bin

a way that is directly comparable to our estimation of the interaction effect. Here, each

main effect represents the sum of squares attributable to that variable after partialling out

the other main effect and the interaction.

To obtain SSA, we will delete the predictor associated with the main effect of Aand cal-

culate. For these data, , producing a drop in R^2 of .532 2 .523 5

.009. In terms of the predictable sum of squares ( ), the elimination of afrom the

model produces a decrement in of

SSA= 3.7555

SSregressionb,ab=203.9500

SSregressiona,b,ab=207.7055

SSregression

SSregression

SSregressionb,ab R^2 b,ab=.523

=SSY(R^2 0(ab.a,b))

=SSY(R^2 a,b,ab 2 R^2 a,b)

SSAB=SSY(R^2 a,b,ab) 2 SSY(R^2 a,b)

SSregressiona,b,ab SSY(Ra^2 ,b,ab)

SSAB=19.2754

SSregressiona,b=188.4301

SSregressiona,b,ab=207.7055

R^2 a,b=.483,

R^2 a,b

Section 16.4 Analysis of Variance with Unequal Sample Sizes 597

(^4) Some people have trouble understanding the concept of nonindependent treatment effects. As an aid, perhaps an
extreme example will help point out how a row effect could cause an apparentcolumn effect, or vice versa. Con-
sider the following two-way table. When we look at differences among means, are we looking at a difference due
to A, B, or AB? There is no way to tell.
B 1 B 2 Means
A (^110)
A (^230)
Means 10 30
n= 0 n= 20
X= 30
n= 20 n= 0
X= 10