Statistical Methods for Psychology

Looking first at the ABinteractions, we see from Exhibit 16.2 that when the interaction

terms were deleted from the model, the sum of squares that could be accounted for by the

model decreased by

This decrement can only be attributable to the predictive value of the interaction terms, and

therefore

By a similar line of reasoning, we can find the other sums of squares.^1

Notice that these values agree exactly with those obtained by the more traditional pro-

cedures. Notice also that the corresponding decrements in R^2 agree with the computed val-

ues of.

As Overall and Spiegel (1969) pointed out, the approach we have taken in testing the

effects of A, B, and ABis not the only one we could have chosen. They presented two alter-

native models that might have been considered in place of this one. Fortunately, however,

the different models all lead to the same conclusions in the case of equal sample sizes,

since in this situation effects are independent of one another and therefore are additive.

When we consider the case of unequal sample sizes, however, the choice of an underlying

model will require careful consideration.

h^2

SSAB=27.344

SSAB=SSregressiona,b,ab 2 SSregressiona,b=231.969 2 204.625=27.344

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

Table 16.3 Regression solution for the data in Table 16.2

Summary Table

Source df SS MS F

A 1 5.282 5.282 , 1

B 3 199.344 66.448 11.452*

AB 3 27.344 9.115 1.571

Error 24 139.250 5.802

31 371.220

*p,.05

SSerror=SSresiduala,b,ab=139.250

SSB=SSregressiona,b,ab 2 SSregressiona,ab=231.969 2 32.625=199.344

SSA=SSregressiona,b,ab 2 SSregressionb,ab=231.969 2 226.687=5.282

SSAB=SSregressiona,b,ab 2 SSregressiona,b=231.969 2 204.625=27.344

SSregressiona,ab=32.625 R^2 =.088

SSregressionb,ab=226.687 R^2 =.611

SSregressiona,b=204.625 R^2 =.551

SSresiduala,b,ab=139.250

SSregressiona,b,ab=231.969 R^2 =.625

(^1) A number of authors (e.g., Judd & McClelland) prefer to use the increase in the error term (rather than the
decrease in SSregression) when an effect is deleted. The result will be the same.