Statistical Methods for Psychology

default method employed by SPSS. Method III sum of squares are the values labeled as

Type III SS in SAS, and now by more recent versions of SPSS.) It is also the method that

is approximated by the unweighted means solutiondiscussed in Chapter 13. (You may

recall that in Chapter 13 we saw that the traditional label “unweighted means solution” re-

ally should be the “equally weighted means solution,” if that name hadn’t been appropri-

ated in the past for a different procedure, since, using it, we are treating all means equally,

regardless of the sample sizes.) Method III essentially assumes that observations are miss-

ing completely at random, so there is no reason that a cell with more observations should

carry any more weight than one with fewer observations. If this is not the case you should

consider a different method.

As an illustration of this method, we will take the data used in the previous example

but add four scores to produce unequal cell sizes. The data are given in Table 16.5, with the

unweighted and weighted row and column means and the values resulting from the various

regression solutions. The unweighted means are the mean of means (therefore, the mean of

row 1 is the mean of the four cell means in that row). The weighted mean of row 1 , for exam-

ple, is just the sum of the scores in row 1 divided by the number of scores in row 1.

From Table 16.5 we see that , indicating that approximately 53% of the

variation can be accounted for by a linear combination of the predictor variables. We do

not know, however, how this variation is to be distributed among A, B, and AB. For that we

need to form and calculate the reduced models.

R^2 a,b,ab=.532

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

Table 16.5 Illustrative calculations for nonorthogonal factorial design

Unweighted Weighted

B 1 B 2 B 3 B 4 Mean Mean

A 1 52811

7 5 11 15

9 7 12 16 8.975 8.944

8 3 14 10

99

A 2 73911

9 8 12 14

10 9 14 10 9.625 9.778

911 812

713

Unweighted Means 8.000 6.475 10.625 12.1 9.300

Weighted Mean 8.000 6.333 10.556 12.1 9.3611

Full Model

Reduced Models

SSregressiona,a,b=29.7499

SSregressionb,a,b=203.9500

R^2 b,ab=.523

SSregressiona,b=188.430

R^2 a,b=.483

SSresidual=182.6001

SSregressiona,b,ab=207.7055

R^2 a,b,ab=.532