Statistical Methods for Psychology

All that we have left to calculate is the sum of squares due to error. Just as in the one-
way analysis, we will obtain this by subtraction. The total variation is represented by SStotal.
Of this total, we know how much can be attributed to A, C, and AC. What is left over repre-
sents unaccountable variation or error. Thus

However, since , it is simpler to write

This provides us with our sum of squares for error, and we now have all of the necessary
sums of squares for our analysis.
A more direct, but tiresome, way to calculate exists, and it makes explicit just
what the error sum of squares is measuring. represents the variation within each cell,
and as such can be calculated by obtaining the sum of squares for each cell separately. For
example,

5 (9 2 7)^21 (8 2 7)^21... 1 (7 2 7)^25 30

We could perform a similar operation on each of the remaining cells, obtaining

The sum of squares within each cell is then summed over the 10 cells to produce.
Although this is the hard way of computing an error term, it demonstrates that is
in fact the sum of within-cell variation. When we come to mean squares, MSerrorwill turn
out to be just the average of the variances within each of the 2 3 5 5 10 cells.
Table 13.2c shows the summary table for the analysis of variance. The source column
and the sum of squares column are fairly obvious from what has already been said. Note,
however, that we could organize the summary table somewhat differently, although we
would seldom do so in practice. Thus, we could have

Source df SS

Between cells 9 1945.49

A 1 240.25

C 4 1514.94

AC 4 190.30

Within cells 90 722.30

(Error)

Total 99 2667.79

This alternative summary table makes it clear that we have partitioned the total variation
into variation among the cell means and variation within the cells. The former is then fur-
ther partitioned into A, C, and AC.
Returning to Table 13.2c, look at the degrees of freedom. The calculation of dfis
straightforward. The total number of degrees of freedom ( ) is always equal to N 2 1.
The degrees of freedom for Age and Condition are the number of levels of the variable
minus 1. Thus, dfA=a 21 = 1 and dfC=c 21 = 4. The number of degrees of

dftotal

SSerror

SSerror

SScell 11 =30.0

SScell 12 =40.9

Á Á

SScell 25

SSerror

=

64.1

722.30

SScell 11

SSerror

SSerror

SSerror=SStotal 2 SScells

SSA 1 SSC 1 SSAC=SScells

SSerror=SStotal 2 (SSA 1 SSC 1 SSAC)

Section 13.1 An Extension of the Eysenck Study 419