Statistical Methods for Psychology

Calculations

The calculations for the sums of squares appear in Table 13.2b. Many of these calculations

should be familiar, since they resemble the procedures used with a one-way. For example,

is computed the same way it was in Chapter 11, which is the way it is always com-

puted. We sum all of the squared deviations of the observations from the grand mean.

The sum of squares for the Age factor ( ) is nothing but the that we would ob-

tain if this were a one-way analysis of variance without the Condition factor. In other

words, we simply sum the squared deviations of the Age means from the grand mean and

multiply by nc. We use ncas the multiplier here because each age has nparticipants at each

of clevels. (There is no need to remember that multiplier as a formula. Just keep in mind

that it is the number of scores upon which the relevant means are based.) The same proce-

dures are followed in the calculation of , except that here we ignore the presence of the

Age variable.

Having obtained , , and , we come to an unfamiliar term,. This term

represents the variability of the individual cell means and is in fact only a dummy term; it

will not appear in the summary table. It is calculated just like any other sum of squares. We

take the deviations of the cell means from the grand mean, square and sum them, and mul-

tiply by n, the number of observations per mean. Although it might not be readily apparent

why we want this term, its usefulness will become clear when we calculate a sum of

squares for the interaction of Age and Condition. (It may be easier to understand the calcu-

lation of if you think of it as what you would have if you viewed this as a study with

10 “groups” and calculated .)

The is a measure of how much the cell means differ. Two cell means may differ

for any of three reasons, other than sampling error: (1) because they come from different

levels of A(Age); (2) because they come from different levels of C(Condition); or (3) be-

cause of an interaction between Aand C. We already have a measure of how much the cells

differ, since we know. tells us how much of this difference can be attributed to

differences in Age, and tells us how much can be attributed to differences in Condi-

tion. Whatever cannot be attributed to Age or Condition must be attributable to the interac-

tion between Age and Condition ( ). Thus, has been partitioned into its three

constituent parts— , , and. To obtain , we simply subtract and

from. Whatever is left over is. In our example,

SSAC=SScells 2 SSA 2 SSC

=1945.49 2 240.25 2 1514.94=190.30

SScells SSAC

SSA SSC SSAC SSAC SSA SSC

SSAC SScells

SSC

SScells SSA

SScells

SSgroups

SScells

SStotalSSA SSC SScells

SSC

SSA SStreat

SStotal

418 Chapter 13 Factorial Analysis of Variance

Table 13.2 (continued)

(c) Summary table

Source df SS MS F

A(Age) 1 240.25 240.250 29.94*

C(Condition) 4 1514.94 378.735 47.19*

AC 4 190.30 47.575 5.93*

Error 90 722.30 8.026

Total 99 2667.79

*p<.05

SSerror=SStotal 2 SScells=2667.79 2 1945.49=722.30

SSAC=SScells 2 SSA 2 SSC=1945.49 2 240.25 2 1514.94=190.30

SScells