Statistical Methods for Psychology

Moreover, the elements of bare the values of m, t 1 , t 2 ,... , tk. In other words, these are

the actual treatment effects in which we are interested.

The design matrix we have been using has certain technical limitations that must be

circumvented. First, it is redundant in the sense that if we are told that a subject is not in

A 1 or A 2 , we know without being told that she must be in A 3. This is another way of say-

ing that there are only 2 dffor treatments. For this reason we will eliminate the column

headed A 3 , leaving only a 2 1 columns for the treatment effects. A second change is nec-

essary if we want to use any computer program that obtains a multiple-regression equa-

tion by way of first calculating the intercorrelation matrix. The column headed mhas no

variance, and therefore cannot enter into a standard multiple-regression program—it

would cause us to attempt division by 0. Thus, it too must be eliminated. This is no real

loss, since our ultimate solution will not be affected. In fact, the software will sneak it

back in.

One further change will be made simply for the sake of allowing us to test the desired

null hypotheses using the method to be later advocated for factorial designs. Since we have

omitted a column dealing with the third (or ath) level of treatments, solutions given our

modified design matrix would produce estimates of treatment effects in relation to

rather than in relation to. In other words, b 1 would turn out to be ( ) rather than

( ). This is fine if that’s what you want, but I would much rather see treatment ef-

fects as deviations from the grand mean. It just seems tidier. So we will modify the design

matrix to make the mean ( ) of each column of Xequal to 0. Under this new system, a

subject is scored 1 in column Aiif she is a member of Treatment Ai; she is scored 2 1 if she

is a member of the ath (last) treatment; and she is scored 0 if neither of these conditions

apply. (This restriction corresponds to the fixed-model analysis of variance requirement

that .)

These modifications have led us from

Although these look like major changes in that the last form of Xappears to be far removed

from where we started, it actually carries all the necessary information. We have merely

eliminated redundant information, removed a constant term, and then caused the treatment

effects to be given as deviations from.

16.2 One-Way Analysis of Variance

At this point a simple example is in order. Table 16.1 contains data for three subjects in

each of four treatments. Table 16.1b shows the summary table for the corresponding analy-

sis of variance, along with the value of (discussed in Chapter 11). Table 16.1c contains

the estimated treatment effects ( ) where Since the fixed-model analysis of

variance imposes the restriction that , t 4 is automatically defined by t 1 , t 2 , and t 3

().t 4 = 02 gtj

gti= 0

Nti tNi=mNi2mN.

h^2

X.

X=F

1100

1100

1010

1010

1001

1001

V to F

110

110

101

101

100

100

V to F

11

10

01

01

00

00

V to F

10

10

01

01

- 1 - 1

- 1 - 1

V

gti= 0

Xi

X 12 X.

X. X 12 X 3

X 3

Section 16.2 One-Way Analysis of Variance 583