Statistical Methods for Psychology

treatment associated with that column, and 0 otherwise. Thus, for three treatments with two

subjects per treatment, the complete design matrix would be

S m A 1 A 2 A 3

Notice that subjects 1 and 2 (who received Treatment A 1 ) are scored 1 on mand A 1 , and 0

onA 2 and A 3 , since they did not receive those treatments. Similarly, subjects 3 and 4 are

scored 1 on mand A 2 , and 0 on A 1 and A 3.

We will now define the vector tof treatment effects as [mt 1 t 2 t 3 ]. Taking Xas the

design matrix, the analysis of variance model can be written in matrix terms as

y 5 Xt1e

which can be seen as being of the same form as the traditional regression equation. The

elements of tare the effects of each dummy treatment variable, just as the elements of bin

the regression equation are the effects of each independent variable. Expanding, we obtain

y 5 X 3t1e

which, following the rules of matrix multiplication, produces

For each subject we now have the model associated with her response. Thus, for the sec-

ond subject in Treatment 2, Y 22 5m1t 21 e 22 , and for the ith subject in Treatment j, we

have Yij5m1tj 1 eij, which is the usual analysis of variance model.

The point is that the design matrix allows us to view the analysis of variance in a multiple-

regression framework, in that it permits us to go from

Yij=m1tj 1 eij to y 5 Xb 1 e

Y 23 =m1t 31 e 23

Y 13 =m1t 31 e 13

Y 22 =m1t 21 e 22

Y 12 =m1t 21 e 12

Y 21 =m1t 11 e 21

Y 11 =m1t 11 e 11

y=F

1100

1100

1010

1010

1001

1001

V 3 D

m

t 1

t 2

t 3

T 1 F

e 11

e 21

e 12

e 22

e 13

e 23

V

a=

11100

21100

31010

41010

51001

61001

V

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

F