Statistical Methods for Psychology

The Full Model

The most common model for a two-way analysis of variance is

As we did before, we can expand the aiand bjterms by using a design matrix. But then

how should the interaction term be handled? The answer to this question relies on the fact

that an interaction represents a multiplicative effect of the component variables. Suppose

we consider the simplest case of a 2 3 2 factorial design. Letting the entries in each row

represent the coefficients for all subjects in the corresponding cellof the design, we can

write our design matrix as

A 1 B 1 AB 11

The first column represents the main effect of A, and distinguishes between those sub-

jects who received A 1 and those who received A 2. The next column represents the main

effect of B, separating B 1 subjects from B 2 subjects. The third column is the interaction of

Aand B. Its elements are obtained by multiplying the corresponding elements of columns

1 and 2. Thus, 1 51 3 1, 21 51 32 1, 21 521 3 1, and 1 521 32 1. Once again,

we have as many columns per effect as we have degrees of freedom for that effect. We have

no entries of 0 simply because with only two levels of each variable a subject must either

be in the first or last level.

Now consider the case of a 2 3 3 factorial. With two levels of Aand three levels of

B, we will have dfA 5 1, dfB 5 2, and dfAB 5 2. This means that our design matrix will

require one column for Aand two columns each for Band AB. This leads to the follow-

ing matrix:

A 1 B 1 B 2 AB 11 AB 12

Column A 1 distinguishes between those subjects who are in treatment level A 1 and

those in treatment level A 2. Column 2 distinguishes level B 1 subjects from those who are

not in B 1 , and Column 3 does the same for level B 2. Once again, subjects in the first a 21

and first b 2 1 treatment levels are scored 1 or 0, depending on whether or not they served

in the treatment level in question. Subjects in the ath or bth treatment level are scored 21

for each column related to that treatment effect. The column labeled AB 11 is simply the

product of columns A 1 and B 1 , and is the product of and.

The analysis for a factorial design is more cumbersome than the one for a simple one-

way design, since we wish to test two or more main effects and one or more interaction

effects. If we consider the relatively simple case of a two-way factorial, however, you

should have no difficulty generalizing it to more complex factorial designs. The basic prin-

ciples are the same—only the arithmetic is messier.

AB 12 A 1 B 2

X=

a 1 b 1

a 1 b 2

a 1 b 3

a 2 b 1

a 2 b 2

a 2 b 3

F

11010

10101

1 21 21 21 21

2110210

2101021

21 21 2111

V

X=

a 1 b 1

a 1 b 2

a 2 b 1

a 2 b 2

D

111

1 21 21

21121

21 211

T

Yijk=m1ai1bj1abij 1 eijk

Section 16.3 Factorial Designs 587