Statistical Methods for Psychology

ran a one-way with young participants and then another one-way with old participants, we

would need twice as many participants overall for each of our experiments to have the

same power to detect Condition differences—that is, each experiment would have to have

20 participants per condition, and we would have two experiments.

Factorial designs are labeled by the number of factors involved. A factorial design with

two independent variables, or factors, is called a two-way factorial, and one with three fac-

tors is called a three-way factorial. An alternative method of labeling designs is in terms of

the number of levels of each factor. Eysenck’s study had two levels of Age and five levels

of Condition. As such, it is a 2 3 5 factorial.A study with three factors, two of them hav-

ing three levels and one having four levels, would be called a 3 33 3 4 factorial. The use

of such terms as “two-way” and “2 3 5” are both common ways of designating designs,

and both will be used throughout this book.

In much of what follows, we will concern ourselves primarily with the two-way analy-

sis. Higher-order analyses follow almost automatically once you understand the two-way,

and many of the related problems we will discuss are most simply explained in terms of

two factors. For most of the chapter, we will also limit our discussion to fixed—as opposed

to random—models, as these were defined in Chapter 11. You should recall that a fixed fac-

tor is one in which the levels of the factor have been specifically chosen by the experi-

menter and are the only levels of interest. A random model involves factors whose levels

have been determined by some random process and the interest focuses on all possible lev-

els of that factor. Gender or “type of therapy” are good examples of fixed factors, whereas

if we want to study the difference in recall between nouns and verbs, the particular verbs

that we use represent a random variable because our interest is in generalizing to all verbs.

Notation

Consider a hypothetical experiment with two variables, Aand B. A design of this type is

illustrated in Table 13.1. The number of levels of Ais designated by a, and the number of

Introduction 415

Table 13.1 Representation of factorial design

B 1 B 2 ... Bb

X 111 X 121 ... X 1 b 1

X 112 X 122 X 1 b 2

A 1 ...... ...

X 11 n X 12 n X 1 bn

X 211 X 221 ... X 2 b 1

X 212 X 222 X 2 b 2

A 2 ...... ...

X 21 n X 22 n X 2 bn

... ............

Xa 11 Xa 21 Xab 1

Xa 12 Xa 22 Xab 2

Aa ...... ...

Xa 1 n Xa 2 n Xabn

X.1 X.2 ... X.b X..

Xa 1 Xa 2 Xab

Xa.

X 21 X 22 X 2 b

X2.

X 11 X 12 X 1 b

X1.

2 3 5 factorial