Statistical Methods for Psychology

Overall and Spiegel (1969) is also worth seeing. Both of these papers appeared in the

Psychological Bulletinand are therefore readily available. Other good discussions can

be found in Overall (1972), Judd and McClelland (1989), and Cohen, Cohen, West, and

Aiken (2003). Cramer and Appelbaum (1980), and Howell and McConaughy (1982)

provide contrasting views on the choice of the underlying model and the procedures to

be followed.

There are two different ways to read this chapter, both legitimate. The first is to look

for general concepts and to go lightly over the actual techniques of calculation. That is the

approach I often tell my students to follow. I want them to understand where the reasoning

leads, and I want them to feel that they could carry out all of the steps if they had to (with

the book in front of them), but I don’t ask them to commit very much of the technical

material to memory. On the other hand, some instructors may want their students to grasp

the material at a deeper level. There are good reasons for doing so. But I would still sug-

gest that the first time you read the chapter, you look for general understanding. To develop

greater expertise, sit down with both a computer and a calculator and work lots and lots of

problems.

The Linear Model

Consider first the traditional multiple-regression problem with a criterion (Y) and three pre-

dictors (X 1 , X 2 , and X 3 ). We can write the usual model

or, in terms of vectornotation

where y, x 1 , x 2 , and x 3 are (n 3 1) vectors (columns) of data, eis a (n 3 1) vector of errors,

and b 0 is a (n 3 1) vector whose elements are the intercept. This equation can be further

reduced to

y 1 Xb 1 e

where Xis a n 3 (p 1 1) matrix of predictors, the first column of which is 1s, and bis a

(p 1 1) 3 1 vector of regression coefficients.

Now consider the traditional model for a one-way analysis of variance:

Here the symbol tjis simply a shorthand way of writing t 1 , t 2 , t 3 ,... , tp, where for any

given subject we are interested in only that value of tjthat pertains to the particular treat-

ment in question. To see the relationship between this model and the traditional regression

model, it is necessary to introduce the concept of a design matrix. Design matrices are used

in a wide variety of situations, not simply the analysis of variance, so it is important to

understand them.

Design Matrices

A design matrixis a matrix of coded, or dummy, or countervariables representing group

membership. The completeform of the design matrix (X) will have p 1 1 columns, repre-

senting the mean (m) and the ptreatment effects. A subject is always scored 1 for m, since

mis part of all observations. In all other columns, she is scored 1 if she is a member of the

Yij=m1tj 1 eij

y=b 01 b 1 x 11 b 2 x 21 b 3 x 31 e

Yi=b 01 b 1 X 1 i 1 b 2 X 2 i 1 b 3 X 3 i 1 ei

Section 16.1 The General Linear Model 581

design matrix