Statistical Methods for Psychology

follows from our definition of Xand t. Moreover, if we were to examine the significance

of the bi, given as the column of t-ratios, we would simultaneously have tests on the

hypothesis (H 0 : tj5mi2m50). Notice further that the intercept (b 0 ) is equal to the grand

mean ( ). This follows directly from the fact that we scored the ath treatment as 2 1 on all

coded variables. Using the ( 2 1) coding, the mean of every column of X( ) is equal to 0

and, as a result, and therefore. This situation

holds only in the case of equal ns, since otherwise would not be 0 for all i. However, in

all cases, b 0 is our best estimate of min a least squares sense.

The value of 5 .626 is equivalent to , since they both estimate the percentage of

variation in the dependent variable accounted for by variation among treatments.

If we test for significance, we have F 5 4.46, p 5 .040. This is the Fvalue we

obtained in the analysis of variance, although this Fcan be found by the formula that we

saw for testing in Chapter 15.

Notice that the sums of squares for Regression, Error, and Total in Exhibit 16.1 are

exactly equivalent to the sums of squares for Between, Error, and Total in Table 16.1. This

equality makes it clear that there is complete correspondence between sums of squares in

regression and the analysis of variance.

The foregoing analysis has shown the marked similarity between the analysis of vari-

ance and multiple regression. This is primarily an illustration of the fact that there is no

important difference between asking whether different treatments produce different means,

and asking whether means are a function of treatments. We are simply looking at two sides

of the same coin.

We have discussed only the most common way of forming a design matrix. This matrix

could take a number of other useful forms. For a good discussion of these, see Cohen (1968).

16.3 Factorial Designs

We can readily extend the analysis of regression of two-way and higher-order factorial

designs, and doing so illustrates some important features of both the analysis of variance

and the analysis of regression. (A good discussion of this approach, and the decisions that

need to be made, can be found in Harris (2005).) We will consider first a two-way analysis

of variance with equal ns.

F(3, 8)=

.626(8)

.374(3)

=4.46

F(p, N 2 p 2 1)=

R^2 (N 2 p 2 1)

(1 2 R^2 )p

R^2

R^2

R^2 h^2

Xi

gb 1 Xj= 0 b 0 =Y 2 gb 1 Xj=Y 20 =Y

Xj

Y

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

Sum of

Model Squares df Mean Square F Sig.

1 Regression 45.667 3 15.222 4.455 .040a

Residual 27.333 8 3.417

Total 73.000 11

ANOVAb

aPredictors: (Constant), X3, X2, X1

bDependent Variable: Y

Exhibit 16.1 (continued)