Statistical Methods for Psychology

The logic of the analysis is straightforward and follows that used in the previous ex-

amples. is the variation attributable to a linear combination of the covariate,

the main effects of Aand B, and the ABinteraction. Similarly, is the variation

attributable to the linear combination of the covariate and the main effects of Aand B. The

difference

is the variation attributable to the ABinteraction, with the covariate and the main effects

partialled out. Since, with equal sample sizes, the two main effects and the interaction are

orthogonal, all that is actuallypartialled out in equal ndesigns is the covariate.

By the same line of reasoning

represents the variation attributable to B, partialling out the covariate, and

represents the variation attributable to the main effect of A, again partialling out the covariate.

The error term represents the variation remaining after controlling for A, B, and AB,

and the covariate. As such it is given by

The general structure of the analysis is presented in Table 16.10. Notice that once again the

error term loses a degree of freedom for each covariate. Since the independent variable and

the covariate account for overlapping portions of the variation, their sums of squares will

not equal.

As an example, consider the study by Spilich et al. (1992) that we examined in Chapter 13

on performance as a function of cigarette smoking. In that study subjects performed either a

Pattern Recognition task, a Cognitive task, or a Driving Simulation task. The subjects were

divided into three groups. One group (Active Smoking) smoked during or just before the task.

A second group (Delayed Smoking) were smokers who had not smoked for three hours, and a

third group (NonSmoking) was composed of NonSmokers. The dependent variable was the

number of errors on the task. To make this suitable for an analysis of covariance I have added

an additional (hypothetical) variable, which is the subject’s measured level of distractibility.

(Higher distractibility scores indicate a greater ease at being distracted.)

The data are presented in Table 16.11 and represent a 3 3 3 factorial design with one

covariate (Distract).

SStotal

SSresidualc,a,b,ab

SSregressionc,a,b,ab 2 SSregressionc,b,ab

SSregressionc,a,b,ab 2 SSregressionc,a,ab

SSregressionc,a,b,ab 2 SSregressionc,a,b

SSregressionc,a,b

SSregressionc,a,b,ab

Section 16.9 The Factorial Analysis of Covariance 613

Table 16.10 Structure of the analysis of covariance for factorial designs

Source df SS

A(adj) a– 1
B(adj) b– 1
AB(adj) (a– 1)(b– 1)
Error N – ab– c
Covariate c

Total N 21

SSregressionc,a,b,ab 2 SSregressiona,b,ab

SSresidualc,a,b,ab

SSregressionc,a,b,ab 2 SSregressionc,a,b

SSregressionc,a,b,ab 2 SSregressionc,a,ab

SSregressionc,a,b,ab 2 SSregressionc,b,ab