Statistical Methods for Psychology

Section 16.5 The One-Way Analysis of Covariance 599

Exhibit 16.3 Abbreviated SAS analysis of the data in Table 16.7

Data Nonorth;

Infile ‘Table16-7.dat’;

Input A B dv;

Run;

Proc GLM Data = Nonorth;

Class A B;

Model dv = A B A*B;

Analysis of Variance

Source DF Type III SS Mean Square F Value Pr > F

A 1 3.7555556 3.7555556 0.58 0.4543

B 3 177.9556246 59.3185415 9.10 0.0002

A*B 3 19.2755003 6.4251668 0.99 0.4139

Error 28 182.6000000 6.5214286

Within the framework of multiple regression, however, it is remarkably simple, requiring

little, if any, more work than does the analysis of variance.

Suppose we wish to compare driving proficiency on three different sizes of cars to test

the experimental hypothesis that small cars are easier to handle. We have available three

different groups of drivers, but we are not able to match individual subjects on driving

experience, which varies considerably within each group. Let us make the simplifying

assumption, which will be discussed in more detail later, that the mean level of driv-

ing experience is equal across groups. Suppose further that using the number of steering

errors as our dependent variable, we obtain the somewhat exaggerated data plotted in

Figure 16.2. In this figure the data have been plotted separately for each group (size of car),

as a function of driving experience (the covariate), and the separate regression lines have

been superimposed.

One of the most striking things about Figure 16.2 is the large variability in both per-

formance and experience within each treatment. This variability is so great that an analysis

of variance on performance scores would almost certainly fail to produce a significant

effect. Most of the variability in performance, however, is directly attributable to differ-

ences in driving experience, which has nothing to do with what we wish to study. If we

could somehow remove (partial out) the variance that can be attributed to experience

X

Driving experience

Large

Medium Cars

Small

Performance

(X 2 ,Y 2 )

(X 1 ,Y 1 )

(X 3 ,Y 3 )

Figure 16.2 Hypothetical data illustrating error-reduction in the analysis of covariance

covariate